KIT REPORT 111
Deriving Inference Rules for Description Logics:
a Rewriting Approach into Sequent Calculi
Véronique Royer
ONERA–CERT
2 AV. EDOUARD BELIN-BP 4025
F-31055 TOULOUSE
e-mail: [email protected]
J. Joachim Quantz
TECHNISCHE UNIVERSITÄT BERLIN
PROJEKT KIT-BACK, FR 5–12
FRANKLINSTR. 28/29, D–10587 BERLIN 10
e-mail: [email protected]
December 1993
Abstract
Description Logics (DL) can be investigated under different perspectives. The
aim of this report is to provide the basis for a tighter combination of theoretical
investigations with issues arising in the actual implementation of DL systems. We
propose to use inference rules, derived via the Sequent Calculus, as a new method
for specifying terminological inference algorithms. This approach combines the
advantages of the tableaux methods and the normalize-compare algorithms that
have been predominant in terminological proof theory so far. In our paper presented at JELIA’92 we proposed a generic method for deriving complete sets of
inference rules for DL. The method relies upon translations into Sequent Calculus
and systematic rewriting of sequent proofs. We illustrated our method on a relatively restricted terminological logic. In this report the approach is extended to
the more expressive logic underlying Back V5. It turns out that concept-forming
operators involving equality and role-forming operators considerably increase the
complexity of our rewriting strategy.
The derived inference rules can be used in two ways for the characterization
of DL systems: first, the incompleteness of systems can be documented by listing
those rules that have not been implemented; second, the reasoning strategy can be
described by specifying which rules are applied forward and which backward.
Contents
1
Introduction
2
2
Description Logics
2.1 The Roots of Description Logics : : : :
2.2 Theoretical Investigations : : : : : : :
2.3 Implemented Systems and Applications
2.4 Extending Description Logics : : : : :
3
Inference Rules and Sequent Calculus
3.1 Motivating Axiomatic Characterizations : : : : : : : :
3.2 Methodology for Deriving Complete Inference Systems
3.3 Technicalities on Sequent Calculi : : : : : : : : : : :
4
Concept Subsumption without Equality
4.1 Conjunction and Disjunction : : : :
4.2 Quantification : : : : : : : : : : :
4.3 Negation : : : : : : : : : : : : : :
5
Concept Subsumption with Equality
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
5.2 Introduction of Complete Equality Assumptions : : : : : : :
5.3 Axiomatization of Numerical Operators : : : : : : : : : : :
5.4 Axiomatization of Extensional Operators : : : : : : : : : :
6
Role Subsumption
6.1 The Propositional Fragment : :
6.2 Domain and Range Restrictions
6.3 Role Composition and Inversion
7
Conclusion and Future Work
7.1 The Impact of Complex Roles on Concept Subsumption
7.2 Object Recognition : : : : : : : : : : : : : : : : : : :
7.3 Flexible Inference Strategies : : : : : : : : : : : : : : :
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References
79
79
82
83
85
A Proofs for Numerical Operators
A.1 Formulae with Only One Creative Term : : : : : : : : : :
A.2 Formulae with exactly two eigen-parameter-creating terms
B Survey of Inference Rules
::::
::::
89
89
96
111
1
1 INTRODUCTION
2
Γ
j=DL iff
iff
iff
Γ `DL
Γ
Γ
j=FOL iff
`SC
Figure 1: The commutative diagram underlying our approach to inference rule
derivation.
1 Introduction
In this paper we conclude our work on deriving inference rules for Description
Logics (DL) started in [Royer, Quantz 92] and [Royer, Quantz 93]. The aim of
our investigation was to provide a complete axiomatization of the expressive DL
underlying the implementation of the DL system BACK V5 [Hoppe et al. 93]. It
turned out that this enterprise was a lot more complex than we had originally
expected. Though we are thus not completely satisfied with the results presented
in this report, we think that they are interesting enough to be published in the current
form. We hope that our work will be continued by others, and will eventually reach
a more stable and complete status.
Let us briefly sketch the main difficulties of our work. The basic idea is to find a
complete set of inference rules for DL which allows a constructive characterization
of the set of consequences of a given set of DL formulae. Thus our inference rules
typically have the form
1 ; : : : ; n * expressing that we can infer the DL formula if we already have inferred the DL
formulae 1 ; : : :; n . Most researchers working on DL will have no problems to
give examples for such inference rules. The main problem is to show that a given
list of inference rules is complete for a particular DL.
In [Royer, Quantz 92] we have demonstrated how the Sequent Calculus (SC)
for First-Order Logic (FOL) can be used to obtain a complete set of inference rules
for a restricted DL. Consider the commutative diagram shown in Figure 1.1 We
know that a DL formula can be translated into an FOL formula ( ). That is we
know
Γ j=DL iff Γ j=FOL
Γ j=FOL iff Γ `SC If we now can show that
Γ `SC
1
iff Γ `DL
The following argument is presented more carefully in Section 3.
3
we immediately get the desired completeness
Γ j=DL
iff Γ `DL
Thus our main task was to prove that for every SC proof of an FOL translation
of a DL formula there is a corresponding DL proof of . We thus had found
a systematic way for deriving inference rules for DL—we just had to look at SC
proofs in order to construct corresponding DL inference rules.
Unfortunately, being systematic does not guarantee simplicity. As it turned
out, SC proofs for some DL constructs showed a high degree of complexity. The
main cause for this complexity is that these DL constructs cannot be translated
into pure FOL, but rather require eqality statements, i.e. are translated into FOL= .
Consequently, we need to consider proofs in SC= to deal with these constructs. One
way of coping with this difficulty is to introduce some simplifying assumptions
about the role-forming operators.
In addition to the problem to the problem of concept subsumption we investigate role subsumption and object recognition, i.e. derivation of descriptions. The
derivation of inference rules performed in this report can thus be divided into five
subtasks with varying degree of complexity:2
1. A DL containing only concept-forming operators without equality and no
role-forming operators is investigated in Section 4. We regard the inference
rules for this DL as complete.
2. A DL containing concept-forming operators with equality but no roleforming operators is investigated in Section 5. This task is almost completed,
but there are still some open questions.
3. Inference rules for role subsumption are derived in Section 6. This derivation
is performed by a straightforward case analysis guaranteeing completeness.
4. Some inference rules for a DL containing role-forming operators and
concept-forming operators involving equality are listed in Section 7.1. The
presentation is only sketchy though, and a thorough investigation is still
lacking.
5. For object recognition we present the most important inference rules (Section 7.2 without specifying the rules needed for complex case reasoning,
however.
As should be obvious from these remarks we do not offer technical proofs regarding
the completeness of our axiomatization. Nevertheless we think that the technique
2
Note that complexity here refers to the task of deriving inference rules and not to the computation of subsumption itself.
4
2 DESCRIPTION LOGICS
used in this report is, due to its systematicity, appropriate for deriving a complete
set of inference rules.
One further issue before we begin our detailed description of the method. The
main motivation for our investigation of inference rules is the implementation of
DL systems. Obviously, the behavior of such systems should reflect the underlying
logic. In other words, given a set of DL formulae Γ we want to query the system
whether a particular DL formula is entailed, and we expect the answer to correlate
with the formal entailment relation Γ j=DL . To achieve this we could also use
tableaux methods, i.e. we could try to prove the inconsistency of Γ [ : . This
approach has the disadvantage, however, that costly computations are performed
at query time. In several application, such as information retrieval, it is more
appropriate to compute consequences already at assertion time (see Section 3 for
more details).
Note that the inference rules of the form sketched above can be used in both
directions: they can be used to infer whenever 1 ; : : : ; n already have been
inferred; or they can be used backward, i.e. when trying to prove one proof
would be to prove 1 ; : : :; n . This is important, because we might choose different
formats of rules, depending on the direction in which we want to apply the rules.
We will address this issue in Section 7.3.
Before we begin the derivation of inference rules we give a short introduction
into Description Logics (Section 2) and into the Sequent Calculus and our rewrite
approach based on it (Section 3).
2 Description Logics
2.1 The Roots of Description Logics
The field of Knowledge Representation (KR) originated in the late 1960’s and
found its first paradigm in Semantic Networks as proposed in [Quillian 68]. Quillian assumed semantic memory to be general memory, underlying such diverse
cognitive activities as language understanding and perception. The intimate connection between his work and research in psychology and linguistics is typical
for this early phase of KR—it was assigned the task of modeling the human
ability to use information for intelligent activities such as natural language understanding, perception, planning, etc. Consequently, psychological experiments
were conducted to determine the cognitive adeqacy of the proposed formalisms
[Collins, Quillian 69].
The second paradigm of KR was created by Minky’s seminal paper on Frames
[Minsky 75]. Minsky’s original goal was to account for the effectiveness of
common-sense reasoning in real-world tasks — to a large degree motivated by
“human” models borrowed from psychology or text linguistics.
2.1 The Roots of Description Logics
5
Though not obvious on first sight, there is a close resemblance between Semantic Nets and Frames. A semantic net consists of nodes that are connected by
labelled links. Some of these links, the so called ISA-Links indicate specialization,
while others stand for other relations holding between the nodes. A key idea of Semantic Nets is the notion of inheritance: the links attached to a node are inherited
to its specializations. Due to their graphical notation and the inheritance principle,
Semantic Nets thus allow a compact representation of the complex dependencies
between pieces of information.
The frames correspond to the nodes in a semantic net, the ISA-Link is represented by the subframe relation, and the labelled links are modeled by slots and
fillers (a slot correponds to the labelled link, the filler to the node to which the link
points). Note that inheritance from frames to subframes mirrors the inheritance
via ISA-Links in Semantic Networks.
Though both representation formats are thus very similar, there are also important differences. Semantic Nets are more appropriate for conveying the general
structure of the represented information, especially interconnections and dependencies; a frame representation, on the other hand, focuses not so much on the
overall structure but on the basic units and the information locally associated
with each frame. Note that this difference between both formalisms is not an
issue of expressive adequacy but one of structural adequacy. Both Semantic Nets
and Frames are ancestors of Description Logics and all three approaches to KR
have much in common. There are, however, essential characteristics of DL that
distinguish them from their ancestors—the basic difference concerns the attitude
towards theoretical foundations and towards the question of what is constitutive
for a representation formalism.
In the second half of the 1970’s representation languages from the area of
Semantic Nets, Frames, or Scripts were seriously attacked in a number of papers
for their apparent lack of formal rigor (e.g., [Woods 75] and [Hayes 77]). The key
issue was the relationship between Knowledge Representation and Formal Logic.
Note that Quillian located his semantic memory between natural languages and
symbolic logic [Quillian 68, p. 230]. This opinion clearly implies that Semantic
Nets are superior to Predicate Logic with respect to expressive adequacy.
One apparent problem we face here is the missing entailment relation for
Semantic Nets or Frames—how do we know whether a semantic net contains
more information than another? Of course, this boils down to the question of how
we know what exactly a semantic net or a frame means?
Hayes answers this question by mapping frame definitions into FOL formulae
[Hayes 80], and thus shows that Frames are not superior to FOL with respect to
expressive adequacy. Two remarks seem in order here. First, the above argumentation is, at least to a certain degree, circular—we do not know what formulae in
a formalism mean unless we define a notion of entailment; given such a notion of
entailment we have a logic; thus all representation formalisms are logics. Second,
6
2 DESCRIPTION LOGICS
if representation formalisms are reducible to FOL it does not mean that they are dispensable. It just means that they are not superior to FOL with respect to expressive
adequacy, but they stil can be with respect to structural or computational adequacy.
Reducibility to FOL is rather an advantage because it guarantees applicability of the
tools for theoretical investigations developed in Formal Logic. To draw an analogy
to Programming Languages—nobody would discard a programming language just
because it can be compiled into an already existing language. On the contrary, the
new language has to be compiled into Machine Code ultimately, and it is confined
to the limits of theoretical computability as expressed by Turing Machines, the
Lambda Calculus, etc.
Brachman endorsed the logic-oriented view on Knowledge Representation in
his early papers on Semantic Nets [Brachman 77] and [Brachman 79]. Instead
of formalizing Semantic Nets in terms of FOL, however, he examined in detail,
what the constructs used in them were supposed to represent. There are two
important results of his investigations: for one thing, Brachman proposes to
distinguish several levels in the discussion of knowledge representation systems,
namely the implementational, the logical, the epistemological, the conceptual,
and the linguistic level. In addition, he proposes KL-ONE as a formalism on the
epistemological level and presents its basic elements or epistemological primitives.
An overview over the basic features of the KL-ONE formalism circulated in
the beginning of the 80’s and was finally published in [Brachman, Schmolze 85].
In the following years several DL systems have been developed incorporating
different dialects but similar with respect to the underlying representation philosophy. The respective formalisms and systems were called KL-ONE alike systems,
term subsumption systems, concept logics, terminological logics, and description
logics.
2.2 Theoretical Investigations
Though Brachman mentioned the importance of a formal semantics in his papers,
it took some years before Description Logics were thoroughly investigated from
a theoretical point of view. Schmolze and Israel presented a formal semantics for
KL-ONE in [Schmolze, Israel 83]. In the same year Brachman and Levesque raised
the question of tractability of KR formalisms [Brachman, Levesque 84]. In the
late 1980’s several results were obtained concerning the tractability of different
DLs. In order to understand these results we now have to take a closer look on the
theoretical foundations of DL.
In DL one typically distinguishes between terms and objects as basic language
entities from which three kinds of formulae can be formed: definitions, descriptions, and rules (see the sample modeling on page 13 below). A definition has the
:
form tn = t and expresses the fact that the name tn is used as an abbreviation for
the term t. A list of such definitions is often called terminology (hence also the
2.2 Theoretical Investigations
7
name Terminological Logics). All DL dialects provide two types of terms, namely
concepts (unary predicates) and roles (binary predicates), but they differ with
respect to the term-forming operators they contain. Common concept-forming
operators are: conjunction (c1 u c2 ), disjunction (c1 t c2 ), and negation (: c), as
well as quantified restrictions [Quantz 92] such as value restrictions (8r:c), which
stipulate that all fillers for a role r must be of type c, or number restricitions (n r:c
or n r:c), stipulating that there are at least or at most n role-fillers of type c for r.
Role-forming operators are, besides conjunction, disjunction, and negation, role
composition (r1 .r2), transitive closure (r+ ), inverse roles (r ) and domain or range
restrictions (cjr or rjc ). In a description, an object is described as being an instance
of a concept (o :: c), or as being related to another object by a role (o1 :: r:o2 ).
Rules have the form c1 ) c2 and stipulate that each instance of the concept c1 is
also an instance of the concept c2.
The following syntax specifies the DL we will ultimately investigate in this
report (cn and rn are concept names and role names respectively):
! c;r
! > ; ? ; cn ; : c ; c1 u c2 ; c1 t c2 ; 8r:c1 ;
n r:c ; n r:c ; r:fo1; :::; on g ; fo1; :::; on g_
r ! >r; ?r ; ; rn ; : r ; r1 u r2 ; r1 t r2 ; r1 ; r1 .r2 ; c jr1 ; r1jc
! t1 v t2 ; o :: c
t
c
For this language a model-theoretic semantics can be given where a model M of
a set of DL-formulae Γ is a pair hD; [[]]I i. [[]]I maps concepts into subsets of D,
roles into subsets of D D, and object-names injectively into D, in accordance
with the following equations (we use r(d) to denote fe : hd; ei 2 rg):
>]]I
I
[[?]]
I
[[t1 u t2 ]]
I
[[t1 t t2 ]]
I
[[:c]]
I
[[8r:c]]
I
[[n r:c]]
I
[[n r:c]]
I
[[r:fo1 ; :::; on g]]
_ I
[[fo1 ; :::; on g ]]
I
[[>r]]
I
[[?r]]
I
[[:r]]
[[
=
D
=
;
=
[[t1 ]]
=
=
=
=
=
=
=
=
=
=
I \ [[t2 ]]I
I
I
[[t1 ]] [ [[t2 ]]
D n [[c]]I
fd 2 D : [[r]]I (d) [[c]]I g
fd 2 D : j[[r]]I (d) \ [[c]]I j ng
fd 2 D : j[[r]]I (d) \ [[c]]I j ng
fd 2 D : f[[o1]]I ; :::; [[on]]I g [[r]]I (d)g
f[[o1]]I ; :::; [[on ]]I g
DD
;
I
(D D) n [[r]]
2 DESCRIPTION LOGICS
8
I
I
[[r1 :r2 ]]
I
[[cjr]]
I
[[rj ]]
[[r1 ]]
c
=
=
=
=
fhd; ei 2 D D : he; di 2 [[r]]I g
I
I
[[r1 ]] [[r2 ]]
I
I
[[r]] \ ([[c]] D)
I
I
[[r]] \ (D [[c]] )
Satisfaction of formulae is then defined as follows:
M j= t2 v t1 iff
M j= o :: c iff
I [[t ]]I
1
I
I
[[o]] 2 [[c]]
[[t2 ]]
A structure M is a model of a formula iff M j= ; it is a model of a set of
formulae Γ iff it is a model of every formula in Γ. A formula is entailed by a set
of formulae Γ (written Γ j=DL ) iff every structure which is a model of Γ is also
a model of .
In addition to the notation defined above, we will use the following notational
conventions:
n r
n r
9r:c
def
=
def
=
def
=
r:o
def
cn = c
c1 ) c2
def
:
=
=
def
=
n r:>
n r:>
1 r:c
r:fog
cn v c; c v
c1 v c2
cn
Clearly, this logic is a subset of First-Order Logic with Equality. We can make
this precise by specifying a translation of DL-formulae into FOL-formulae (cf. also
[Schmolze, Israel 83] and [Royer, Quantz 92]). The set of concept-names (resp.
role-names) is bijectively mapped into a set of unary predicate symbols (resp.
binary predicate symbols); object-names are bijectively mapped into constants. Let
j be the corresponding bijection. We define the translation function as follows:
a concept-name (or a primitive concept component) c is translated into xC (x),
where C denotes the unary predicate symbol representing c (C = j (c)); similarily,
a role-name (or a primitive role component) r is translated into xyR(x; y ), where
R denotes the binary predicate symbol representing r (R = j (r)); object-names
are translated into constants. We then define translation of arbitrary terms and
DL-formulae:
>
?
:c
def
=
def
=
def
=
x(True) (True is as special propositional letter)
x(False) (False is a special propositional letter)
x(:C (x))
2.2 Theoretical Investigations
c1 u c2
c1 t c2
8r:c
def
=
def
=
def
=
n r:c
def
diff(y1 ; : : :; yn )
def
=
=
n r:c
def
equ(y1 ; : : :; yn )
def
r:o
o1 ; :::; on _
>r
?r
:r
r1 u r2
r1 t r2
r
r1:r2
c jr
rjc
v c2
r1 v r2
c1
o :: c
=
x(c1(x) ^ c2 (x))
x(c1(x) _ c2 (x))
x(8yr(x)(y) ! c(y))
x(9y1 : : :yn r(x)(y1) ^ c(y1) : : : r(x)(yn) ^
c(yn ) ^ diff(y1; : : : ; yn ))
^i6 j (:(yi = yj ))
x(9y1 : : :yn 1 r(x)(y1) ^ c(y1) : : : r(x)(yn 1 ) ^
c(yn 1 ) ! equ(y1 ; : : :; yn 1 ))
_i6 j (yi = yj )
x(r(x; o))
x(x = o1 _ ::: _ x = on)
xy(True)
xy(False)
xy(:R(x)(y))
xy(r1 (x)(y) ^ r2 (x)(y))
xy(r1 (x)(y) _ r2 (x)(y))
xy(r(y; x))
xy(9zr1(x)(z) ^ r2(z)(y))
xy(r(x)(y) ^ c(x))
xy(r(x)(y) ^ c(y))
8x(c1(x) ! c2(x))
8x8y(r1(x)(y) ! r2(x)(y))
=
+
+
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
9
+
+
=
c(o)
Note that, by beta-reduction, the lambda abstractions are eliminated from the
translation of DL formulae. Thus DL formulae are translated into pure FOL formulae.
There are basically two main reasoning components in a DL system: the
classifier computes subsumption between terms, i.e., whether Γ j=DL t2 v t1; the
recognizer computes whether an object is an instance of a concept, i.e., whether
Γ j=DL o :: c. Since recognition is partly reducible to classification theoretical
research focuses usually on the problem of subsumption.
The first classifiers were specified as structural subsumption algorithms
[Schmolze, Israel 83]. The basic idea underlying structural subsumption is to
transform terms into canonical normal forms and then structurally comparing them.
2 DESCRIPTION LOGICS
10
Structural subsumption algorithms are therefore also referred to as normalizecompare algorithms.
Let us take a closer look at these algorithms following the presentation in
[Nebel 90]. The algorithm SUBSUMES has to decide whether a term t1 subsumes a
term t2 with respect to a given terminology. A terminology is a list of definitions
satisfying the following restrictions:
1. for each term-name there is at most one definition;
2. each term-name used in a definition is either primitive or has been defined
in a preceding definition.
The second condition ensures the non-cyclicity of terminologies.3
Now given two terms t1 and t2, and a terminology Γ a first normalization is
obtained by replacing each term-name occurring in t1 and t2 by its definition until
both terms contain only primitive components and quantified restrictions. Note
that this is done recursively, so that also terms occurring in the restrictions do
not contain any names. This normalization step is called abstraction from term
introductions in [Nebel 90, p. 60]. Note that this normalization step performs the
inheritance of information from terms to its subterms.
In a second step normalization rules capturing the semantics of term-forming
operators are applied. For example, all value-restrictions for the same role are
unified to obtain a single value-restriction:
8r:c1 u 8r:c2 !
7 N 8r:c1 u c2
Other examples for normalization rules include the computation of incoherence
due to conflicting number-restrictions or the inheritance of restrictions to subroles and superroles according to the monotonic properties of the quantifiers
(cf. [Quantz 92]):
m r u n r !
7N ?
8r1:c !
7 N 8r2:c
n r1 !
7 N n r2
(if m > n)
(if r2 v r1 )
(if r1 v r2 )
The application of these rules yields a canonical normalform for each term. In
order to determine subsumption, these normalforms are structurally compared.
Note that there is a general tradeoff between normalization and comparison: the
more inferences are drawn in normalization, the less inferences have to be drawn
in comparison, and vice versa. In BACK V5 the strategy is to determine for each
3
The problem of cyclic definitions is discussed in [Nebel 90, Ch. 5], [Baader 90], and
[Dionne et al. 93].
2.2 Theoretical Investigations
11
relevant role the most specific number and value-restriction in the normalization
phase. Thus it can be guaranteed that in the comparison phase the restrictions have
to be compared only locally.
There is one severe drawback of normalize-compare algorithms—though it
is in general straightforward to prove the correctness of such algorithms there
is no method for proving their completeness. In fact, most normalize-compare
algorithms are incomplete which is usually demonstrated by giving examples for
subsumption relations which are not detected by the algorithm.
At the end of the 1980’s tableaux methods, as known from FOL,
cf. [Sundholm 83, p. 180ff], were applied to DL, e.g. [Donini et al. 91a,
Donini et al. 91b]. The resulting subsumption algorithms had the advantage of
providing an excellent basis for theoretical investigations. Not only was their
correctness and completeness easy to prove, they also allowed a systematic study
of the decidability and the tractability of different DL dialects.
The main disadvantage of tableaux-based subsumption algorithms is that they
are not constructive but rather employ refutation techniques. Thus in order to prove
the subsumption c2 v c1 it is proven that the term c1 u : c2 is inconsistent, i.e.
that o :: c1 u : c2 is not satisfiable. In most existing systems, on the other hand,
inference rules are more seen as production rules, which are used to pre-compute
part of the consequences of the initial information. This corresponds more closely
to Natural Deduction or Sequent Calculi, two deduction systems also developed
in the context of FOL.
In [Royer, Quantz 92] we have shown how sequent rules for DL can be derived
by rewriting the corresponding FOL proofs. We thereby obtained a set of inference
rules which can be used as a basis for specifying normalize-compare algorithms,
for proving their correctness and completeness, and to characterize the inferential
behavior of (even incomplete) DL systems. In the following sections we will apply
this framework to the expressive DL underlying BACK V5.
To summarize these theoretical issues, we can say that DLs are characterized
by a particular stance towards the essentials of KR-formalisms—in order to call
something a KR formalism
1. it has to be a formal language in the sense that there is a formal specification
of its syntax;
2. it has to have a formal semantics which defines an entailment relation on
formulae;
3. there have to be (efficient) algorithms computing entailment between formulae.
2 DESCRIPTION LOGICS
12
2.3 Implemented Systems and Applications
From the beginning on, research in DL was praxis-oriented in the sense that the
development of DL systems and their use in applications was one of the primary
interests. In the first half of the 1980’s several systems were developed that might
be called in retrospection first-generation DL systems.
In the second half of the 1980’s three systems were developed which are still
in use, namely BACK, CLASSIC, and LOOM. The LOOM system [MacGregor 91]
is being developed at USC/ISI and focuses on the integration of a variety of
programming paradigms aiming at a general purpose knowledge representation
system. CLASSIC [Brachman et al. 91] is an ongoing AT&T development. Favoring limited expressiveness for the central component it is attempted to keep the
system compact and simple so that it potentially fits into a larger, more expressive
system. The final goal is the development of a deductive, object-oriented database
manager. BACK [Hoppe et al. 93] is intended to serve as the kernel representation
system of AIMS (Advanced Information Management System), in which tools for
semantic modeling, defining schemata, manipulating data, and querying, will be
replaced by a single high-level description interface. To avoid a “tool-box-like”
approach, all interaction with the information repository occurs through a uniform
knowledge representation system, namely BACK, which thus acts as a mediating
layer between the domain-oriented description level and the persistency level. The
cited systems share the notion of DL knowledge representation as being the appropriate basis for expressive and efficient information systems [Patel-Schneider 87].
In contrast to the systems of the first generation, these second generation DL systems are full-fledged systems developed in long-term projects and used in various
applications.
The systems of the second generation take an explicit stance to the problem that
subsumption determination is at least NP-hard or even undecidable for sufficiently
expressive languages: CLASSIC offers a very restricted DL and almost complete
inference algorithms, whereas LOOM provides a very expressive language but is
incomplete in many respects. Recently, the KRIS system has been developed,
which uses tableaux-based algorithms and provides complete algorithms for a
very expressive DL [Baader et al. 92]. KRIS might thus be the first representative
of a third generation of DL systems, though there are not yet enough experiences
with realistic applications to judge the adequacy of this new approach.4
In order to get a better understanding of these systems let us take a look at
the modeling scenario assumed for applications. An application in DL is basically
a domain modeling, i.e. a list of definitions, rules, and descriptions (a set of DLformulae Γ). Consider the highly simplified domain modeling below, whose net
representation is shown in Figure 2. One role and five concepts are defined, out of
4
The missing constructiveness of the refutation oriented tableaux algorithms (see above) leads
to problems with respect to object recognition and retrieval.
2.3 Implemented Systems and Applications
1..in
company
13
product
produces
high risk
company
conc_1
1..
in
pro
du
ce
s
chemical
product
all
chemical
company
chemoplant
biological
product
ces
produ
toxiplant
toxipharm
biograin
1..1
produces
Figure 2: The net representation of the sample domain. ‘conc 1’ is the concept
9produces:chemical product.
which four are primitive (only necessary, but no sufficient conditions are given).
Furthermore, the modeling contains one rule and four object descriptions.
product
chemical product
biological product
company
produces
chemical company
9produces:chemical product
toxipharm
biograin
chemoplant
toxiplant
v >
v product
v product u : chemical product
v 1 produces:product
v: domain(company)
= company u 8produces:chemical product
! high risk company
::
::
::
::
chemical product
biological product
chemical company
1 produces u produces:toxipharm
In DL, such a modeling is regarded as a set of formulae Γ. Given the formal
semantics of a DL, such a set of formulae will entail other formulae, i.e., there
is an entailment relation Γ j= . Now the service provided by DL systems is
basically to answer queries whether some formula is entailed by a modeling Γ.
The following list contains examples for the types of queries that can be answered
2 DESCRIPTION LOGICS
14
by a DL system like BACK:
Γ j= t1 v t2
Is a term t1 more specific than a term t2, i.e., is t1 subsumed by t2? In the
sample modeling, the concept ‘chemical company’ is subsumed by ‘high
risk company’, i.e., every chemical company is a high risk company.
Γ j= t1 u t2 v ?
Are two terms t1 and t2 incompatible or disjoint? In the sample modeling,
the concepts ‘chemical product’ and ‘biological product’ are disjoint, i.e.,
no object can be both a chemical and a biological product.
Γ j= o :: c
Is an object o an instance of concept c (object classification)? In the sample
modeling, ‘toxiplant’ is recognized as a ‘chemical company’.
Γ j= o1 :: r:o2
Are two objects o1,o2 related by a role r, i.e., is o2 a role-filler for r at o1 ? In
the sample modeling, ‘toxipharm’ is a role-filler for the role ‘produces’ at
‘toxiplant’.
Γ j= X :: c
Which objects are instances of a concept c (retrieval)? In the sample modeling, ‘chemoplant’ and ‘toxiplant’ are retrieved as instances of the concept
‘high risk plant’.
Γ[ fg j= ?
Is a description inconsistent with the modeling (consistency
check)?
With respect to the sample modeling, the description
chemoplant :: produces:biograin is inconsistent, i.e., ‘biograin’ cannot be
produced by ‘chemoplant’.
This very general scenario can be refined by considering generic application tasks
such as information retrieval, diagnosis, or configuration.
2.4 Extending Description Logics
Research in DL was originally concerned with the set of epistemological primitives presented in [Brachman, Schmolze 85], but the interest in extending DL
to integrate results from other fields of AI increased considerably in the last
years. The integration of rule-based reasoning into DL was suggested very early in
[Owsnicki-Klewe 88]. It turned out, however, that the so-called implication-links
or rules which have been implemented in BACK, CLASSIC, and LOOM, differ from
the rules of rule-based system with respect to their logical, monotonic character
15
and thus function more as constraints then as production rules [Schild 89]. As
a consequence, no conflict resolution strategies have to be devised. Thus the
integration of rules kept very close to the original paradigm of DL.
The epistemological primitives of KL-ONE were meant to be ontologically neutral, i.e., it should be possible to use them for the definition of concepts belonging
to arbitrary ontological categories. It became quickly obvious, however, that certain ontological areas need additional term-forming operators which specifically
reflect the ontological structure of these areas. Schmiedel proposed an integration
of DL, Shoham’s temporal logic, and Allen’s interval calculus to support the modeling of temporal knowledge in a DL system [Schmiedel 90]. Schmiedel’s ideas
were taken up by Schild, who developed a restricted tense-logical extension of DL,
which constituted an optimal tradeoff between expressiveness and computational
complexity [Schild 91]. An integration of this tense-logical extension into the
BACK system is described in [Fischer 92].
Another topic of discussion concerned the integration of defeasible knowledge
into DL. In the traditional framework of DL all information is regarded as strict.
This is obviously a restriction much too strong for most domain: most of our
knowledge is expressable only as rules that allow for exceptions or as rules with
a certain degree of reliability. In order to capture this kind of information the
integration of probabilistic rules [Heinsohn, Owsnicki-Klewe 88, Heinsohn 91]
and of defaults [Baader, Hollunder 92, Quantz, Royer 92, Baader, Hollunder 93,
Quantz, Ryan 93] was suggested.
Other envisaged extensions of DL include the integration of generalized quantifiers [Quantz 92], of part-whole relations and collective entities in general
[Franconi 92], of epistemic operators [Donini et al. 92], and of test/compute functions [Kortüm 93].
In this report we will contend ourselves with an axiomatization of the DL
presented on page 7. Our method can in principle be applied to all extensions
which are translatable into First-Order Logic.
3 Inference Rules and Sequent Calculus
3.1 Motivating Axiomatic Characterizations
In the AIMS project that underlies the development of the BACK system, typical
application scenarios involve first modeling the domain knowledge by terminological definitions. In a second step, the concepts and roles defined in the terminology
are used to enter descriptions of domain objects and to retrieve objects matching
given descriptions. Given such a framework, it is advantageous to precompute
information that is implicitly given in order to speed up the query-answering in the
retrieval phase. Thus in the current BACK implementation (cf. [Hoppe et al. 93]
16
3 INFERENCE RULES AND SEQUENT CALCULUS
) the subsumption relations between all concepts and roles of the terminology are
computed and stored, as well as the most specific description for each object. This
way, most queries can be answered by simple table-look-ups, and only in some
cases reasoning is necessary to answer a query (see [Kindermann 90] for details).
Inference systems provide a syntactic characterization for precomputing implicitly given information. They allow the presentation and the derivation of
information in a constructive and modular way. So inference systems have been
heavily used not only for logical Proof Theory but also in Software Engineering
(specification methods are most often based on inference rules [Jones 90]).
Similar motivations also inspire the tradition of least fixed point semantics
in Deductive Databases. The idea is to provide an axiomatic semantics by
defining the closure of the extensional part of the database under an operator
(immediate consequence operator) which formalizes the execution in forward
chaining of the database intensional rules, seen as production or inference rules
[Kowalski, van Emden 76, Apt et al. 86].
More recently, Borgida also promoted the idea of an axiomatic definition of
DL by analogy with the type systems of programming languages [Borgida 92].
He shows how to define DL in terms of Natural Deduction rules. However, his
approach goes only part of the way—no effort is made in the paper to prove the
completeness of the inference system wrt the classical denotational semantics of
DL.
In a quite different context, namely consistency checking in knowledge
validation, Hors and Rousset stress the need for complete axiomatizations
of DL [Hors, Rousset 93]. Finally, papers by Dionne et al., for example
[Dionne et al. 93], inspired by the theory of programming languages, develop
algebraic semantics for simple DL (no negation, no number restriction). Though
not an explicit axiomatization, the approach also emphasizes the idea of “computable” semantics, i.e. semantics which can be interpreted as specifications of
subsumption algorithms.
In summary, we briefly list some of the advantages of defining axiomatic
semantics for DL:
Modularity: This is one classical advantage of inference systems, such as Natural Deduction or Sequent Calculi, which obey the principle of axiomatizing
each language constructor by one specific set of rules. This is particularly interesting in the view of an incremental development of knowledge
representation languages.
Deductive Capacity: This is in the spirit of a constructive, generative approach
to knowledge base systems, where part of the consequences of the initial
knowledge is pre-computed by means of inference rules seen as production
rules. This essentially contrasts with the approach of Tableaux Methods
3.2 Methodology for Deriving Complete Inference Systems
17
[Hollunder, Nutt 90, Donini et al. 91a], which exploit refutation techniques
(reducing subsumption provability to unsatisfiability). These methods are
appropriate for proof-checking, i.e. verifying at query-time that some formula is entailed by some others. They are not cut out for deriving relevant
consequences at assert-time.
Guidelines for Implementation: We also think that implementations can benefit
from axiomatic semantics of DL. It can help to derive so-called NormalizeCompare algorithms [Nebel 90]. Implementations based on NormalizeCompare algorithms are in general ad-hoc and lack a methodology for
defining a right notion of “normal forms”. An axiomatic semantics can help
to choose an appropriate notion of normal forms. Intuitively “normal forms”
are most likely to be produced by the inference rules deriving formulae of
the form c1 u c2 v c3 . This corresponds to some kind of “vivification” of
the knowledge base [Etherington et al. 89]. Then the “compare” part of the
algorithm can be computed by using the “introduction” rules of the form
c1 v c2* q(c1 ) v q(c2), where q denotes some role restriction quantifier
(see [Royer, Quantz 92] for more details).
Characterization of Implemented Systems: In order to guarantee efficient performance, implemented algorithms are often incomplete. At least the degree
of incompleteness should be indicated by specifying which inference rules
are applied by the algorithms and which are not. Furthermore, it can be
indicated for each rule whether it is applied at assert-time or at query-time.
Finally, the inference rules can serve as a basis for flexible implementations
in which the inference process can be guided from the outside. We see this
as one of the major research issues in DL in the near future.
3.2 Methodology for Deriving Complete Inference Systems
Motivating the Use of Sequent Calculi
We present briefly the main characteristics of Gentzen’s Sequent Calculi. For complete and excellent presentations the reader is referred to [Sundholm 83, Gallier 86,
Girard et al. 89] .
A sequent is a meta-formula Γ ) ∆, where both the “antecedent” Γ and the
“succedent” ∆ are sets of first-order formulae. Intuitively, the sequent Γ ) ∆ is
meant to encode the entailment relation j=L Γ^ ! ∆_ , where Γ^ is the conjunction
of the formulae of Γ and ∆_ is the disjunction of the formulae of ∆ .
Formally, the notion of (classical) satisfaction for the sequent Γ ) ∆ is
defined as the satisfaction in First-Order Logic of the formula Γ^ ! ∆_ . Γ ) ∆
is said valid iff every first-order structure satisfies the formula Γ^ ! ∆_ ; it is
called falsifiable otherwise. As usual, these notions can be generalized to obtain
18
3 INFERENCE RULES AND SEQUENT CALCULUS
satisfiability and falsifiability wrt a given set of assumption (proper axioms). If Σ
is a set of sequents and is an individual sequent, then is said valid wrt Σ iff
every first-order structure satisfying all the sequents of Σ also satisfies .
Sequent Calculi are deduction systems (systems of inference rules) aiming at
axiomatizing the above notion of validity. Their main interest lies in a systematic
and principled approach to Proof Theory. First, the proof process in Sequent
Calculi is seen as a morphism of the set of formulae into the set of proofs. In
classical first-order Sequent Calculi this gives rise to the Subformula Property—
one can restrict to proofs built only with subformulae of either axiom sequents or
the initial sequent to be proved. Due to this property, the proofs can be searched
in a closed universe of formulae known in advance.
Furthermore, there is a systematic search procedure for proofs. It consists
in simplifying the sequents by eliminating successively the top-most connectors
and quantifiers of the formulae appearing in the sequents, until axiom schemata
or assumptions are reached. The strategic aspects concern the choices of the
subformulae on which the connector elimination will be performed and in what
ordering. The connector elimination is made possible just because the Sequent
Calculi axiomatize them individually (as opposed to the Hilbert-style calculi). For
each connector two specific rules are provided, the so-called left and right rules,
respectively giving necessary and sufficient provability conditions when the formula containing the given connector appears as antecedent or succedent. All these
rules can be applied either as elimination rules (backwards from the conclusion to
the assumptions) or as introduction rules (forwards from the assumptions to the
conclusion) of the corresponding connectors. In Section 4.2 we will show how a
sequent proof proceeds.
Note that these remarkable properties of Sequent Calculi have been already
recognized as extremely useful in the domain of Software Engineering. Sequent
Calculi or Natural Deduction systems (which are intuitionistic Sequent Calculi)
are widely used for the specification of programming languages, program verification, and type checking [Paulson 87]; there the proofs are interpreted as correct
constructions of formulae from an initial set of axioms. Here, we want to promote
the same kind of approach for knowledge representation languages.
How to Guarantee Completeness?
The idea is to rely on sound and complete Sequent Calculi axiomatizations of FirstOrder Logic without (FOL) or with Equality (FOL= ) [Sundholm 83, Gallier 86]. We
use SC to refer to some sound and complete Sequent Calculus for FOL.
A general methodology for obtaining sound and complete inference systems
for any logical language L is to translate the formulae of L (L-formulae) into
first-order formulae (provided L is translatable into FOL), and then to identify
necessary and sufficient conditions of provability in SC of the FOL translations
3.3 Technicalities on Sequent Calculi
19
from the formulae encoding L-formulae [Girard et al. 89].
More precisely, let (L,j=L ) be some logical language L together with its entailment relation j=L. Let us also suppose that L is translatable into FOL,5 i.e. there
exists a first-order language FL and a translation function 7! from L to FL
such that:
Γ j=L
, Γ j=FL Given some translation = , and some proof tree T( ) of in SC, let us assume
that T can be rewritten into a tree T’, whose root is still but whose nodes are
all formulae encoding L-formulae (formulae of the form f for some L-formula
f). Then, up to decoding, the resulting tree T’ is a proof tree of in an inference
system constituted of all the inference rules encoding the elementary construction
steps of T’. If this rewriting process succeeds systematically, i.e. for all formulae
of L and one proof tree for each of them, then the collection of all inference rules,
used in all the rewritten trees T, is a complete set of inference rules for L.
This key idea is exploited methodically for deriving complete inference systems
for subsumption in DL. Technically, the task is to rewrite systematically SC proofs
of (the translation of) subsumption formulae into proofs containing only sequents
encoding terminological formulae.
In the following sections we show how to apply the method on classical
language constructs. Depending on the complexity of the first-order fragment
needed to encode terminological formulae, different Sequent Calculi will be used.
We first present the derivation method for language constructs translatable into
First-Order Logic without Negation and Equality. Then we show how to proceed
with constructs involving Negation or Equality.
Remarks: In [Cas91] a translation of DL in the sequent formalism was already
used, but in a quite different perspective. Linear Sequent Calculus6 was used to
formalize a “tractable” notion of subsumption, i.e. a more “intuitionistic” notion
of subsumption relying on Linear Logic rather than classical First-Order Logic.
3.3 Technicalities on Sequent Calculi
Figure 3 shows the particular Sequent Calculus SC on which this paper is based.
It is sound and complete for FOL, as shown in [Gallier 86]. A Sequent Calculus
for FOL= , SC= , is obtained from SC by adding the equality axioms for identity,
substitution, and functional substitution:
`SC
5
Γ)a=a
This condition is fulfilled by every DL we know. They all admit natural translations into
first-order languages.
6
A sequent calculus for Linear Logic [Girard 87].
3 INFERENCE RULES AND SEQUENT CALCULUS
20
Operator
left-rule
Weakening
Γ ∆
Γ; ∆
Contraction
Γ;; ∆
Γ; ∆
right-rule
)
)
)
)
Γ ∆
Γ ∆;
)
)
);;∆
);∆
Γ
Γ
Axiom/Cut-Rule Γ; ) ∆; _
Γ; ∆ ; Γ; ∆
Γ; ∆
^
Γ;; ∆
Γ; ∆
!
Γ; ∆ ; Γ ;∆
Γ; ∆
:
Γ; ∆
Γ ∆; 8
Γ;
9
Γ;A(a) ∆
Γ; xA(x) ∆
)
_ )
)
)
)
) _
)∆; ;Γ)∆;
Γ)∆;^
Γ
)
! )
)
) !
Γ; ∆;
Γ ∆; )
) :
)∆;
:)∆
Γ
Γ;
8xA(x);A(t))∆
Γ;8xA(x))∆
9
)
Γ ∆;;
Γ ∆; )
^ )
)
)
Γ1 ; ∆1 ; Γ2 ∆2 ;
Γ1 ;Γ2 ∆1 ;∆2
)
)
)
) 8
Γ ∆;A(a)
Γ ∆; xA(x)
Γ)∆;9xA(x)
)∆;A(t);9xA(x)
Γ
Figure 3: The Sequent Calculus SC. “a” stands for a parameter (called “eigenparameter” or “eigen-variable”) which does not appear in the lower sequent; “t”
stands for a term appearing in the lower sequent (we thus consider pure proofs
[Gallier 86]).
`SC Γ; s1 = t1; :::; sn = tn; P (s1; :::; sn) ) P (t1; :::; tn)
`SC s1 = t1; :::; sn = tn ) f (s1; :::; sn) = f (t1; :::; tn)
In this paper we will also use SCNNF , a sequent calculus for formulae in Negation
Normal Form (NNF).7 SCNNF is obtained from SC by eliminating the negation
rules and replacing the logical axiom schema Γ; ) ∆; by the following four
schemata:
) ∆; Γ; ; : ) ∆
Γ; Γ; :
Γ
) ∆; :
) ∆; ; :
Two essential properties of SC are of strategic importance in our axiomatization
work: the Subformula Property, stating that the Cut Rule can be eliminated unless
7
A formula is in NNF iff the negation symbols appear only in front of atomic subformulae.
21
it is applied to initial assumptions, and the Normalization Property, which allows
the restriction to “normal” proofs where the quantification rules are applied before
the propositional rules. This latter property requires some syntactic conditions,
for example that the sequents are in NNF.8
Proposition 1 Cut Elimination Theorem [Girard et al. 89]
Let S be a set of sequents (proper axioms) and s an individual sequent. S `SC s
iff there is a proof in SCof s whose leaves are either logical axioms or (instances
of) proper axioms of S, and where the Cut Rule is only applied with one premise
being a proper axiom.
Proposition 2 Normalization Property [Gallier 86]
Any NNF sequent provable in SCNNF from some set of assumptions ∆ has a normal
proof, i.e. a proof consisting, top-down from the root to the leaves, of:
first-order steps involving applications of the quantification rules or the
structural rules, except from Weakening,
propositional steps,
Cut-Rule applications with one proper axiom of ∆ and one sequent whose
complexity (in terms of height of a normal proof) is less than the root sequent.
4 Concept Subsumption without Equality
We will split our investigation of concept subsumption into three parts. We
start with a fragment which can be translated into pure First-Order Logic, i.e.
without using equality axioms. In the next section we will then investigate concept
forming operators whose translation involves equality. Both sections deal with
DL fragments in which only primitive roles are allowed. After having discussed
role subsumption in Section 6, we will briefly address the impact of complex role
terms on concept subsumption (Section 7.1).
In this section we will thus study the following fragment:
! > ; ? ; cn ; : c ; c1 u c2 ; c1 t c2
8r:c ; 9r:c ; :9r:c
r ! rn ; r1 u r2
! c1 v c2 ; r1 v r2
c
8
One other condition is that the sequents contain only prenex formulae, which is not the case
for the sequent translations of terminological formulae. The NNF condition will be true for
terminological languages without negations or negation restricted to primitive concepts.
22
4 CONCEPT SUBSUMPTION WITHOUT EQUALITY
anything >
and
c1 u c2
or
c1 t c2
not
:c
nothing
all
some
no
?
8r:c
9r:c
:9r:c
Figure 4: Concrete names for the term-forming operators of TL.
Note that this DL fragment can be translated into pure First-Order Logic, i.e.
without using equality axioms. In addition to the abstract syntax used for DL
we will also use names to refer to the various term-forming operators. The
correspondence between these names and the concrete syntax is shown in Figure 4
for the fragment under consideration in this section. Thus we will call terms of
the form 8r:c all terms, for example.
In this section we will restrict our attention to inference rules relevant for
determining concept subsumption, i.e. for deriving formulae c1 v c2 . We start
by considering concept conjunction and disjunction (Section 4.1), then take into
account the quantifiers all and some(Section 4.2), and finally investigate negation
and no(Section 4.3).
4.1 Conjunction and Disjunction
The translation of a subsumption formula c1 v c2 results in the following equivalent
sequents:
; ) 8(x)c1(x) ! c2(x)
; ) c1(a) ! c2(a)
c1(a) ) c2 (a)
by definition
8 right
! right
Note that the elimination of the right universal quantifier introduces a new parameter “a”, called an eigen-variable. Generally, the rules 9-left and 8-right,
introducing new eigen-variables, contribute the so-called proof lexique, i.e. the set
of parameters used for further instantiations via the 9-right and 8-left rules.
To simplify the notation, we will usually omit the translation function. Typically, we will simply write :::; c1 (a) ) c2 (a) instead of :::; c1 (a) ) c2 (a). The
arrow * will be used for terminological inference rules. The correct syntax is:
Prem * Con, where Prem represents the list of premises (separated simply by
commas, to be taken conjunctively) and Con the conclusion of the rule. The symbol Prem *
) Con is the abbreviation for Prem *Con and Con * Prem (meaning
that the rule is reversible).
For the propositional fragment of the terminological language TL without
negation, the subsumption formulae are trivially isomorphic to the propositional
4.2 Quantification
23
fragment of SC without its two negation rules. No significant rewriting is necessary
here. The following complete set of rules is immediately obtained, for concept
subsumption.9 From the axiom group of SC we get the logical axiom (2); the
Weakening rules give (3 and (4); from the ^-right rule and _-left rule we obtain
the conjunction (5) and disjunction (6) rules respectively; the Cut-Rule of SC gives
the Cumulative Transitivity (7):
c1
c1
c1 v c2; c1
c2 v c1 ; c3
c1 v c2 ; c2 u c3
v
v
v
v
v
c2
c2
c3
c1
c4
* cv c
* c1 u c2 v c1 t c3
* c1 u c3 v c2
* c1 v c2 t c3
*
) c1 v c2 u c3
*
) c2 t c3 v c1
* c1 u c3 v c4
* ?v c
* cv >
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Note that we have assumed implicitly that c u c = c and c t c = c. So we can omit
the explicit Contraction rules of SC.10 Consequently, the axiom *c v c is implied
by (2). Similarly, simple transitivity c1 v c2 , c2 v c3 * c1 v c3 is just a special
case of (7), i.e. Cumulative Transitivity.
The following equivalences can also be derived, using Cut rules and the axioms
for ? and > for the left to right direction, and simple weakenings for the converse
direction:
c1
c1
v
v
c2
c2
*
)
*
)
c1 u > v c2
c1 v c2 t ?
4.2 Quantification
We now consider the quantified part of the language TL without negation, i.e. formulae containing all and some. They are translated into sequents with quantifiers.
The derivation of inferences rules for these constructs will require more rewriting
efforts. The general idea is to use classical proof strategy called “lifting”—any
first-order proof can be obtained by “lifting” a propositional proof, i.e. by adding
quantification steps on top of it. In SCNNF , the formal justification of lifting is
given by the Normalization Property (cf. Section 3.3). We ignore negation in the
beginning since this guarantees trivially that all formulae are in NNF.
9
Note that we do not specify the rules for commutativity and associativity of conjunction and
disjunction.
10
The sequents are seen as sets of formulae rather than lists.
4 CONCEPT SUBSUMPTION WITHOUT EQUALITY
24
(10)
(9)
(8)
(7)
(6)
(5)
(4)
(3)
(2)
(1)
`SC c1(b) ) c2(b)
`SC r1(a; b) ) r2(a; b)
`SC r1(a; b); c1(b) ) c2(b)
`SC r1(a; b); c1(b) ) r2(a; b)
`SC r1(a; b); c1(b) ) r2(a; b) ^ c2(b)
`SC r1(a; b); c1(b) ) r2(a; b) ^ c2(b); 9r2:c2(a)
`SC r1(a; b); c1(b) ) 9r2:c2(a)
`SC 9r1:c1(a) ) 9r2:c2(a)
`SC ) 9r1:c1(a) ! 9r2:c2(a)
`SC ) 8x9r1:c1(x) ! 9r2:c2(x))
`TL 9r1:c1 v 9r2:c2
Figure 5:
SC
assumption c1 v c2
assumption r1 v r2
left-Weakening
left-Weakening
^-right
right-Weakening
9-right
9-left
!-right
8-right
decoding
derivation of 9r1 :c1 v 9r2 :c2 .
Finally, we will use one more technicality, purely specific to the terminological
context. Indeed, as there is no way in terminological theories to express mutual
propositional interactions between roles and concepts, one can separate the proofs
of role terms and concept terms in the propositional fragment.
Lemma 1 (Separation Lemma) Let be a propositional sequent of the form
r; c ) c; r where c, c are formulae corresponding to the translations of
concept terms different from ? and > and r , r are formulae corresponding to
the translations of role terms. Let ∆ be a set of sequents representing terminological
formulae (assumptions).
Then ∆ `SC iff ∆ `SC c ) c or ∆ `SC r ) r .
Example of Derivations
In Figure 5, we show the derivation of the inference rule corresponding to the
monotonicity property of some:
c1
v c2; r1 v
r2
* 9r1 :c1 v 9r2 :c2
It is important to note that the derivation trees may be searched either top-down
or bottom-up. They are obtained by a systematic (sometimes long) case analysis
about the possible ways of reducing the (first-order) conclusion sequent to propositional ones (corresponding to admissible axioms in the terminological setting).
Let us give the intuition of the process. We are searching for necessary and
sufficient conditions for formulae of the form 9r1 :c1 v 9r2 :c2 . The corresponding
sequent is (1) in Figure 5. The only SC rule applicable to (1) is the 8-right rule,
4.2 Quantification
25
giving the equivalent sequent (2). Then, the only rule applicable to (2) is !-right.
This gives the equivalent sequent (3) of the form:
9yr1(a; y) ^ c1(y) ) 9zr2(a; z) ^ c2(z)
At this step, two rules are applicable to (3): either 9-left for eliminating 9y or
9-right for eliminating 9z.
Let us examine the second choice. By 9-right, the variable z can only be
substituted by a parameter t already appearing in the current sequent (in the
current “proof lexique”), i.e. by a, y or z ; then the succedent of (3) becomes
r2(a; t) ^ c2 (t). The ^-rule gives two sequents to prove, one being
9y(r1(a; y) ^ c1(y)) ) r2(a; t); 9z(r2(a; z) ^ c2(z))
Now the only rule applicable is 9-left, eliminating y with a new eigen-variable b.
We end up with the sequent:
r1 (a; b); c1(b)
)
r2 (a; t); 9z (r2 (a; z ) ^ c2(z ))
where t = a, y or z . One can show that this sequent is unprovable propositionally,
unless c1 v ?. Indeed, trying to prove the propositional subsequent
r1 (a; b); c1 (b)
)
r2 (a; t)
is only possible, due to the Separation Lemma, by proving one of the following
sequents
) ;
r1 (a; b) ) r2 (a; t)
c1 (b)
The first sequent is provable by the assumption c1 v ?. The latter sequent can
only be proved using assumptions about roles (no insconsistent role is allowed in
TL). But all role assumptions in TL give sequents of the form r1 (x; y ) ) r2 (x; y ),
i.e. with the same first and second argument.11 This shows that the sequent is only
provable independently of the particular instance r2(a; t) ^ c2(t).
This leads back to the initial choice step for (3). Applying systematically
the second choice for quantifier elimination leads to an infinite loop and no proof.
This dead-end illustrates a wrong strategy in Sequent Calculi—the use of quantifier
elimination rules which do not create parameters before rules contributing to the
creation of the proof lexique.
Going back to the first choice, the 9-left rule eliminates y by replacing it by a
new parameter b. We get the sequent (4). This gives a new substitution possibility
11
This would be different with more complex role operators like composition or inversion.
26
4 CONCEPT SUBSUMPTION WITHOUT EQUALITY
for z , z := b, giving (5). The lifting strategy corresponds to the passage from
(5) to (6): reducing the provability of the first-order sequent (5) to the provability
of the propositional sequent (6). (7) and (8) result from (6) by the ^-right rule.
Then the Separation Lemma gives the sequents (9) and (10), each one provable by
correct terminological assumptions.
In summary, the case analysis gives the following necessary and sufficient
conditions: 9r1 :c1 v 9r2 :c2 is provable iff either (c1 v c2 and r1 v r2) or c1 v ?
are provable.
Complete Characterization of TL (First-Order Fragment)
Considering all possible ways of introducing quantifiers into propositional sequents given by axioms of the form c1 v c2 , c1 u c2 v c3 , c3 v c1 t c2 or c3 u c4
v c1 t c2, in order to recover quantified subsumption formulae, yields the following complete characterization of first-order terminological entailment in TL1
(non-propositional rules only):
>v c
cv ?
c1 v c2; r2 v r1
c1 v c2; r1 v r2
c2 u c1 v ?; r1 v r2
> v c2 t c1; r1 v r2
c1 u c2 v c3 ; r3 v r1; r3 v r2
c3 v c1 t c2 ; r3 v r1; r3 v r2
c1 u c2 v c3 ; r2 v r1; r2 v r3
c3 v c1 t c2 ; r2 v r1; r2 v r3
*
)
*
)
*
*
*
*
*
*
*
*
> v 8r:c
9r:c v ?
8r1:c1 v 8r2:c2
9r1:c1 v 9r2:c2
8 r2:c2 u 9 r1:c1 v ?
> v 9r2:c2 t 8r1:c1
8r1:c1 u 8r2:c2 v 8r3:c3
9r3:c3 v 9r1:c1 t 9r2:c2
8r1:c1 u 9r2:c2 v 9r3:c3
8r3:c3 v 9r1:c1 t 8r2:c2
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
Let us say a word about the complexity and effectiveness of the method. In SC, it
is possible to create infinite proof lexiques, by arbitrary long quantifier elimination
chains. This is typically the case with sequents of the form 8x9yP (x; y ); ) .
Every substitution of 8x, (by 8-left) can be followed by an elimination of the
9y, which creates a new eigen-parameter (by 9-left). The process can be repeated
infinitely, by substituting again the initial 8x with the new parameter. This situation
is also possible with our terminological sequents, for example with formulae like
8r1:(9r2:c) .
Here, the axiomatization work is effective, because one can foresee the size
of the proof lexique needed to prove a given subsumption formula. The key
point is that quantification in terminological formulae arises only through role
restriction, thus non-quantified concept terms always come with chains of role
terms. One can then prove that the number of quantifier eliminations needed
4.3 Negation
27
for proving a given formula is bounded by the maximal length of role restriction
chains. Here, in absence of role composition (or constructs able to “shrink” or
modify the role restriction chains), it is enough to eliminate quantifiers uniformly
in the subformulae participating in the proof. This explains why the inference
rules in TL lift uniformly one quantifier at a time. For role composition, which can
shrink role chains, e.g. r1 (a,b) ^ r2 (b,c) ! r1 .r2 (a,c), non-uniform lifting becomes
possible, however:
c1
v c2 * 8r1:8r2:c1 v 8r1.r2:c2
4.3 Negation
We will now consider the effect of including negation into the DL fragment on the
derivation process. There are three possibilities to integrate negation into DL:
1. negation restricted to atomic concepts,
2. implicit negation,
3. full negation.
Negation Restricted to Primitive Concepts
For tractability reasons, an explicit negation constructor, whose translation is given
def
by :c = x:c(x),12 is often restricted to primitive concepts (atomic concept
terms) only. Adding “primitive negation” does not change neither the preceding
approach nor the results. The derivation method, as well as its formal justification
in the sequent formalism, still works for exactly the same reasons as before (the
translations into sequents will be in NNF). Essentially, the only inference rules to
be added correspond to the additional logical axiom schemata of SCNNF .
* cu:cv ?
* >v ct:c
(20)
(21)
Then, for example, the contraposition rule c1 v c2 * : c2 v : c1 can be proved
from the assumption c1 v c2 by cutting successively with the logical axioms
c2 u : c2 v ? and > v c1 t : c1 .
Note that we thus use the symbol ‘:’ both for the negation of DL terms and for the negation of
formulae.
12
FOL
4 CONCEPT SUBSUMPTION WITHOUT EQUALITY
28
Implicit Negations
If the fragment does not contain full negation, but implicit negation due to operators
like no, the sequent translations are no more guaranteed to be in in NNF. As the
Normalization Property holds no more, the arguments made above fail—the lifting
process is not necessarilyy complete. The difficulty is overcome by exploiting the
:
terminological equivalence :9r:c = 8r:: c and by axiomatizing no indirectly, by
the set of rules for all. The method for obtaining a complete axiomatization for no
is to take a complete set of inference rules for all, to replace uniformly c by : c in
every inference rule, and to rewrite them without explicit negations.13 Technically,
the rewriting process is also achieved in the sequent formalism, using the negation
rules of SC, i.e. once again up to the translation of terminological formulae into
sequents.
For example, let us consider the following rule (for the sake of simplicity, the
restrictions are for the same role ):
c1 u c2
v
c3
* 8r:c1 u 8r:c2 v 8r:c3
and replace 8r:c1 by :9r:c1. Substituting : c1 for c1 gives the rule
: c1 u c2 v
c3
* :9r:c1 u 8r:c2 v 8r:c3
Eliminating negation is done by the negation rules of SC (on the sequent translation), giving the new terminological rule:
c2
v c t c1 * :9r:c1 u 8r:c2 v 8r:c
This kind of rewriting will produce the three next rules from (16), i.e. the additivity
rule of all.
c2 v c3 t c1 ; r3
c3 v c2 t c1 ; r3
c1 u c3 v c2 ; r3
v r1 ; r3 v
v r1 ; r3 v
v r1 ; r3 v
r2
r2
r2
* :9r1:c1 u 8r2 :c2 v 8r3 :c3
* :9r1:c1 u :9r2:c2 v :9r3 :c3
* 8r1:c1 u :9r2 :c2 v :9r3 :c3
The systematic rewriting ends up in a complete axiomatization of no consisting
of 11 rules. The large number of rules is due to the lack of explicit negation. All
the De Morgan rules and the equivalences between no, all, and some need to be
encoded implicitly in the rules.
Full Negation
If full negation is allowed, tractability may become problematic, but axiomatization
is far simpler! But, formally, one can no more rely on SCNNF (the arguments
13
Note that this approach does not work for languages containing full negation.
4.3 Negation
29
exploiting the Normalization Property are false outside SCNNF ). As usual, the
difficulty is overcome by showing that the language with full negation is equivalent
to the language with negation restricted to atomic terms. The trick is to give a
complete set of inference rules for transforming formulae with arbitrary negations
into formulae in Negation Normal Form.
First, the propositional fragment of TL is extended by the negation rules from
SC:
c1 u c2 v c3
c1 v c3 t c2
*
)
*
)
c1 v c3 t : c2
c1 u : c2 v c3
(22)
(23)
From these new rules, together with the rules of TL1 one obtains easily (propositionally) the classical De Morgan rules and the duality rules between all and
some:
*
*
*
*
*
:: c =: c
:(c1 u c2) =: : c1 t : c2
:(c1 t c2) =: : c1 u : c2
:9r:c =: 8r:: c
:8r:c =: 9r:: c
(24)
(25)
(26)
(27)
(28)
These five rules are exactly what is needed to put arbitrary terminological terms
into NNF, i.e. terms having negations only in front of primitive (atomic) concept terms. Using them as conversion rules (in the left to right direction), one
gets a decidable procedure for converting arbitrary terminological terms into NNF
(obviously the conversion preserves the logical semantics).14 However, the five
conversion rules above cannot be directly taken as an inference system for transforming subsumption formulae into formulae in NNF. It is further necessary to
:
turn every conversion rule for terms, c1 =c2 , into a left-rule, c3 v c1 t c4 *
) c3
v c2 t c4, and a right-rule, c1 u c3 v c4 *
c
u
c
v
c
.
Then
the
resulting
set
) 2 3 4
of rules yields an inference system, `NNF , for transforming arbitrary formulae
into NNF. Finally, a complete axiomatization for a DL containing full negation
is obtained by adding the rules of `NNF to any complete axiomatization of the
language without negation.
In Figure 6 we list the inference rules for concept subsumption in the language
investigated in this section.
14
The expression “Negation Normal Form” is here used in a generic sense, for the sake of
simplicity. Strictly speaking, NNF for sequents, NNF for terminological terms, or NNF for
subsumption formulae are three different notions, applying to different syntactic entities.
4 CONCEPT SUBSUMPTION WITHOUT EQUALITY
30
c
c
>
(1)
(9)
?
c1 u c2
c1 t c2
(5)
(4)
(22)
>
?
(8)
c1 u c2
(3)
(7)
(23)
c1 t c2
(6)
:c
(24)
(14)
(20)
(24)
(21)
(15)
(10)
(2)
(26) (16) (18)
(25)
(26)
8r:c
9rc
: c 8r:c 9r:c
(11)
(25)
(27) (28)
(19)
(27) (12)
(17)
(28)
(13)
Figure 6: Inference rules for concept subsumption. The operator in the rows is
the outmost operator of the subsumed concept term, whereas the operator in the
columns is the outmost operator of the subsuming concept term.
31
n r:c
n r:c
atleast
atmost
fills
oneof
R:S
S_
Figure 7: Names of the additional concept-forming operators in TL=
5 Concept Subsumption with Equality
In this section we derive inference rules for a DL fragment containing operators
which cannot be translated into pure FOL but require a First-Order Logic with
Equality. We call this fragment TL= :15
! > ; ? ; cn ; : c ; c1 u c2 ; c1 t c2 ; 8r:c ;
n r:c ; n r:c ; r:fo1; :::; on g ; fo1; :::; on g_
r ! rn ; r1 u r2
! c1 v c2 ; r1 v r2
c
The names of the additional concept-forming operators are shown in Figure 4.3.
Note that the quantifiers some and no investigated in the previous section are just
special cases of atleast and atmost. We can define them as follows:
9r:c
:9r : c
def
=
def
=
1 r:c
0 r:c
Consequently, we will not consider some and no terms in our derivations, but
restrict ourselves to the quantifiers all, atleast, and atmost.
All the provability conditions are necessarily to be found by following a standard procedure:
1. Creation of a finite proof lexique by eliminating left-existentials and rightuniversals (see Section 5.1).
2. Putting all possible, finitely many instantiation cases in the sequent by eliminating right-existentials and left-universals systematically wrt the proof
lexique (see Section 5.1).
3. Introduction of finitely many equality assumptions (see Section 5.2).
4. Decidable provability test for the resulting propositional sub-sequents (see
Sections 5.3 and 5.4).
15
Note that the n in n r:c must be greater than 0.
5 CONCEPT SUBSUMPTION WITH EQUALITY
32
Creative
Instantiation
atleast-left atleast-right
atmost-right atmost-left
all-right
all-left
Figure 8: Contribution of operators to the proof lexique.
The first three steps can thus be seen as preparations for the complex case analysis performed in the fourth step. They allow to restrict the case analysis to
propositional subsequents.
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
We begin with some remarks concerning the terminology we will use in the
following. We call terms occurring in the subsumed term of a subsumption
formula left terms and terms occurring in the subsuming term right terms. Thus,
in the sequent translations left terms occur in the antecedent and right terms in the
succedent of a sequent.
In analogy to the standard FOL terminology, a concept term is in prenex form
iff it is of the form q1:q2::::qn(c), where c is a propositional concept term, and
qi denotes arbitrary role or numerical restriction operators. Then the integer n is
called the quantification degree of the term. The quantification degree counts the
number of role restrictions in a given prenex term, not the total number of FOL
quantifiers necessary to the FOL translation.
For non-prenex concept terms, the notion of quantification degree is defined
as the maximal length of role restriction chains. Formally, it is defined by the
following conditions:
degree(c) = 0 if c is a propositional concept term;
degree(q.r.(c1 c2 )) = 1 + max(degree(c1),degree(c2 )), where q.r... stands
for any role restriction operator and (: :) for any boolean operator.
Finally, we distinguish creation rules and instantiation rules in the process of
quantifier elimination. We call the SC rules 8-right and 9-left creation rules because
they indroduce new parameters in the sequents, the so-called eigen-parameters.
Similarly we call the SC rules 8-left and 9-right instantiation rules because they
do not create new eigen-parameters, but instead use the parameters of the current
proof lexique in order to instantiate the variables of the quantified expressions (left
universals and right existentials).
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
33
For eliminating quantifiers in the numerical restrictions atmost or atleast several applications of quantification rules of the same type are needed. Typically, n
successive creation (resp. instantiation) rules are needed to eliminate quantification in a left n r:c (resp. a left term n 1 r:c). To simplify our presentation we
will often refer to such a sequence of creation (or instantiation) rules as if it were
a single creation (or instantiation) rule. Figure 8 show how the concept-forming
operators investigated in the following contribute to the proof lexique.
On Substitutions
We recall that the substitution axioms for the calculus SC=
NNF are obtained from
the following classical substitution axiom, by applying all possible negation rules
on x = y , (x) or (y ):
(x); x = y ) (y)
(classical substitution axiom)
In the following, we will call substitution the process of applying one Cut wrt
one substitution axiom. For example, if some sequent contains, up to negation
rules, a left equality x = y and a left term (x), then is provable if the sequent
0 obtained from by replacing (x) by (y) is provable.
= ; x = y; (x) ) 0 = ; x = y; (y) ) is obtained by a Cut Rule applied to 0 and one substitution axiom.
Substitution is relevant only wrt two eigen-variables (parameters coming from
creative rules 9-left and 8-right) not for parameters coming from instantiation
rules (9-right and 8-left). In the latter case, the effect of the substitution can be
reproduced simply by an adequate instantiation.
Lemma 2 Substitution is useless when applied to parameters coming from a noncreative quantifier elimination rule (i.e. 9-right or 8-left).
On Relevant Instantiations
We now investigate general properties of the instantiation rules 8-left and 9-right
in the terminological sequents. Basically, we will show that under some assumptions about the proof strategy, it is useless to apply the instantiations rules if the
assignment function of variables to actual parameters is not injective or does
not use exclusively eigen-parameters. The assumption is that only one level of
quantifier elimination is considered.
Let be some terminological sequent. Let us call B the set of eigen-parameters
occurring in (parameters either produced by previous creative quantifier elimination or by terms with constant parameters like oneof). Thus the current proof
5 CONCEPT SUBSUMPTION WITH EQUALITY
34
lexique is B [ X where X is the set of all variables occurring in the quantified
terms of . Let us assume that the proof lexique will remain constant in the subsequent proof steps. This means no further creative quantifier elimination will be
done during the proof.
Definition 1 Let be a terminological sequent. An instantiation closure of is
a sequent C obtained from by
1. unfolding first some left atleast, right atmost, left some, or right all terms,
thus creating a set B of eigen-parameters;
2. replacing every left n r:c by a finite list (conjunction) of expressions of the
n+1 (r(a; x ) ^ c(x )) ! equ(~x), where ~x is an injective instantiation
form ^ii=
i
i
=1
into B ;
3. replacing every right n r:c by a finite list (disjunction) of expressions of the
=n+1
form ^ii=
(r(a; xi) ^ c(xi )) ^ dif (~
x), where ~x is an injective instantiation
1
into B ;
4. replacing every left 8r:c by a finite list (conjunction) of expressions of the
form r(a; x) ! c(x) where x 2 B ;
5. replacing every right 9r:c by a finite list (disjunction) of expressions of the
form r(a; x) ^ c(x), where x 2 B .
Proposition 3 A terminological sequent is provable in SC=
NNF by considering
only quantifier elimination of degree 1 iff one instantiation closure of is provable
in SC=
NNF .
Before presenting the proof details, we give some preliminary remarks. One
can restrict to normal proofs, by the Normalization Property of SC=
NNF . Then
the quantification steps, coming after the preliminary creative elimination rules,
involve only eliminations of left universals or right existentials. By Weakenings,
one can always assume that all the possible rules for eliminating the left universals
or right existentials (a finite number of instantiation rules) are applied to , with
possible repetitions in the application of one given instantiation rule (as many as
required for the proof16). Let ) be the resulting sequent.
Note that the instantiation rules concern uniquely left atmost, right atleast, left
all, or right some terms. Important logical symmetries exist between these different
constructors, which justify that in all the forthcoming proofs we will restrict to left
all and left atmost terms. This is justified, since left atmost and right atleast terms
behave identically: after quantifier elimination and some boolean manipulations,
16
By finiteness of the proofs, for each rule there are only finitely many repetitions.
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
35
they both introduce right role and concept terms. Left atmost terms introduce left
equalities, right atleast terms introduce right inequalities. As we reason in the NNF
fragment of SC, we do not have the negation rules explicitly to make left equalities
and right inequalities equivalent. But they behave nevertheless identically because
negation rules are implicitly encoded into the identity and equality axioms (in
particular, the substitution axioms are given in both forms: with left equality and
right inequality, see above p. 33).
Left all and right some do behave identically only wrt role terms (they both
produce right role terms); they behave identically for concept terms up to negation.
They are not equivalent in SC=
NNF , but any proof for degree-1 quantified formulae
with one right some can be rewritten into a proof with one left all and vice versa.
atmost
Notation: We write (
n:r:c)(~x)(a) for the expression resulting from
quantifier elimination upon the left term n r:c(a), wrt the instantiation function
~x (a is the eigen-parameter used to eliminate the top-level universal in the FOLtranslation of the given subsumption formula).
atmost n:r:c)(~x)(a)
(
=
^ii
n r(a; x ) ^ c(x )) ! equ(~x)
i
i
=
=1 (
Proof: After these preliminary remarks we can now deliver the proof for Proposition 3.
I. Non injective instantiation functions: Let us first assume that one left atmost
is instantiated non-injectively (non-injective instantiation functions concern only
numerical operators). We get to prove a sequent ) of the form
1 ; (atmost n:r:c)(~x) ) 1
where one disjunct of equ(~x) is an identity x = x . If in a subsequent proof
step the !-left rule is applied to that term (
n:r:c)(~x), then the sequent 1; equ(~x) ) 1 must be proved. By applying _-left rules, one must
prove in particular the sequent 1 ; x = x ) 1 . This sequent is provable iff
1 ) 1 is provable!17 This shows that the proof is independent of the term
(
n:r:c)(~x).
II. Instantiations by non-eigen-parameters: It means that some instantiation parameters do not belong to the set of current eigen-parameters B . One proves, by
induction about the number of such parameters, that the same proof conditions can
atmost
atmost
Proofs by substitutions wrt x=x or by cuts wrt the identity axiom ; ) x = x give directly
1 ) 1 . Other possibilities could be by using an equality axiom x = x ) x = x or x = x; x 6=
x ) ;, which implies that x 6= x occurs in 1 or x = x occurs in 1 . In the first case, x 6= x may
only come from a left atleast or the negation of a left oneof; in the second case, x = x may only
17
come from right atmost or right oneof terms. In both cases, the (in)equality involves necessarily
two syntactically distinct eigen-parameters (they come from creative quantifier eliminations).
5 CONCEPT SUBSUMPTION WITH EQUALITY
36
be obtained by considering only terms instantiated with eigen-parameters from B
(i.e. instantiation rules by eigen-parameters).
II.1 Induction Basis: The proof involves terms for which all instantiation parameters, but one variable parameter x1 , are taken from B . We assume that the
formula is not provable without terms in x1 . In the case where only left all terms
8ri:ci, for i = 1; : : : ; n, every one with instantiation parameter x1, may contribute
to the proof, then one can assume that no left equality x1 = b; b 2 B occurs during
the proof. (Otherwise, no equality being introduced by atmost terms, the possible
occurences of x1 = b may come only through inessential cuts; then, let us consider
the dual sequents with the corresponding inequality x1 6= b). Then, there is no
means to link, through substitutions, terms with the argument x1 with other terms
having parameter in B .
We get to prove a collection of sequents of the form:
n )
n ; ^i2I (ci(x1 )) )
r1 (a; x1); :::; rn (a; x1); n
^i21;nnI (rk (a; x1)); n
for any indexing subsetI 1; n
n; c1 (x1); :::; cn(x1 ) ) ;
where n ) n is ) without the left terms 8ri :ci (x1).
Therefore x1 does not occur in n ) n . The proof proceeds necessarily
by Constant Separation, proving either subsequents containing terms in x1 or
subsequents containing terms in B . The first alternative holds for all the sequents,
otherwise the proof is independent of x1 . In particular the first sequent must be
proved by the subsequent
; )
r1 (a; x1); :::; rn (a; x1)
From this sequent one gets a proof condition of the form: >r v rj t ... t rk , where
>r means the top-most role, a constructor similar to > for concepts. This is not a
legal condition in the languages so far considered which contain no >r.
NB: If the language would allow >r and disjunctive roles, the proposition holds
nevertheless, only in this case. The provability of the initial formula results from
some inconsistency property between the all terms solely. An equivalent proof
could be obtained by instantiating x1 by any eigen-parameter of B 18 as well.
Now, assuming that at least one left (
n:r:c)(~x) contributes to the proof,
the sequent 1; x1 = xi ) 1 must be proved, where 1 ) 1 is not provable.
One can assume that no proof working with x1 = xi may arise by direct equality
axioms because this would require a left inequality x1 6= xi in 1 or a right equality
x1 = xi in 1. (Indeed, such (in)equalities come necessarily from left atleast or
atmost
B is always non-empty as it contains at least the paramater created by the elimination of the
outer-most right universal used in the translation of the initial subsumption formula.
18
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
37
right atmost terms, which have been consequently unfolded by creative quantifier
elimination rules, so come with eigen-parameters of B . If one inessential cut wrt
a left x1 6= xi has been introduced, then let us consider the dual sequent with
the left x1 = xi .) The remaining possibility is by a substitution axiom. This
means taking another left equality x1 = yj (or right inequality) from another left
(
p:r’:c’)(~y) (or a right (
n:r’:c’)(~y)). Then the sequent
atmost
atleast
1 ; equ(~x); equ(~y) ) 1
must be proved. This implies that x1 = ~x \ ~y and both ~x n x1 and ~y n x1 take
elements from the subset B 0 B constituted of the eigen-parameters created by
some m r1 :c1. (So that the corresponding dif (B ) provides with appropriate
inequalities. Again we can ignore the inessential cuts that could have been introduced, because they always give rise to sequents with no more equalities than
those provided by the dif terms). Then, we get the condition n + p + 1 m.
Unfolding the two atmost terms gives in particular the sequents (together with
similar sequents with role terms instead of concept terms):
2; x1 = yj ) 2; c(x1 )
2 ; x1 = xi ) 2; c’(x1)
2 ) 2; c(x1 ); c’(x1 )
where 2 ) 2 is the sequent obtained from ) by eliminating
(atmost n:r:c)(~
x) and (atmost p:r’:c’)(~y). Note also that 2 contains the
terms c1 (xi) and c1 (yj ) (resulting from the unfolding of m r1 :c1). Assuming
that 2 ) 2 contains no term in x1 anymore, the first sequent gives the proof
condition c1 v c and a similar condition for the role term. This shows that the
initial sequent contains a subsequent corresponding to the simple inconsistency
formula:
n + 1 m; c1 v c; r1 v
r
* m r1 :c1 u n r:c v ?
Then it is easy to see that a simpler proof can be done without any instantiation
parameter outside B and with the unfolding of one unique atmost (see Section
(A.1) for technical details). Note that the condition > v c t c’ given by the third
sequent is not significant for the proof.
If 2 ) 2 still contains other terms in x1, we can assume that these terms
come either from left atmost or left all. Then it is easy to check that one gets
sequents like
n ; x1 = xi ) c(x1); n
where n contains no term in x1, and where n contains a list of terms ci (x1 ) ,
only one of which comes from a left atleast, the others coming from all terms
exclusively.
5 CONCEPT SUBSUMPTION WITH EQUALITY
38
Then the proof conditions correspond to inconsistency schemata of the following form, which can also be proved by choosing instantiation rules wrt B
exclusively (see also (A.1)):
n + 1 m; c1 u c2 v c; : : : * m r1:c1 u 8r2 :c2 u ... u n r:c v ?
II.2 Induction Step: In the case where several instantiation parameters,
x1; x2 ; :::, are non-eigen-parameters, then again the significant case is
when these parameters come from instantiation rules applied to left terms
(
ni:ri:ci)(x~i), for i = 1; : : :; n. Let us consider the proof conditions
for the sequent
atmost
n; equ(x~1); :::; equ(x~n) ) n
resulting from the unfolding of all the left atmost terms having instantiation
parameters outside B , xi 2 x~i. The proof being dependent on the xi ’s, there must
be a chain b = x1; x1 = x2; x2 = x3 :::; xp = b0 such that the sequent
n; b = x1 ; x1 = x2; x2 = x3:::; xp = b0 ) n
resulting from the precedent one by _-rules, is provable but not n ) n (n and
n contain no parameter outside B ). Let us consider one such length-minimal
chain. Then one can easily check that all the parameters of the x~i except from the
xi’s, b and b0 in particular, are to be taken from the same subset B 0 of the eigenparameters created by one left n r:c (same as case II.1). Hence the cardinality
conditions ni < m. In particular, assuming the equation b = x1 belongs to
equ(x~1), by the unfolding of n1 r1 :c1 , one gets at least to prove the sequent:
n ; x1 = x2; x3 = x4:::; xp = b0 ) n; c1 (x1)
and one similar sequent with r1 (x1). The situation becomes basically the same as
in II.1; the proof simply reduces to one inconsistency schema between one atleast
(with possibles all terms) and one atmost.
2
In the general situation where the proof lexique may grow during the proof process
(which means elimination of nested quantifications), the same arguments still hold
for proving that the instantiation functions can be chosen to be injective (cf. case
I of the previous proof).
Corollary 4 For proving terminological sequents, one can always assume that
the instantiation rules are applied wrt injective instantiation functions.
When no complex role operators are considered, then Case II of the precedent
proof can be generalized by induction over the quantification degree (cf. next
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
39
section). The major difficulties however for extending Case II arise when heterogeneous quantification degrees are possible between the different concept terms
of the subsumption formula. As quantification depth is related to role restriction, the changes concern only the role conditions. Possible generalizations of
the proposition need specific discussions according to what kinds of role operators are actually contained in the terminological language. The extension to role
composition seems to cause no problem. (But no formal proof is given here.)
Finally, the results here show that the right strategy for quantifier elimination
is to apply the creation rules before the instantiation rules. This is a quite classical
strategy in Sequent Calculus, justified here by Proposition 3 for quantified formulae
of degree 1.
On Breadth-First Proof Strategy
The depth of quantifier eliminations and the global proof strategy are now discussed. The point is to show that one degree of quantification at a time needs to
be considered for the axiomatization work, in absence of any complex roles (i.e.
without role operators which are able to modify the structure of role chains). This
property is of extreme importance because it will allow to restrict to search for
inference rules of homogeneously quantified formulae with quantification degree
equal to 1.
In this section, we will prove a preliminary property about the way quantifier
eliminations can be applied. Namely, we will show that one can restrict to proofs
where the elimination of top-level quantifiers by instantiation rules can be done
wrt the set of eigen-parameters created by the creation rules applying to top-level
terms. This justifies proof strategies where the first steps consist in eliminating
the top-level role restriction operators, for all the terms which need quantifier
elimination in the proof. Intuitively, this leads to some kind of breadth-first
strategy for quantifier elimination. In the next section, we will show that one can
furthermore restrict to “homogeneous quantifier elimination” strategies, where the
same quantification degree is eliminated homogeneously for all the concept terms.
First, we introduce the notion of level of a concept term in a given subsumption
formula as being the total number of role restriction operators it is bound to in the
formula.
Definition 2 Given a subsumption formula or a concept term and an occurrence
of a concept term c in , the level of c in , written level(c,) is the integer defined
inductively by the following conditions:
level(c, c2 v c1 ) = level(c, c1 ) if c occurs in c1 , level(c, c2
c2 ) if c occurs in c2,
v c1) = level(c,
level(c, c) = 0, level(c, : c) = 0, level(c, c u c’) = 0, level(c, c t c’) = 0
5 CONCEPT SUBSUMPTION WITH EQUALITY
40
level(c, q.r.c’) = 1 + level(c, c’) where q denotes any role restriction operator.
The level of a role term r occurring in a subsumption formula is the level of the
encapsulating concept term (q.r.c) in which it occurs in .
Proposition 5 (Breadth-First Strategy) Let be the terminological sequent corresponding to some subsumption formula . is provable in SC=
NNF wrt some
terminological theory Γ if there exists a finite collection of sequents (i )i2I such
that, for every i 2 I :
1. the FOL formulae of i are the FOL translations of concept terms occurring
in with level 1 or 0;
2. i is provable in SC=
NNF wrt Γ.
Proof: Similarly as the definition of level for concept terms, the notion of level
of one eigen-parameter in a given terminological sequent proof is defined as being
the total number of role restriction operators in the field of which it occurs at
creation time (i.e. at application of the creative rule introducing it). For example,
the sequent translation of the left term 8r1 :(n r2 :c)(x) will provide with terms
r1(x,y), r2 (y; bi) and c(bi ) where the eigen-parameters bi ’s will be of common level
2.
As usual, we assume normal proofs for NNF sequents. We write ) to
denote the sequent obtained from the initial subsumption formula (sequent ) after
all the quantifier elimination steps have been performed. Note that ) is
obtained from by syntactic rewritings similar to those described in Definition 1.
Let us now state some preliminary remarks about the form taken by ) .
1. The creation rules replace the quantified term q r : c(x), by adding both role
and concept terms r(x,b) and c(b), where b is a new eigen-parameter, to the
original sequent. It is easy to see that the terms are added to the left-hand
side of the sequent, except from concept terms created by right all terms.
Typically, ; n r:c(x) ) gives the equivalent sequent
; r(x; b1); c(b1); :::; r(x; bn); c(bn); dif (~b) ) where the bi ’s are new eigen-parameters.
2. On the other hand, instantiation rules applied to quantified terms q r : c(x),
result in splitting the original sequents into several new ones: some containing the role terms r(x,y), some others containing the concept terms c(y), for
every parameter y . (In general, the instantiated role and concept terms go
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
to the right-hand side of the sequent.) Typically, ; n r:c(x)
collection of sequents, in particular the sequents
where
41
) gives a
; n r:c(x) ) ; r(x; y)
; n r:c(x) ) ; c(y)
y
is instantiated by a parameter occurring in the prof lexique of
; n r:c(x) ) .
Let us now consider one top-level left atmost term (of level 0) in the sequent and let us assume that this term has been unfolded into ) . Thus n r:c(a)
contributes a term (
n:r:c)(~x)(a), where c is possibly quantified.19 We
write 1 ) 1 for the sequent obtained by eliminating (
n:r:c)(~x)(a) in
) :
atmost
atmost
)
(atmost n:r:c)(~
x)(a)
=
=
1 ; (atmost n:r:c)(~x)(a) ) 1
equ(~x) ! ^ii n1 (r(a; xi) ^ c(xi))
=
=
Let us assume that the quantifier elimination is done by instantiating some variable
of (
n:r:c)(~x), say x1 , by the eigen-parameter c of level l, where l > 1.
Eliminating the boolean connectors in (
n:r:c)(~x), we get to prove in
particular
atmost
atmost
1 )
r(a; c); 1
NB: By construction, c has been introduced by a creation rule upon a term of
degree at least 2, (qn... (q2.r2 .(q1.r1.c))..), where q1 corresponds to one creative
quantifier (left atleast left some or right atmost).
In the following, the role or concept terms introduced by creation rules will
be called creative terms. They are of the form ri (u; v ) or ci (v ) where v is an
eigen-parameter. Note also that the creative role terms occur necessarily in 1 .
Let us now reason about the different proof possibilities for 1 ) r(a; c); 1
assuming that 1 ) 1 is not itself provable. The next (necessarily propositional) steps must lead to at least one role subsequent which cannot be proved
independently of r(a; c).
1. Let us assume that there is one subsequent, occurring at some step in the
proof for ) , whose proof depends on r(a; c) and which involves
necessarily the contribution of at least one role term introduced by creative
quantifier elimination. (No subsequent containing r(a; c) can be proved
without creative role terms.)
Two cases arise according to whether the proof depends on the creative role
term r1 (x1 ; c) introduced at the creation of c (b) or not (a).
19
The argument a is the eigen-parameter created by the elimination of the top-level right universal
variable used in the FOL translation of the considered subsumption formula.
42
5 CONCEPT SUBSUMPTION WITH EQUALITY
(a) Then the relevant creative terms for proving r(a; c) are of the form
r’(u; v ) where v is syntactically different from c. r’(u; v ) belongs to
1 . Let us assume that one subsequent is provable, with both r(a; c)
and r’(u; v ) (but the sequent without one of them is not provable). The
proof succeeds if and only if one left equality c = v is introduced20.
One possibility is when a preliminary inessential cut wrt c = v has
been performed. This means that the above sequent yields the two
sequents
1 ; v = c )
1 ; v 6= c )
r(a; c); 1 ; r’(u; v ) 2 1
r(a; c); 1
In the second sequent no substitution is possible to make v and c
logically equal, hence it must be proved without r’(u; v ). As there are
only finitely many other candidates for creative role terms in 1 , by the
same kind of reasoning one eventually gets to prove the sequent:
1 ; u1 6= a; :::; un 6= a )
r(a; c); 1
where no more proofs are possible by subsequents containing r(a; c)
and at least one creative role term of 1 .
The second possibility is when the equality is introduced by means of
the left term equ(~x) coming from the unfolding of some left atmost,
where c = v is one disjunct of equ(~x). Then, the unfolding gives
other sequents which do not contain equ(~x) but contain instead role
or concept terms with parameters different from c (because one may
restrict to injective ~x, so ~x has at least two distinct elements).
Finally, this shows that there must be one subsequent which must be
proved with r(a; c) but without any creative role term of 1 .
(b) Let us assume now that there is one subsequent whose proof depends on r(a; c) and involves the role term r1 (x1; c) introduced at the
creation of c. c being of level at least 2, it comes with a role chain
r1(x1 ; c); r2 (x2; x1). If x1 is also an eigen-parameter, x1 = b1, then the
same kind of reasoning as previously applies to r(a; c) and r1(b1 ; c).
Since level(b1) = level(c)-1, b1 cannot be syntactically equal to a, therefore one inessential cut wrt b1 = a or one equ(~x) containing b1 = a
is necessary. In both cases, there is one other sequent, almost similar
to , but with b1 6= a, which gets unprovable with r(a; c) and r1 (x1; c)
together. We get the same conclusion as precedently.
20
The case of right inequality, being quite similar, is omitted.
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
43
2. Let us now consider a subsequent whose proof depends on r(a; c) and
non-creative role terms exclusively. Being non-creative, these role terms go
to the right-hand side (cf. preliminary remarks).
p )
p )
r’(u; y ); r(a; c); p
c’(y ); r(a; c); p
(In the second sequent the concept term goes possibly to the left-hand side
if it comes from a left all.) For the same reasons as in the precedent cases,
one can assume that the instantiation rules for introducing r’(u; y ) and c’(y )
have been donedirectly wrt (u; y ) := (a; c).
Let us consider a minimal set of left terms ni ri :ci or 8ri :ci , i = 1:::k ,
which contribute to the proof of wrt r(a; c). We use p ) p to denote
the sequent obtained from ) by eliminating these terms. One gets to
prove the sequent
p )
r1 (a; c); :::; ri (a; c); :::; rk (a; c); r(a; c); p;
such that it is provable without creative role terms, hence without p , from
; )
r1(a; c); :::; ri (a; c); :::; rk (a; c); r(a; c)
If the language does not contain >r, this sequent is unprovable because it
requires some forbidden proof condition: >r v r1 t ... t rk .
2
Corollary 6 Any terminological sequent can be proved by applying the quantifier
elimination rules of degree 1 (quantifier elimination of the first quantification
degree for level-0 terms) and the rules for level-0 role terms, before the quantifier
elimination rules for the concept terms of level greater or equal to 1.
Proof: The instantiation rules for the top-level terms (level-0 terms, corresponding to the first degree of quantifier elimination) can be done by means of level-1
eigen-parameters, i.e. parameters created by the first degree of creative quantifier
elimination. As a result, using the Separation Principle between concept and
role sequents, the propositional steps which apply to role terms of level 0 can
be permuted with any inference rule (quantifier elimination or propositional rule)
applied to any concept term of level greater than 1. Consequently, the proof can be
rewritten into a non-normal proof, where all the proof steps for concept terms of
level greater than 1 are delayed, coming after the degree one quantifier elimination
44
5 CONCEPT SUBSUMPTION WITH EQUALITY
rules (for level-0 concept terms) and the propositional steps for level-0 role terms.
2
Corollary 7 The following strategy is sound and complete for proving any subsumption formula .
1. Applications of degree-1 creation rules on level-0 terms of ;
2. Applications of degree-1 instantiation rules on level-0 terms of , with
instantiation functions which are injective in the set of eigen-parameters
created in step (1);
3. Applications of inessential cuts wrt the set of eigen-parameters created at
step (1);
4. Propositional proofs for role terms of level 0;
5. Proofs of subsumption formulae (i )i2I , where each i is a subsumption formula constituted of level-1 or level-0 terms from . No quantifier elimination
applies in the ongoing proof to level-0 terms of i .
Proof: The last steps in the proof of Proposition 5 have shown that the instantiation rules of degree 1 (elimination of the first degree of quantification for level-0
terms) and the rules applied to level-0 role terms can be permuted with the other
steps of the proof, i.e. the quantifier elimination rules for the concept terms of
level greater than 1. The same property obviously holds for the creative quantifier
elimination for level-0 terms (because the creative rules are independent from the
others). This shows that one can rewrite the original proof into a proof where all
eliminations of degree-1 quantification, creative or non-creative, are performed at
the beginning of the proof. Then, the resulting proof gives rise to a collection Σ of
sequents containing level-1 concept terms and possibly remaining level-0 terms,
the terms needing no quantifier elimination in the original proof. The point is to
show that the sequents of Σ correspond to terminological subsumption formulae.
Let us now consider one such sequent .
The concept terms of
; c1(a1 ); c2(a2); ci (ai), etc. all have eigen-parameters of level 0 or 1 as arguments. has the form: :::; cj (aj ); ::: ) :::; ci (ai); ::: Every subsequent quantifier
elimination step applied to a term ci (ai ) in will create a new proof lexique B (ai).
One shows that the proof steps of the subterms in ai can be separated from the
proof steps of the subterms in aj , whenever these can be considered “distinct”,
i.e. whenever the inequality ai 6= aj can be assumed. The result will be that the
instantiation rules applied to subterms of ci (ai ) can be done wrt B (ai), i.e. the
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
45
forthcoming quantification steps for subterms in ci (ai) are independent of quantification steps of cj (aj ).(More precisely, one can show that any instantiation rule
for a top-level term of ci can be done wrt B (ai ) and so on by induction.)
The formal justification is to show that the inessential cuts occurring in the
propositional part of the initial proof tree for can be permuted with the quantification rules occurring in . The key point is to show that whenever a subterm
t1 in some c1(a) is instantiated by an eigen-parameter created by a subterm t2
of some c2 (b), then the proof succeeds for the corresponding role terms iff some
equality assumption between a and b holds. If this equality assumption a = b
is introduced after the instantiation of t1, then t1 cannot contribute to any proof
of the dual sequent obtained by the inessential cut which contains the inequality
a 6= b. For that sequent, other subterms of c1 than t1 must be unfolded. Therefore,
the inessential cut wrt a = b can be permuted with all the quantification steps for
c1 (a) and c2 (b).
Consequently, the initial proof tree for can be rewritten into an equivalent
proof tree where:
1. all the inessential cuts wrt the level-0 or level-1 eigen-parameters occurring
in are performed at first,
2. the remaining steps correspond to normal proof steps, without inessential
cuts wrt level-1 or level-0 eigen-parameters.
Then any subtree obtained at step (2) corresponds to a sequent of the following
form (written in rather free notation):
^i;j2I (ai = aj )); ci(ai); ck (ak) ) cj (aj ); cn (an); :::
The equalities in the list (^ii 1p (ai = bi ) induce an equivalence relation R between
the set of parameters (ai; i 2 I ). It is easy to prove that the above is provable
(
=
=
iff there is a subsequent containing only concept terms belonging to the same
equivalence class C of the relation R (because of the form of terminological
axioms which corresponds to subsumption formulae between concepts or roles):
^i;j2C (ai = aj )); ci (ai); ::: )
(
cj (aj ); :::
By substitution axioms, this sequent is equivalent to the following one for any
representative a of the equivalence class C :
^i;j2C (ai = aj )); ci(a); ::: )
cj (a); :::
This sequent corresponds exactly to the subsumption formula ci
(
u ... v cj t .... 2
The preceding results are unfortunately false, when more expressive languages are
considered. This is the case for languages with role operators acting as “role-chainshrinking” operators, like composition or >r. (Counterexamples to Proposition 5
are easily constructed from the preceding proof).
5 CONCEPT SUBSUMPTION WITH EQUALITY
46
On Homogeneous Lifting
We are now ready to prove the final property about the strategical aspects, namely
that homogeneous lifting, or, in this case, homogeneous quantifier elimination, is a
sound and complete strategy for terminological languages with Equality and without complex roles. Indeed, in absence of any “role-chain-shrinking” operator, no
proof possibility may arise between terms having different quantification degree,
except from pure propositional proofs.
Definition 3 A subsumption formula is called homogeneously quantified iff all
its level-0 concept terms have the same degree of quantification.
Definition 4 Given an arbitrary subsumption formula , is a homogeneous
subformula of iff is a subsumption formula constituted of concept terms
occurring in , whose levels in are the same.
Proposition 8 (Homogeneous Lifting) Let Γ be a terminological theory; let be a subsumption formula. is provable wrt Γ iff there exists a finite collection of
subsumption formulae (i)i2I such that, for every i 2 I :
1.
i is a homogeneous subformula of , constituted of concept terms either of
level 0 or of level 1;
2.
i is provable wrt Γ.
Corollary 9 The quantified fragment of TL= can be axiomatized by homogeneous
inference rules of degree 1.
The benefit of the Corollary is clearly a significant reduction in the axiomatization
effort. As already mentioned in the previous section, homogeneous lifting is
not valid for languages possessing complex role operators. For example, the
following valid inference rule could not be proved by a homogeneous lifting
strategy. The peculiarity comes from the role condition which is “free” while the
concept condition depends on the creative terms.
c2
v c; >r v
r
* 9r1 :(2 r2:c2 ) u 1 r:c v ?
Non-boolean role operators, like role composition, invalidate also Proposition 5.
One typical example is the following inference rule for role composition:
* (8r1 .r2:c) v 8r1 :8r2:c
In general one cannot even estimate a general bound for the depth of quantifier
elimination if the role operators can shrink arbitrarily the length of role chains
(like, for example, the transitive closure role).
5.1 Elimination of Quantifiers and Creation of the Proof Lexique
47
Proof: We now prove Proposition 8. The point is to show that in Corollary 7,
the subsumption formulae i ’s can be chosen to be homogeneous subformulae of
the initial subsumption. This means that the i ’s must contain exclusively either
level-0 terms or level-1 terms of the initial formula. In order to prove that, we will
show that no necessary proof condition can arise between level-0 concept terms
and level-1 concept terms.
Let us again consider a normal proof of the initial subsumption formula .
can be rewritten into a proof 0 satisfying Corollary 5, where all eliminations of
degree-1 quantifier are performed at first. Let us assume that the initial formula
contains at least one level-0 term t1, to which no quantifier elimination will ever
apply in , and one quantified term t2 which needs quantifier elimination in . Let
us assume that derives some proof condition between t1 and (subterms of) t2 . In
the following, we assume that t1 = c’(a) and that t2 provides at least two terms
written r(a; b) and c(b).
We will show that whenever some terminological sequent containing a concept
term c’ is provable without applying any quantification rule to c’ then it is provable
propositionally or independently of c’. This is proved by induction about the
complexity of sequent proofs, in terms of the total number of quantifier elimination
steps in the given proof. Only the arguments for the induction step are presented;
the induction basis is quite similar. Two cases arise according to whether the
alleged proof condition between t1 and t2 involves terms coming exclusively from
creation rules (Case I) or not (Case II).
I. Creative Terms Only (Induction Step): There is a proof condition where no
non-creative quantified term is involved. Basically we get to prove sequents like
n ; r1 (a; b1); c1(b1 ); :::; rn(a; bn); cn(bn) )
c’(a); n:::
where no non-creative term comes in, where one possible negation rule has been
applied to c’, and where n ) n denotes the sequent obtained from the initial
subsumption formula by eliminating the n creative terms necessary to obtain the
proof condition for t1. By Separation of role and concept conditions, 1 is either
provable by the role subsequent or the concept subsequent. We exclude the case
where r v ?r or ci v ? or c’ v ?, because in each case the proof works by the
inconsistency of one single term t1 or t2 (every creative term with inconsistent role
or concept is an inconsistent term).
Let us assume that 1 is proved by the concept subsequent. Let us also assume
that the set fci , i = 1 to ng is a minimal set such that the condition c1 u ... u cn v c’
holds. To get this condition, preliminary left equalities of the form bi = a must
be introduced. As non-creative terms may only contribute, the required equalities
come from inessential cuts. Then, one sequent in which no equality bi = a is
available must be proved. This sequent is no more provable by the previous
concept condition. Thus we have to prove 2
48
5 CONCEPT SUBSUMPTION WITH EQUALITY
2 n; b1 6= a; ::::; bn 6= a; r1(a; b1); :::; rn (a; bn) )
c’(a); n:::
No proof condition may now arise by the role subsequents (because the possible
role condition are forbidden conditions like ri v ?r). Thus 2 is provable by 3
3 n ; b1 6= a; ::::; bn 6= a; ) c’(a); n:::
In 3 we can also get rid of the inequalities bi 6= a. Indeed, they cannot be used
in substitution axioms. They possibly lead to identity axioms like bi 6= a; a = bi
in the two following cases. The first case, is when the equality a = bi comes
from a left atmost; then, there is a sequent without the equality, but with left role
or concept terms instead, which has the form 3 and must be proved without the
inequalities bi = a. The second case is when a left oneof introduces a left equality
a = c; oneof being a creative term, c is syntactically different from b; even if one
inessential cut is introduced to match b and c, there is a similar sequent where the
proof can no more proceed by the equality axiom.
Eventually, 3 is provable by a sequent corresponding to a sequent of smaller
complexity:
n ; )
c’(a); n:::
The rest follows by induction hypothesis.
II. Non-Creative Terms (Induction Step): At least one non-creative term t2 contributes to the proof. Let us assume that t2 = (
n:r:c)(~x)(a). We get to
prove sequents 1 , 2 , 3
atmost
1 n ; equ(~x); ::: )
2
n ; ::: )
3
n ; ::: )
c’(a); ::::; n
c’(a); r(a; x); :::; n
c’(a); c(x); ::::; n
where n ) n denotes the sequent obtained by eliminating the non-creative
terms necessary to get the proof condition for c’.
Let us first assume that the proof condition for c’is obtained from a sequent
like 1. If the proof is independent of equ(~x), the situation reduces to a formula
of smaller complexity. Whenever some equality from equ(~x) is necessary, it must
be of the form x = a which means that a 2 ~x; all the sequents of the form 2
must be proved with the role terms r(a; x) otherwise we get again to prove nonhomogeneous formulae of smaller complexity. For x = a, the proof of 2 reduces
to the proof of one role subsequent contained in n ) r(a; a); n. Such a sequent
cannot be proved by admissible role conditions unless a left equality b = a is
available.
If b = a comes from an inessential cut, then the dual sequent with the inequality
b 6= a gets unprovable with r(a; a). Instead we have to prove
n; :::; b 6= a )
c’(a); :::; n
5.2 Introduction of Complete Equality Assumptions
49
which corresponds to a sequent of smaller complexity. As shown in the precedent
case, one can get rid of the inequality b 6= a, so that the proof reduces to a sequent
of smaller complexity. Otherwise b = a comes from another non creative term
: : : r1:c1. The sequent like 3, obtained in the unfolding of this term by choosing
the concept term c1(a), cannot be proved by its role part; it must be proved by
its concept part, which again corresponds to a sequent of smaller complexity. In
all these cases, the initial sequent is provable by a sequent of smaller complexity.
Then, by induction hypothesis, it gets provable propositionally or independently
of c’.
Let us now assume that the proof condition for c’ comes from sequents of the
form 2 or 3 . If the proof condition for c’ comes from the sequent 2, then 2 is
provable without r(a; x) and the proof reduces to a sequent of smaller complexity.
The last case is when the proof condition for c’ comes exclusively from a
sequent of the form 3, where all the non-creative terms contribute by concept
terms only (no right role term, no left equality). The condition is of the form
c1 u ... v c’ t c t .... Let us consider a minimal set of concept terms leading to
such a condition. This supposes that the arguments x of the terms like c(x) are
made equal to a, through appropriate substitutions. To do this, as no equality from
the equ(~x) components is assumed available, at least one inessential cut must be
introduced. Then, the dual sequent, with the corresponding inequality, failed to
be proved unless the proof is independent of c’. Then it is easy to see that all the
sequents 1, 2 , and 3 are provable independently from c’.
2
5.2 Introduction of Complete Equality Assumptions
Now that the quantifiers steps have been performed, we end up with a proof lexique
which will remain constant in all the remaining steps occurring before final cuts
with proper axioms (assumptions).
The major change from the languages without Equality concerns the use of
inessential cuts in the proof trees. This makes the search space more complex
in the sense that the Subformula Property remains true up to the extension of the
notion of subformulae to equalities or inequalities between parameters of the proof
lexique. The problem is how to perform the inessential cuts, in other words what
equalities or inequalities are to be introduced in order to manage the proof.
First we will show how the presence of equality assumptions in the sequents
influence the proof strategy. Then, we will show that one has to introduce equality
and inequality assumptions which are complete, i.e. giving complete knowledge
about whether or not two arbitrary parameters are equal.
50
5 CONCEPT SUBSUMPTION WITH EQUALITY
Proof Strategy with Equality Assumptions
Let us start with a preliminary remark on inessential cuts. When applied in
backward chaining (i.e. from the conclusion to be proved to premisses as sufficient
conditions), an inessential cut is:
1 ; b = d ) 1; 2 ) b = d; 2
1 ; 2 ) 1; 2
Considering normal proofs21, in the propositional fragment only, there is no loss
of generality in applying inessential cuts as follows:
; b = d ) ; ; b 6= d ); )
Indeed, by weakenings, any proof of 1 ; b = d ) 1 is a proof of ; b = d ) where 1 and 1 (the same holds with b 6= d). Conversely, all the
normal proofs of ; b = d ) are necessarily proofs of one of its subsequents
1 ; b = d ) 1 (because assuming normal proofs, before application of “terminal”
cuts i.e. cuts with proper axioms, the only formulae in the proof which are not
subformulae of the initial sequent are equality or inequality formulae). In all the
following , we will always consider inessential cuts of this type.
Case Where a Single Term Creates Eigen-Parameters. We first show that
inessential cuts are useless for searching inference rules for formulae containing
one unique term creating eigen-parameters (one left atleast or one right atmost).
Introduction of new equality assumptions makes no change in the provability
conditions of the original sequent.
Let us assume that = ; n r:c ) is a sequent containing one
unique left atleast and no right atmost. Let us also assume that ) is not provable. By the 9-left rule, is equivalent to ) where =
i=n (r(a; b ) ^ c(b )); dif (b ; :::; b ) .
i
i
1
n
i=1
Any introduction of new equality or inequality formulae is made through an
inessential cut on parameters of the proof lexique, say (b1; b2 ). (Inessential cut upon
parameters not in the proof lexique is nonsense: it could only contribute to a proof
not involving the atleast term; which is contrary to the initial assumption.) Thus
we get: is provable if both ; b1 = b2 ) and ) b1 = b2; are provable. The
former sequent is verified because its subsequent dif (b1 ; :::; bn); b1 = b2 ) ; is
trivially provable (no condition about the other terms arises). The latter sequent is
equivalent to the initial sequent (by :-right and left contraction on the inequality
:(b1 = b2) in dif (b1; :::; bn)). Thus, inessential cuts are useless. They don’t give
any new proof condition.
P
21
I.e. proofs without non-inessential cuts. By the Normalization Property in Sequent Calculus
with Equality, one can restrict to normal proofs.
5.2 Introduction of Complete Equality Assumptions
51
Lemma 3 Subsumption formulae with one single term creating eigen-parameters
can be proved without inessential cuts.
Case Where At Least Two Terms Create Eigen-Parameters. Things are different when two atleast terms are simultaneously present as antecedents in the
subsumption formula (sequent). Indeed, inessential cuts will now introduce equality or inequality assumptions not redundant nor inconsistent with the information
contained in the subterms dif (:::) coming from the atleast terms.
Typically, let us assume that the antecedent of = ) contains two
terms p r1 :c1 and q r2:c2 . Two sets of parameters are introduced by the 9-left
rules: B = b1 ; :::; bp and D = d1 ; :::; dq coming with respective conditions dif (B )
and dif (D). Inessential cuts can contribute with new sequents, not equivalent to
the initial ones and with consistent antecedents, only if they bear on couples of
parameters (bi ; di ).
Then ) is provable if both ; bi = dj ) and ; bi 6= dj ) are
provable.
Then new proof conditions may arise because the assumption bi = dj in
the former sequent allows for possible substitution of parameters. For example,
if = c1 (b); c2(d); b = d ) c(b), then by applying a Cut (backwards) with
the substitutivity axiom c2 (b); b = d ) c2 (d) one obtains the sufficient proof
condition c1 u c2 v c (through the sequent c1 (b); c2 (b) ) c(b)).
This proof strategy fails for sequents containing no equality assumptions. For
example, the sequent c1 (b); c2(d); (b 6= d) ) c(b) is (propositionally) provable
only by separating the subsequents according to the different constant names, i.e.
either c1(b) ) c(b) or c2 (d) ) ; are provable. Here no appeal to substitution
axioms is possible for making concept terms telling of the same objects.
The preceding remarks justify the following lemma for propositional proofs
in linear sequents, i.e. sequents containing neither left disjunctions nor right conjunctions.
Definition 5 A sequent ) is linear iff there is no disjunctive formula in and no conjunctive formula in .
Lemma 4 (Constant Separation Principle) If a terminological sequent is linear
and does not contain any equality assumption in the antecedent (symmetrically,
any inequality assumption in the succedent) then the sequent is provable propositionally only if one of its subsequents constituted of terms telling of the same
parameter name is provable.
Proof: If the sequent is both disjunction- and conjunction-free, and no substitution axiom is applicable, then a propositional proof (in the NNF version) arises
only through cuts with terminological axioms which “tell” necessarily about the
52
5 CONCEPT SUBSUMPTION WITH EQUALITY
same individual. (NB: the proposition is obviously in non terminological context!)
2
This becomes false with left disjunctions and right conjunctions, because the _left and ^-right rule make the sequents split into two new ones, with different
antecedents or succedents. The ^-right rule makes the sequents split into two new
ones, with the same antecedents. For example, the sequent
C1 (a); C2(b) ) C3 (a) ^ C4 (b); C5(b)
is equivalent (by ^-right) to the sequents
C1(a); C2(b) ) C3(a); C5(b)
C1(a); C2(b) ) C4(b); C5(b)
If C2 (b) ) C5(b) is not valid, the first sequent must be proved by C1 (a) ) C3(a)
and the second one by C2 (b) ) C4(b); C5(b).
Complete Equality Assumptions
Definition 6 Let be a sequent. Let b and d be two distinct parameters in . We
say that (b; d) is decidable in iff either b = d or b 6= d occur in the antecedent
list or the succedent list of the sequent . Otherwise (b; d) is indecidable.
We say that a sequent is equality-complete (in short, complete) iff every
couple of distinct parameters (b,d) of is decidable in .
An equality complete extension of ) is an equality complete sequent
containing and as antecedent and succedent, respectively, and differing from
) by equalities or inequalities only.
By the negation rules, one can always assume that equalities or inequalities appear
in the antecedent list only. We now show that the search space for proving a given
sequent can be restricted to the space of all the equality complete sequents containing it, instead of all the sequents resulting from it by arbitrary equality/inequality
assumptions.
Proposition 10 A sequent is propositionally provable iff a subset of equality
complete sequents extending are provable. Thus the search space for provability
conditions of can be restricted to all its equality complete extensions.
Proof: The proof simply proceeds by showing that proving a sequent ) ,
in which b = d is indecidable, can be done by proving both ; b = d ) and
; b 6= d ) . In other words, there is no loss of generality in performing the
inessential cut wrt b = d.
5.3 Axiomatization of Numerical Operators
53
In the following we consider only propositional proof trees and backward proof
strategies. Thus cut rules are meant backwards, from conclusion to premisses.
Proof trees for ) can be classified according to whether or not they use an
inessential cut upon b = d. Precisely, ) is either provable without using an
inessential cut upon b = d or the two sequents ; b = d ) and ; b 6= d ) are both provable. We show that the proofs without inessential cuts will be also
encountered by assuming the inessential cut.
If the proof for ) does not use inessential cuts upon b = d, then it
proceeds necessarily without substitution wrt b = d (c(b); b = d ) c(d)) and
without the identity axiom (or equivalent version) b = d; b 6= d ) ; (because
neither b = d nor b 6= d may ever appear in the antecedent).
The same proof conditions necessarily also arise by considering both sequents
; b 6= d ) and ; b = d ) . Indeed, with ; b 6= d ) , no cut with
a substitutivity axiom on b = d is possible, because this would require that the
top (goal to be proved) sequent already contained the equation b = d, contrarily
to the indecidability assumption for (b; d). If ; b 6= d ) enables a proof
possibility by an identity axiom b = d; b 6= d ) ;, this means that a left term
b = d (up to negation rules) emerges during the proof after some connector
elimination (typically, from left atmost). The same term would also emerge
during the proof (assuming that formula is necessary for the proof) of the dual
sequent ; b = d ) , but then no proof succeeds with the identity axiom.
In other words, proof conditions for ) without inessential cut on b = d
will also be found by proving ; b 6= d ) and ; b 6= d ) . This shows that
) is provable iff ; b 6= d ) and ; b 6= d ) are both provable. In each
case, provability conditions are found by examining extension of ) in which
b = d is decidable. The proposition is obtained by generalizing the reasoning to
more parameters.
2
5.3 Axiomatization of Numerical Operators
After these preparing sections we will now derive the inference rules for the numerical operators atleast and atmost. We obtain the following rules for inconsistency
and redundancy:
cv
cv
? * > v n r:c
? * n r:c v ?
(29)
(30)
Some of these rules can be taken over from the corresponding rules for some and
its negation. In fact we have the following equivalencies between atleast and
5 CONCEPT SUBSUMPTION WITH EQUALITY
54
atmost, on the one hand, and some and all on the other:
*
*
*
*
1 r:c =: 9r:c
1 r:c =: :8r:: c
0 r:c =: 8r:: c
0 r:c =: :9r:c
(31)
(32)
(33)
(34)
These rules already indicate the negation duality between atleast and atmost:
m = n 1 * :n r:c =: m r:c
m = n + 1 * : n r:c =: m r:c
(35)
(36)
Due to this negation duality we have various formats for most inference rules
involving atleast and atmost. Consider, for example, the monotonicity rule for
atleast:
q p; c1 v c2 ; r1 v
r2
* p r1 :c1 v q r2 :c2
(37)
We can move the negated right-hand atleast to the left (38), the negated left-hand
atleast to the right (39), or both (40).
q < p; c1 v c2; r1 v
q < p; c1 v c2; r1 v
p q; c2 v c1; r2 v
r2
r2
r1
* q r2 :c2 u p r1 :c1 v ?
* > v q r2 :c2 t p r1:c1
* p r1:c1 v q r2 :c2
(38)
(39)
(40)
In the following we will only specify rules in the inconsistency format, i.e. with ?
on the right-hand side. To find all inconsistency schemes, we have to systematically
investigate all conjunctive combinations of atleast, atmost, and all. Given the
equivalence between atmost and all (33) we obtain the following inconsistency
between atleast and all:
c2 u c1
v ?; r1 v
r2
* n r1:c1 u 8r2 :c2 v ?
(41)
In addition to these rules we obtain inference rules from the interaction of atleast,
atmost, and all in conjunctive terms. The detailed derivations for these rules are
given in the appendix (Section A). Here we just summarize the resulting rules.
Below are the rules for Modus Ponens in inconsistency form (42), as well as
Modus Ponens for atleast (43) and for atmost(44):
q < p; r
q p; r
q p; r
v
1 v
1 v
1
;
r ;r
r ;r
v
1 v
1 v
r2 r1
2
2
;
r ;c
r ;c
r3 c1 u c3
3
3
v
1 u c3 v
1 u c3 v
c2
c2
c2
* q r :c
* p r :c
* q r :c
2
1
2
u p r1:c1 u 8r3:c3 v ? (42)
(43)
1 u 8r3 :c3 v q r2 :c2
(44)
2 u 8r3 :c3 v p r1 :c1
2
5.4 Axiomatization of Extensional Operators
55
The following rule is the terminological part of the object-level abstraction over
closed role-filler sets:
r1
v r2; , v
r1r3 ; c1
v
* n r1:c1 u n r2 :> v 8r3 :c2
c2
(45)
Furthermore, we have the general additivity rule wrt a single atleast term
Pi k (pi) < n; ri v r; c v c t c t : : : t ck
i
=
=1
1
*
2
(46)
n r:c u p1 r1:c1 u p2 r2:c2 u : : : u pk rk :ck v ?
and the additivity for atmost wrt disjunction:
r v r1 ; r v r2
* p r1 :c1 u q r2 :c2 v p + q r:c1 t c2
(47)
Then there is additive Modus Ponens:
p + q < n; r v r1 ; r v r2; r3 v r1 ; r4 v r2 ; c3 u d1 v c1 ; c4 u d2 v
*
n r:d1 t d2 u q r1:c1 u p r2:c2 u 8r3:c3 u 8r4:c4 v ?
c2
(48)
There are a number of additional rules derivable, which are all reducible, however,
to the Union/Intersection rule:
r1 u r2
v r3 u r4; r1 t r2 v r3 t r4; c1 u c2 v
c3 u c4 ; c1 t c2
k+n<p+q
*
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
v
c3 t c4 ;
(49)
Note that this rule can in fact be generalized to an arbitrary number of numerical
restrictions (see Section A).
5.4 Axiomatization of Extensional Operators
We repeat the definition of the extensional operators oneof and fills.
fo1; :::; on g_
r:fo1 ; :::; on g
def
=
def
=
x(x = o1 _ : : : x = on)
x(r(x)(o1 ) ^ ::: ^ r(x)(on))
Furthermore most systems offer the notational convention
r:o
def
=
r:fog
56
5 CONCEPT SUBSUMPTION WITH EQUALITY
We can immediately note the following equivalence:
n = jS j *
r:S
: n r:S _
=
(50)
which allows us to restrict our axiomatization effort to oneof.
The following rules can be derived for oneof terms not occurring in quantified
terms:
S T * S_ v T _
* S _ u T _ =: (S \ T )_
(51)
(52)
oneof differs notably from the numerical restrictions because it is not quantified. By the property of homogeneous lifting, one can at first state that there is no
proof possibility arising between a non-quantified oneof term and a role restriction
term, except from pure propositional proofs.
The question is now whether there exist new inference rules for quantified oneof
terms, i.e. terms where oneof is encapsulated in some role restriction operator, and
rules others than the rules inherited from the fragment with numerical restrictions.
When encapsulated in numerical restrictions qn r : S _ , oneofintroduces equalities between the elements of S and the role fillers of r. This is always a creative
introduction (elements of S are constant parameters—they contribute to the creation of the proof lexique). Thus S _ modifies the proof lexique and the notion
of equality complete sequents. In particular the equality assumptions, not only
between the parameters created by quantifier eliminations in the atleast terms,
but also between the constant parameters of the oneof terms, must be taken into
account. Below we examine the proof conditions for quantified oneof terms.
Proof Conditions For a Single Encapsulated Oneof Term
Let us assume that the oneof term occurs in an atmost term (e.g. p r:S _) and
is essential to the proof. Regarding the concept terms, the contributions of the
p r:S _ term is by means of right equalities: x = oi, for oi 2 S . There is at
least one sequent which is not provable without the right x = oi . By considering
the particular sequents where no equality between oi and the other parameters
of the lexique is assumed, then the proof can only be between x = oi and one
identical equality coming from one left term directly (i.e. not via substitutions).
This means that there must be one other oneof term in the formula, providing with
left equalities, i.e. one oneof encapsulated within an atleast or an all term.
The reasoning is similar for other quantified terms. In general, we can show
that no formula containing only one single quantified oneof term is provable,
unless this formula is inconsistent or the formula is provable independent of the
oneof term.
5.4 Axiomatization of Extensional Operators
57
It is easy to see that an inconsistency can only arise from the demands of an
atleast term for n distinct fillers and the maximal cardinality of a oneof term:
jS j < n * n r:S _ v ?
(53)
Correspondingly we have the following rule for redundancy of atmost restrictions:
jS j n * > v n r:S _
(54)
A oneof term occurring in a value restriction implies a maximum restriction:
n jS j; r2 v
r1
* 8r1 :S _ v n r2 :>
(55)
Oneof Terms Occurring in one Atleast and one Atmost Term
To prove inconsistency of p r:S _ u q r:T _, where S = fs1; : : : ; sm g and
T = ft1; : : : ; tng we get to prove sequents like (_-left rules being already applied):
^ii n1 bi = ti; equ(~x) ) ;
^ii 1n bi = ti ) xj = s1; :::; xj = smforj = 1:::p + 1:
As shown above, q r:T _ is only consistent if q n. Immediately, one sees that
there must be at least n distinct elements of T, otherwise n r:T _ is inconsistent.
Assuming the contrary, i.e. n jS j m , then the proof proceeds by case analysis
+
=
+
=
on ~x and the possible equality completions. By assuming no equality assumption
between B [ T and S , but B = T , one proves that ~x must be necessarily included in
B [ T = B , that T must be included in S . Eventually, the inference reduces to the
composition of the monotonicity rule between oneof terms and the inconsistency
schema between atleast and atmost. Hence, no new inference rule arises!
The property generalizes easily with one oneof term encapsulated in an atmost
and several non-oneof terms.
Arbitrary Proofs with Quantified Oneof
Let us consider formulae with mixed terms, some quantified oneof terms together
with some quantified non-oneof terms. Again we will prove some “homogeneity
property”, namely that the proof depends either only on the quantified oneof terms
or only on the non-oneof terms. Intuitively, this is because “equalities prove
equalities” and “concept terms prove concept terms”.
In the following we assume that the contribution of the oneof term duely
involves Equality reasoning, i.e. that the formula could not be proved in the
fragment without Equality. This means that somehow the equalities coming from
the oneof terms are effectively used for substitutions, equality axioms or equality
assumptions in the introduction of inessential cuts.
5 CONCEPT SUBSUMPTION WITH EQUALITY
58
Let B be the set of parameters created by the non-oneof terms, let D be the set
of eigen-parameters created by the oneof terms. Let us consider the collection of
equality complete sequents in which no equality assumption is made between B
and D (hence intuitively B and D are kept disjoint).
Atmost Terms. Let us first consider the coexistence of one p r1 :c and one
q r2:S _ in an arbitrary formula. Let us assume that the formula is not provable
without p r1:c. Then there is one (equality complete) sequent in which unfolding the term p r1 :c and q r2 :S _ results either in a proof by p r1 :c alone—i.e.
one sequent of type 2 shown on page 99 when equ(~x) is unprovable—or in a
proof by one sequent of type 4 shown on page 99 where both equ(~x) and equ(~y)
are provable.
~y takes one element into B , and because no equality between B and
D exists, then the sequent ) c(y) must be proved by the subformula
B constituted of the terms telling only on the parameters from B , that
1. If
is equalities coming from the oneof terms. But c(y) is an equality term,
therefore such sequents are unprovable. This means that ~y is necessarily
included into D and the proof is independent of the oneof terms.
2. Then there is at least one sequent ) _s2S (x = s); c(y ) to be proved
and ) (_s2S (x = s)) is unprovable (otherwise the proof is independent
from p r1 :c). As the left equalities cannot contribute by substitutions, the
former sequent is equivalent to ) c(y ). Assuming that is disjunction
free, then by Constant Separation, it comes that ) c(y ) must be proved
for all y 6= x. For y = x, again because the right x = s cannot be used for
substitutions, then also ) c(x). Moreover, by the same argments as in
the previous case one shows that ~y must be necessarily included in D. Then
the formula is provable without q r2:S _ .
The same reasoning holds also wrt several atmost terms, by generalizing to (~y),
one non-empty right expression obtained after the quantifier elimination of several
p r1:c (i.e. some part not exclusively made of left equ(~z)).
All and Atleast Terms. Now let us assume that contains only quantified oneof
contributing by left equalities, i.e. 8r2 :T _ or q r2 :T _. In one case, the equality
means x 2 T , in the other B T . Again we assume that no equality assumption
is made between the parameters created by the oneof terms (B ) and the others (D).
Then we get to prove sequents of the form: ; 0 ) (~y) where contains the
equalities from the T _ term and 0 only tells on parameters from D. Assuming
again that is disjunction free, then by Constant Separation the proof proceeds
by either one subsequent telling on the parameters of B [ T or one subsequent
59
telling on the parameters of D: ) c(~yB;T ) or 0 ) c(~yD ). But the first sequent
is unprovable because contains no left concept term and was just chosen to
contained ones. Thus the proof is necessarily independent from the oneof term.
6 Role Subsumption
In this section we will derive inference rules for role subsumption. We consider
the following role-forming operators:
>r
?r
r1 u r2
r1 t r2
:r
c jr
rjc
r
r1 :r2
def
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
def
=
xyTrue
xyFalse
xy(r1 (x)(y) ^ r2 (x)(y))
xy(r1 (x)(y) _ r2 (x)(y))
xy(:r(x)(y))
xy(r(x)(y) ^ c(x))
xy(r(x; y) ^ c(y))
xy(r(y)(x))
xy(9zr1(x)(z) ^ r2 (z)(y))
Note that we do not consider the transitive closure of roles for the obvious reason that it is not translatable into FOL. Note further that the normalize-compare
algorithm for role subsumption presented in [Quantz 90] is not complete as conjectured in that paper. Besides incompleteness due to the trans operator there are
additional incompletenesses wrt the operators of the fragment considered here.
We will first analyze the propositional fragment of roles, then we will consider
the domain (c jr) and the range operator (rjc ), and finally turn our attention towards
role composition (r1 .r2 ) and inversion (r ). We will proceed systematically by
analyzing the different possibilities for having a subsumption r1 v r2 —we look at
every combination between the outmost role-forming operators in r1 and r2 (treating
the constants as 0-ary role-forming operators). We start with a translation of the
DL subsumption formula under consideration to which !-right, 8-right, ^-left,
and _-right have already been applied. For illustration consider the subsumption
scheme r1 u r2 v r3 t r4 . We would start with the sequent
R1 (a)(b); R2(a)(b) ) R3 (a)(b); R4(a)(b)
We thus skip the following derivation:
; ) 8x8yR1(x)(y) ^ R2(x)(y) ! R3(x)(y) _ R4(x)(y)
6 ROLE SUBSUMPTION
60
; )
R1 (a)(b) ^ R2 (a)(b) )
R1 (a)(b); R2(a)(b) )
R1 (a)(b); R2(a)(b) )
R1 (a)(b) ^ R2(a)(b) ! R3 (a)(b) _ R4 (a)(b)
R3 (a)(b) _ R4(a)(b)
R3 (a)(b) _ R4(a)(b)
R3 (a)(b); R4(a)(b)
Note that the Ri used in the sequents are translations of possibly complex DL roles,
i.e. they may contain terms themselves (therefore our writing R(a)(b) instead
of R(a; b)). Our proofs will look for the substitutions of the Ri which make the
sequent valid.
Note finally that we are interested in a somewhat minimal list of inference rules.
We are thus not interested in deriving inference rules which are already covered
by more general inference rules. We will call such inference rules redundant and
the corresponding proofs trivial. For example an inference rule like
*
r1
u r2 v
r1
does not say anything interesting about the interaction between an outmost inv
operator on the right-hand side, and an outmost and operator on the left-hand side,
since it is just a special instance of the inference rule
*
r1
u r2 v
r1
We will, however, list inference rules which are redundant in the sense that a set
of other inference rules taken together will also produce the desired inference.
Though these rules are redundant they are important from the perspective of an
efficient implementation of DL systems. We will usually indicate the rules which
are redundant in this sense.
Figure 9, at the end of this section on page 78, contains all inference rules
derived for role subsumption.
6.1 The Propositional Fragment
For the propositional fragment, i.e. the top role (>r), the bottom role (?r), role
conjunction (r1 u r2 ), role disjunction (r1 t r2 ), and role negation (: r), we can
simply take over the results from concept subsumption. For convenience we here
list again all the rules without giving their derivations, however:22
*
*
22
:
r= r
r v >r
(56)
(57)
Again, we do not give the rules for commutativity and associativity of conjunction and
disjunction.
6.2 Domain and Range Restrictions
r1
r1
v
v r2; r1 v
r1 v
r3
r2
v
v
r2
r4
r1
v r3; r2 v
r4
r2
r1
v r1; r3 v
v r3; r2 v
r1
r4
r1
r2; r2 u r3
61
*
)
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
r1 v r2 u r3
r1 v r2 t r3
:
r= ::r
>r =: r t : r
>r =: : ?r
?r v r
r1 u r3 v r2
r1 u r3 v r4
:
r u : r = ?r
r1 u r2 v r3 u r4
r1 u r2 v r1 t r3
: r1 u : r2 =: : (r1 t r2)
r2 t r3 v r1
r1 t r2 v r3 t r4
: r1 t : r2 =: : (r1 u r2)
: >r =: ?r
(58)
(59)
(60)
(61)
(62)
(63)
(64)
(65)
(66)
(67)
(68)
(69)
(70)
(71)
(72)
(73)
6.2 Domain and Range Restrictions
The domain operator (cjr) restricts the first element of a role, the range operator
(rjc) restricts the second. These role-forming operators thus allow the use of
concepts in the definitions of roles. We will now consider the interaction of
these operators with the operators from the propositional fragment. Since domain
and range behave very similar (their translations are almost identical), we will
consider their interaction in parallel. Usually, the derivations for domain carry
over to range.
r1 v c jr2
This is translated into SC as
R1(a)(b) ) R2 (a)(b) ^ C (a)
We thus have to prove the sequents
R1 (a)(b) ) R2 (a)(b)
R1 (a)(b) ) C (a)
If we assume that there is no further information available about R1 , the second
sequent is only provable if C = xTrue(x). We thus obtain the inference rule
> v c; r1 v
r2
*
r1
v c jr2
(74)
6 ROLE SUBSUMPTION
62
r1 v r2 jc
In analogy to the derivation for domain we obtain
> v c; r1 v
r2
*
r1
v r2jc
(75)
c jr1 v r2
The translation of this subsumption scheme is:
R1 (a)(b); C (a) ) R2 (a)(b)
If we do not take into account further information about R2 this is provable if
R1 (a)(b) ) R2 (a)(b)
Note that the corresponding inference rule
r1
v
r2
* cjr1 v
r2
(76)
is redundant, since it follows from (87), (95), and (65).
r1 jc v r2
Again in analogy to the domain derivation we obtain
r1
v
r2
*
r 1 jc v r 2
(77)
which is redundant, since it follows from (88), (98), and (65).
>r v c jr
Translates into
True ) C (a) ^ R(a)(b)
which is provable only if C = xTrue(x) and R = xyTrue(x)(y ).
We thus
obtain the inference rule
> v c; >r v
r
* >r v cjr
(78)
which follows also from (74) and (56), and is thus redundant.
>r v rjc
In analogy to the domain case we obtain
> v c; >r v
r
* >r v rjc
which follows from (75) and (56), and is thus redundant.
(79)
6.2 Domain and Range Restrictions
63
c jr v >r
Is covered by (57).
rjc v >r
Is covered by (57).
?r v cjr
Is covered by (63).
?r v rjc
Is covered by (63).
c jr v ?r
Is translated into
R(a)(b); C (a) ) False
which is provable if R = xyFalse or C
rv
cv
=
xFalse. We thus obtain
?r * cjr v ?r
? * cjr v ?r
(80)
(81)
But we also have to check whether the information in R and C taken together can
be incoherent. Clearly, this can only be the case if C contains information about
there being no b such that R(a)(b). Thus if C = x:9zR(x)(z ) we obtain:
R(a)(b); :9zR(a)(z) ) False
This yields the inference rule
cv
0 r1:>; r2 v
r1
* cjr2 v ?r
(82)
rjc v ?r
In analogy to the domain case we obtain the inference rules
rv
cv
?r * rjc v ?
?r * rjc v ?
(83)
(84)
6 ROLE SUBSUMPTION
64
Furthermore we have to check the sequent
R(a)(b); C (b) ) False
for interactions between R and C . But here we have to find a substitute for
which negates the existence of an a such that R(a)(b). We thus assume C
x:9zR(z)(x) and obtain:
C
=
R(a)(b); :9zR(z)(a) ) False
yielding the inference rule
cv
0 r1:>; r2 v
r1
r2 jc
*
v ?r
(85)
Note that we can also chose to substitute C by x:9zR1 (x)(z ), i.e. without role
inversion, and instead assume R = xyR1 (y )(x). We then get the inference rule
cv
0 r1 :>; r2 v
r1
r2 jc
*
v ?r
(86)
r1 u r2 v c jr3
This subsumption translates into the two sequents
R1 (a)(b); R2(a)(b) ) R3 (a)(b)
R1 (a)(b); R2(a)(b) ) C (a)
To prove these sequent we have to assume that either R1 or R2 contain a domain
restriction. Assume R1 = xyR4 (x)(y ) ^ C1 (x). Then the strongest way to
prove R3 is to use both R4 and R2 , thus letting R3 = xyR5 (x)(y ) ^ R6 (x)(y ):
R4 (a)(b) ) R5 (a)(b)
R2 (a)(b) ) R6 (a)(b)
C1(a) ) C (a)
We thus obtain the inference rule
r1
v r2; r3 v r4; c1 v
c2
*
c1 jr1 u r3
v c2 j(r2 u r4)
(87)
r1 u r2 v r3 jc
The proof is analogous to the domain case and yields:
r1
v r2; r3 v r4; c1 v
c2
*
r1 jc1
u r3 v
(r2 u r4 )jc2
(88)
6.2 Domain and Range Restrictions
65
c jr1 v r2 u r3
This subsumption translates into
C (a); R1(a)(b) ) R2 (a)(b) ^ R3(a)(b)
We can thus use C (a) to prove a domain term in R2, and R1 to prove the remaining
role term in R2 and R3 . In other words, we can prove the sequent
C (a); R4(a)(b); R5(a)(b) ) C1 (a) ^ R6 (a)(b) ^ R3 (a)(b)
from the following sequents
C (a) ) C1 (a)
R4 (a)(b) ) R6 (a)(b)
R5 (a)(b) ) R3 (a)(b)
We thereby obtain the inference rule
r1
v r2; r3 v r4; c1 v
c2
*
c1 j(r1 u r3 ) v c2 jr2 u r4
(89)
r1 jc v r2 u r3
In analogy to the domain case we obtain the inference rule rule
r1
v r2; r3 v r4; c1 v
c2
*
(r1 u r3 )jc1
v r2jc2 u r4
(90)
r1 t r2 v c jr3
We obtain the translation
R1 (a)(b) _ R2 (a)(b) ) R3 (a)(b) ^ C (a)
Splitting this sequent we obtain four sequents to prove
R1 (a)(b)
R1 (a)(b)
R2 (a)(b)
R2 (a)(b)
)
)
)
)
R3 (a)(b)
C (a)
R3 (a)(b)
C (a)
Thus both R1 and R2 must contain domain information and must also support R3 .
The resulting inference rule is thus just a special case of (70) and hence redundant.
r1 t r2 v r3jc
No inference rule in analogy to the domain case.
6 ROLE SUBSUMPTION
66
c jr1 v r2 t r3
This is translated into
R1 (a)(b); C (a) ) R2 (a)(b); R3(a)(b)
We can prove this non-trivially if we subsitute R1 = xyR4 (x)(y ) _ R5 (x)(y )
and R2 = xyR6 (x)(y ) ^ C1 (x). This yields the sequent
R4 (a)(b) _ R5(a)(b); C (a) ) R6 (a)(b) ^ C1(a); R3(a)(b)
Applying the _-left rule we obtain
R4 (a)(b); C (a) ) R6(a)(b) ^ C1 (a); R3(a)(b)
R5 (a)(b); C (a) ) R6(a)(b) ^ C1 (a); R3(a)(b)
Applying the ^-right rule twice we finally obtain
R4 (a)(b); C (a)
R4 (a)(b); C (a)
R5 (a)(b); C (a)
R5 (a)(b); C (a)
)
)
)
)
R6 (a)(b); R3(a)(b)
C1(a); R3 (a)(b)
R6 (a)(b); R3(a)(b)
C1(a); R3 (a)(b)
These seqents are provable from the sequents
R4 (a)(b) ) R6 (a)(b)
C (a) ) C1(a)
R5 (a)(b) ) R6 (a)(b)
We can thus state the following inference rule
r1
v r2; r3 v r4; c1 v
c2
*
c1 j(r1 t r3) v c2 jr2 t r4
(91)
r1 jc v r2 t r3
In analogy to the domain case we obtain the inference rule
r1
v r2; r3 v r4; c1 v
c2
*
(r1 t r3 )jc1
v r2jc2 t r4
(92)
: r1 v cjr2
Translates into
:R1(a)(b) ) R2(a)(b) ^ C (a)
R1 would thus have to contain C as a negated range and the negated R2 . But since
the external : will change the internal ^ into a _, we cannot prove this sequent
nontrivially.
6.2 Domain and Range Restrictions
67
: r1 v r2jc
In analogy to the domain case there is no inference rule.
c jr1 v : r2
Translates into
R1(a)(b); C (a) ) :R2 (a)(b)
To prove this nontrivially R2 had to contain the negative domain C and the negative
R1 . Thus letting R2 = xy:R3 (x)(y) ^ :C1 (x) we obtain the sequent
R1 (a)(b); C (a) ) :(:R3 (a)(b) ^ :C1 (a))
whose negation normal form is
R1 (a)(b); C (a) ) R3 (a)(b); C1(a))
which is provable from
R1 (a)(b) ) R3 (a)(b)
C (a) ) C1 (a))
We thus obtain the inference rule
r1
v r2; c1 v
c2
*
c1 jr1
v : (: c2 j: r2)
(93)
r1 jc v : r2
In analogy to the domain case we obtain the rule
r1
v r2; c1 v
c2
*
r1 jc1
v : (: r2j: c2 )
c1 jr1 v c2 jr2
This translates into
R1 (a)(b); C1(a) ) R2 (a)(b) ^ C2(a)
We thus obtain by the separation lemma
R1 (a)(b) ) R2 (a)(b)
C1 (a) ) C2 (a)
(94)
6 ROLE SUBSUMPTION
68
and hence
r1
v r2; c1 v
c2
*
c1 jr1
v c2 jr2
But if R1 contains itself domain information, i.e. if R1
we can prove the sequent for C2 = xC4 (x) ^ C5(x):
=
(95)
xyR3 (x)(y) ^ C3 (x),
R3(a)(b); C3(a); C1(a) ) R2 (a)(b) ^ C4(a) ^ C5 (a)
from the sequents
R3 (a)(b) ) R2 (a)(b)
C3(a) ) C4(a)
C1(a) ) C5(a)
Note that the above sequent can be translated into DL formulae in two ways, and
that we thus obtain two inference rules
v r2; c1 v c2; c3 v
r1 v r2; c1 v c2 ; c3 v
r1
r1 jc1
c4
c4
*
*
c3 j(c1 jr1) v (c2 u c4 ) jr2
(c1 u c3) jr1 v c4 j(c2 jr2)
(96)
(97)
v r2jc2
In analogy to the domain case we obtain the inference rules
r1 v r2 ; c1
r2; c1 v c2 ; c3
v
r1 v
v
r1 v r2; c1 v c2 ; c3 v
c2
c4
c4
*
*
*
r1 jc1 v r2 jc2
(r1 jc1 )jc3 v r2 j(c2 u c4)
r1j(c1 u c3 )
v (r2jc2 )jc4
(98)
(99)
(100)
6.3 Role Composition and Inversion
We now turn our attention towards role composition (r1 .r2 ) and role inversion (r ).
As will become obvious these role-forming operators interact especially with the
domain and range operators. We will first derive the rules for inv and then for
comp.
r1 v r2
Is translated into
R1 (a)(b) ) R2 (b)(a)
6.3 Role Composition and Inversion
69
which is provable if either R1 or R2 contains an inverse role, yielding two possible
provability conditions:
R3 (b)(a) ) R2 (b)(a)
R1 (a)(b) ) R4 (a)(b)
We thereby obtain the inference rules
r1
r1
r1
v
v
r2
r2
*
*
r1
r1
v
v
r2
r2
(101)
(102)
v r2
Is translated into
R1 (b)(a) ) R2 (a)(b)
We thus have the same choice as in the previous case and obtain similar inference
rules:
r1 v r2
r1 v r2
*
*
r1
r1
v r2
v r2
(103)
(104)
>r v r
Translates into
True ) R(b)(a)
which is provable if R = xyTrue. We thus obtain the inference rule
>r v r * >r v r
r
(105)
v >r
Is covered by (57).
?r v r
Is covered by (63).
r
v ?r
Translates into
R(b)(a) ) False
which is provable if R = xyFalse. We thus obtain the inference rule
r v ?r * r v ?r
(106)
6 ROLE SUBSUMPTION
70
r1 u r2 v r3
Is translated into
R1 (a)(b); R2(a)(b) ) R3 (b)(a)
To prove this we have to substitute: R3 = xyR6 (x)(y ) ^ R7 (x)(y ), R1
xyR4 (y)(x), R2 = xyR5 (y)(x):
R4 (b)(a); R5(b)(a) ) R6 (b)(a) ^ R7 (b)(a)
=
which is provable from the sequents
R4 (b)(a) ) R6 (b)(a)
R5 (b)(a) ) R7 (b)(a)
We thus obtain the inferenc rule
r1
r1
v r2; r3 v
r4
*
r1
u r3 v
(r2
u r4 )
(107)
v r2 u r3
Translates into
R1 (b)(a) ) R2 (a)(b) ^ R3 (a)(b)
In analogy to the previous case we obtain the inference rule
r1
v r2; r3 v
r4
*
(r1 u r3 )
v
r2
u r4
(108)
r1 t r2 v r3
Translates into
R1 (a)(b) _ R2(a)(b) ) R3 (b)(a)
We can then substitute R1 = xyR4 (y )(x), R2 = xyR5 (y )(x), and finally
R3 = xyR6 (x)(y) _ R7 (x)(y) thus having to prove
R4 (b)(a) _ R5(b)(a) ) R6(b)(a); R7(b)(a)
which is, according to _-left, reducible to
R4 (b)(a) ) R6 (b)(a); R7(b)(a)
R5 (b)(a) ) R6 (b)(a); R7(b)(a)
We thus obtain the prove conditions
R4 (b)(a) ) R6 (b)(a)
R5 (b)(a) ) R7 (b)(a)
and thus the inferenc rule
r1
v r2; r3 v
r4
*
r1
t r3 v
(r2
t r4 )
(109)
6.3 Role Composition and Inversion
r3
71
v r1 t r2
Translates into
R1 (b)(a) ) R2 (a)(b); R3(a)(b)
In analogy to the previous case we obtain the inference rule
r1
v r2; r3 v
r4
*
(r1 t r3 )
v
r2
t r4
(110)
: r1 v r2
Gives
which is provable if R1
:R1(a)(b) ) R2(b)(a)
=
xyR3 (y)(x) and R2 = xy:R4 (x)(y):
:R3(b)(a) ) :R4(b)(a)
We thus obtain the inference rule
r1
r1
v
r2
* : r1 v (: r2 )
(111)
v : r2
Translates into
R1(b)(a) ) :R2 (a)(b)
In analogy to the previous case we obtain the inference rule
r1
v
r2
* (: r1 ) v : r2
(112)
c jr1 v r2
Translates into
R1 (a)(b); C (a) ) R2 (b)(a)
The C (a) can be used to prove information stemming from a range restriction in
R2 . Note that the arguments in R1 and R2 are inverted, and that we thus have to
substitute R2 = xyR3 (y )(x) ^ C1(y ) yielding the sequent
R1 (a)(b); C (a) ) R3 (a)(b) ^ C1 (a)
which is provable from the sequents
R1 (a)(b) ) R3 (a)(b)
C (a) ) C1 (a)
This gives the inference rule
r1
v r2; c1 v
c2
*
c1 jr1
v
(r2 jc2 )
(113)
6 ROLE SUBSUMPTION
72
r1
v cjr2
Is translated into
R1 (b)(a) ) R2 (a)(b); C (a)
In analogy to the previous case we obtain the inference rule
r1
v r2; c1 v
c2
*
v c2 jr2
(r1 jc1 )
(114)
r1 jc v r2
In analogy to the domain case we obtain the inference rule
r1
r1
v r2; c1 v
c2
*
r1 jc1
v
(c2 jr2 )
(115)
v r2jc
In analogy to the domain case we obtain the inference rule
r1
r1
v r2; c1 v
c2
*
(c1 jr1 )
v r2jc2
(116)
v r2
Translates into
R1 (b)(a) ) R2 (b)(a)
which can be reconverted into
) 8x8yR1(x)(y) ! R2(x)(y)
We thus obtain the inference rule
r1
v
r2
*
r1
v
r2
r1 v r2 .r3
Translates into
R1 (a)(b) ) 9zR2(a)(z) ^ R3 (z)(b)
for which no non-trivial proof exists.
(117)
6.3 Role Composition and Inversion
73
r1 .r2 v r3
Translates into
9zR1(a)(z) ^ R2(z)(b) ) R3(a)(b)
Again there is no non-trivial proof for this sequent.
>r v r1.r2
Translates into
True ) 9zR1(a)(z) ^ R2 (z)(b)
which is provable if we substitute R1
=
xyTrue and R2 = xyTrue, giving
True ) 9zTrue
We thus obtain the inference rule
>r v r1; >r v
r2
* >r v
r1 .r2
(118)
r1 .r2 v >r
Is covered by (57).
?r v r1.r2
Is covered by (63).
r1 .r2 v ?r
Translates into
9zR1(a)(z) ^ R2(z)(b) ) False
which is provable form one of the following sequents:
R1(a)(c) ) False
R2 (c)(b) ) False
We thus obtain the inference rules:
r1
r2
v ?r *
v ?r *
r1 .r2
r1 .r2
v ?r
v ?r
(119)
(120)
6 ROLE SUBSUMPTION
74
But we also have to check whether information in R1 and R2 can interact in a way
which makes this sequent provable. This is clearly the case if R1 and R2 contain
incompatible information about z . If we substitute R1 = xyR3 (x)(y ) ^ C1(y )
and R2 = xyR4 (x)(y ) ^ C2 (x) we obtain
9zR3(a)(z) ^ C1(z) ^ R4(z)(b) ^ C2(z) ) False
which is provable if we have an axiom
C1 (a); C2(a) ) False
We thus obtain the inference rule
c1 u c2
v ? * (>rjc1 ).(c2 j>r) =: ?r
r1 u r2 v r3 .r4
Translates into
R1 (a)(b); R2(a)(b) ) 9zR3(a)(z) ^ R4 (z)(b)
for which no non-trivial proof exists.
r1 .r2 v r3 u r4
9zR1(a)(z) ^ R2(z)(b) ) R3(a)(b) ^ R4(a)(b)
No non-trivial proof exists.
r1 t r2 v r3 .r4
Translates into
R1(a)(b) _ R2 (a)(b) ) 9zR3(a)(z) ^ R4 (z)(b)
for which no non-trivial proof exists.
r1 .r2 v r3 t r4
9zR1(a)(z) ^ R2(z)(b) ) R3(a)(b); R4(a)(b)
No non-trivial proof exists.
(121)
6.3 Role Composition and Inversion
75
: r1 v r2.r3
Translates into
:R1(a)(b) ) 9zR2(a)(z) ^ R3(z)(b)
for which no non-trivial proof exists.
r1 .r2 v : r3
9zR1(a)(z) ^ R2(z)(b) ) :R3(a)(b)
Again, no non-trivial proof exists.
c jr1 v r2.r3
Translates into
R1(a)(b); C (a) ) 9zR2 (a)(z) ^ R3 (z)(b)
which is provable if we substitute R1 = xy 9zR4 (x)(z ) ^ R5 (z )(y ) and also
R2 = xyR6 (x)(y) ^ C1 (x). We then obtain the sequent
9zR4(a)(z) ^ R5(z)(b); C (a) ) 9zR6(a)(z) ^ C1(a) ^ R3(z)(b)
Applying 9-left and ^-left yields
R4(a)(c); R5 (c)(b); C (a) ) 9zR6(a)(z) ^ C1(a) ^ R3 (z)(b)
We now have to choose an appropriate instantiation for the existentially quantified
z on the right-hand side. Obviously c is the correct choice:
R4 (a)(c); R5(c)(b); C (a) ) R6 (a)(c) ^ C1(a) ^ R3 (c)(b)
This sequent is provable from the sequents
R4 (a)(c) ) R6 (a)(c)
R5 (c)(b) ) R3 (c)(b)
C (a) ) C1 (a)
We thus obtain the inference rule
r1
v r2; r3 v r4; c1 v
c2
*
c1 j(r1 .r3 ) v c2 jr2 .r4
(122)
6 ROLE SUBSUMPTION
76
r1 .r2 v c jr3
Translates into
R1 (a)(c); R2(c)(b) ) R3 (a)(c) ^ C (a)
In analogy to the previous case we obtain the inference rule
r1
v r2; r3 v r4; c1 v
c2
*
c1 jr1 .r3
v c2 j(r2.r4)
(123)
r1 jc v r2.r3
In analogy to the domain case we obtain the equivalence
r1
v r2; r3 v r4; c1 v
c2
*
c1 j(r1 .r3) v c2 jr2 .r4
(124)
r1 .r2 v r3jc
In analogy to the domain case we obtain the inference rule
r1
r1
v r2; r3 v r4; c1 v
c2
*
r1jc1 .r3
v
(r2.r4)jc2
(125)
v r2.r3
Translates into
R1 (b)(a) ) 9zR2(a)(z) ^ R3 (z)(b)
In order to prove this we have to split R1 into a composition, for example by
R1 = xy9zR4 (x)(z) ^ R5(z)(y). Note that on the left-hand side we thus have a
“chain” from b to a. To get a similar chain on the right-hand side we have to inverse
both R2 and R3 , thus substituting R2 = xyR6 (y )(x) and R3 = xyR7 (y )(x).
This gives the sequent
9zR4(b)(z) ^ R5(z)(a) ) 9zR6(z)(a) ^ R7(b)(z)
Application of 9-left, ^-left, and 9-right yield
R4(b)(c); R5 (c)(a) ) R6 (c)(a) ^ R7(b)(c)
which is provable from
R4(b)(c) ) R7 (b)(c)
R5(c)(a) ) R6 (c)(a)
We thus obtain the inference rule
r1
v r2 ; r3 v
r4
*
(r1 .r3 )
v
r4 .r2
(126)
6.3 Role Composition and Inversion
77
r1 .r2 v r3
In analogy to the previous case we obtain the inference rule
r1
v r2; r3 v
r4
*
r1 .r3
v
(r4.r2 )
(127)
r1 .r2 v r3 .r4
Translates into
9zR1(a)(z) ^ R2(z)(b) ) 9zR3(a)(z) ^ R4(z)(b)
Applying 9-left, ^-left, 9-right, and ^-right, we obtain two sequents
R1 (a)(c); R2(c)(b) ) R3(a)(c)
R1 (a)(c); R2(c)(b) ) R4(c)(b)
which are provable from the axioms
R1 (a)(c) ) R3 (a)(c)
R2 (c)(b) ) R4 (c)(b)
We thus obtain the inference rule
* r1 .r2 v r3 .r4
(128)
Note that we can also substitute R1 = xy 9wR5 (x)(w) ^ R6 (w)(y ), R3 =
xyR7 (x)(y), and R4 = xy9wR8 (x)(w) ^ R9 (w)(y). The resulting sequent
r1
v r3 ; r2 v
r4
is
9z9wR5(a)(w) ^ R6(w)(z) ^ R2(z)(b) ) 9zR7(a)(z) ^ 9wR8(z)(w)
^R9(w)(b)
Applying first 9-left and ^-left twice, and then 9-right twice, we obtain
R5 (a)(c); R6(c)(d); R2 (d)(b) ) 9zR7(a)(z) ^ 9wR8(z)(w) ^ R9(w)(b)
R5 (a)(c); R6(c)(d); R2 (d)(b) ) R7(a)(c) ^ R8 (c)(d) ^ R9(d)(b)
This sequent is provable from
R5 (a)(c) ) R7 (a)(c)
R6 (c)(d) ) R8 (c)(d)
R2 (d)(b) ) R9 (d)(b)
Obviously the above sequent is translatable into two DL formulae, thus giving two
inference rules:
r1
r1
v r2; r3 v r4; r5 v
v r2; r3 v r4; r5 v
r6
r6
*
*
(r1 .r3 ).r5 v r2.(r4 .r6)
r1.(r3 .r5) v (r2 .r4 ).r6
(129)
(130)
6 ROLE SUBSUMPTION
78
r
r
>r
(56)
(57)
?r
r1 u r2
r1 t r2
:r
c jr
rjc
(58)
(59)
(60)
(74)
(75)
(101)
(102)
(61)
(62)
(78)
(79)
(105)
(68)
(69)
(87)
(88)
(107)
(71)
(72)
>r
?r
r1 u r2
(64)
(65)
(70)
:r
(60)
rjc
r
r1 .r2
r1.r2
(118)
(63)
r1 t r2
c jr
r
(76)
(77)
(103)
(104)
(66)
(73)
(80)
(81)
(82)
(83)
–
(86)
(106)
(119)
(120)
(121)
(67)
(69)
(89)
(109)
(72)
(91)
(111)
(93)
(90)
(92)
(94)
(108)
(110)
(112)
(95)
(96)
(97)
(113)
(122)
(98)
(99)
(100)
(115)
(124)
(114)
(116)
(117)
(126)
(123)
(125)
(127)
(128)
(129)
(130)
Figure 9: Inference rules for role subsumption. The operator in the rows is the
outmost operator of the subsumed role term, whereas the operator in the columns
is the outmost operator of the subsuming role term.
79
7 Conclusion and Future Work
We have presented a systematic approach for deriving DL inference rules via the
Sequent Calculus. Whereas this approach is feasible for DL fragments translatable
into plain First-Order Logic, derivations for a fragment containing numerical
operators turn out to be rather complex. Furthermore, the impact of complex roles
on concept subsumption and the derivation of inference rules for object recognition
remain to be investigated.
We see our work as a solid basis for the development of flexible DL systems.
To continue this work it would be necessary to devise adequate normalforms and
to transform our inference rules into corresponding normalization and comparison
rules. This has already been done for a restricted fragment in [Royer, Quantz 92,
Sect. 5].
In this section we will briefly sketch those issues concerning inference rules
which we did not cover in the previous sections. We begin with the impact of
complex roles on concept subsumption, then consider the task of deriving inference
rules for object recognition, and finally address the problem of how to construct
flexible DL systems on the basis of the derived inference rules.
7.1 The Impact of Complex Roles on Concept Subsumption
To take into account the complex role-forming operators investigated in the previous section we have to reconsider the analysis of the constructs 8r:c, 9r:c, n r:c,
n r:c, and r:S , since in Sections 4 and 5 we have assumed that the roles r are not
complex. In the following we will consider for each of these constructs the impact
of each role-forming opererator. We will not execute a complete case analysis,
however, and we have thus no proof for the completeness of the inference rules
presented in this section. Furthermore we restrict to a simple presentation of the
inference rules omitting all redundancies.
all
A term containing a value restriction for the top role is subsumed by this value
restriction.
c1
v
c2
* 8>r:c1 v
c2
(131)
A value restriction for the bottom role is redundant since there can be no fillers for
this role anyhow:
* > v 8?r:c
(132)
7 CONCLUSION AND FUTURE WORK
80
The value restriction of a role disjunction is implied by the disjunction of their
respective value restrictions.
c1 t c2
v c3; r3 v r1 t r2 * 8r1:c1 u 8r2:c2 v 8r3:c3
(133)
Value restrictions subsuming the range of a role are redundant, i.e. the range of a
role can be taken as the “default” value restriction.
* > v 8rjc:c
(134)
As already mentioned in the previous section value restrictions for role compositions can be transformed into embedded value restrictions and vice versa.
* 8r1.r2 :c =: 8r1 :(8r2 :c)
(135)
A value restriction for a role chain r.r is subsumed by that value restriction or it
has no fillers for r:
* 8r.r
:c v c t 0 r:>
(136)
some
An existential restriction for the top role is redundant, an existential restriction for
the bottom role is inconsistent:
* > v 9>r:c
* 9?r:c v ?
(137)
(138)
An existential restriction for a role is subsumed by the domain of that role.
* 9cjr:> v
c
(139)
As for value restrictions existential restrictions for role compositions can be transformed into embedded existential restrictions and vice versa.
* 9r1.r2 :c =: 9r1 :(9r2 :c)
(140)
atleast
We obtain rules corresponding to the ones derived for the some operator:
* > v n >r:c
* n ?r:c v ?
* n c jr:> v c
(141)
(142)
(143)
An embedded minimum restriction gives a minimum restriction for role composition:
* m r1:(n r2 :c) v n r1 .r2:c
(144)
7.1 The Impact of Complex Roles on Concept Subsumption
81
atmost
Due to the negation duality between atleast and atmost both operators behave
symmetrically wrt redundancy and inconsistency. Thus a maximum restriction is
redundant for the bottom role and can be inconsistent for the top role:
* 0 >r:c v ?
* > v n ?r:c
(145)
(146)
A combination of a maximum restriction and a maximum restriction embedded in
a value restriction gives an upper bound for role compositions:
* m r1 :> u 8r1:(n r2:c) v m n r1 .r2 :c
(147)
fills
fills terms are redundant for the top role and inconsistent for the bottom role:
* > v >r :S
* ?r :S v ?
(148)
(149)
Fillers occurring at two roles are also fillers for the conjunction of these roles:
S3 = S1 \ S2 *
r1 :S1 u r2 :S2
v r1 u r2:S3
(150)
Negation of fills terms is equivalent to a filler term for the negated role, and fillers
occurring at two disjoint roles yield inconsistency.
* (:r):o =: : (r:o)
S1 \ S2 6= ; * r:S1 u :r:S2 v ?
(151)
(152)
fills terms for a role are subsumed by the domain of that role.
* c jr:S v
c
(153)
Similarly, concepts which are subsumed by some concepts and have fillers for a
role, also have the fillers for the role with the domain being that concept:
*
c u r:o v c jr:o
(154)
fills terms for role compositions are subsumed by an existential restriction with an
embedded atomic fills term:
o
2 S * r1:r2:S v 9r1:(r2:o)
(155)
A filler for an inverse role implies an existential restriction:
*
r :o v
9r :(9r:(r
:o))
(156)
7 CONCLUSION AND FUTURE WORK
82
7.2 Object Recognition
Whereas so far we have only considered subsumption formulae, we will now
present inference rules required for object recognition, i.e. for deriving description
o :: c. Note that DL systems generally do not use object information for determining subsumption. We will therefore not consider the impact of descriptions on
subsumption formulae in the following. Formally it would thus be more adequate
to define the entailment relation for subsumptions and descriptions separately, e.g.
as Γ j= t1 v t2 iff all models of the subsumption formulae in Γ are models of
t1 v t2 ; Γ j= o :: c iff all models of Γ are models of o :: c.
First we need to establish a connection between object recognition and concept
subsumption in general:
o :: c1; c1
v
*
c2
o :: c2
(157)
The we have to analyze the various concept-forming operators for the existence of
object-specific roles. Obviously, every object is an instance of the top concept:23
*
o ::
>
(158)
Different formulae describing the same object are interpreted conjunctively, therefore they can be merged into a single description by using concept conjunction:
o :: c1; o :: c2
*
)
o :: c1 u c2
(159)
There are two obvious rules for instanceship wrt extensional concepts, namely
o2S
o 62 S
*
*
o ::
o ::
S_
: S_
(160)
(161)
The following rule, usually called forward propagation, captures the connection
between value restrictions and typing of role fillers.
o1 :: 8r:c; o1 :: r:o2
*
o2 :: c
(162)
Note that this rule can be transformed into the rules
o1 :: 8r:c; o2 :: : c
o1 :: r:o2 ; o2 :: c
*
*
o1 :: : (r:o2 )
o1 :: 9r:c
(163)
(164)
The second rule can be further generalized to
o :: r:fo1; : : : ; on g; o1 :: c; : : :; on :: c
23
*
o ::
n r:c
(165)
Symmetrically, no object is an instance of the bottom concpet. We will not address the issue
of inconsistent descriptions in the following, however.
7.3 Flexible Inference Strategies
83
We also need a rule for taking into account the impact of role-filler types on roles
containing range restrictions:
o1 :: r:o2 ; o2 :: c
o1 :: rjc :o2
*
(166)
Note that the corresponding rule for domain is redundant since we already have
the subsumption rule
*
c u r:o v c jr:o
Finally we have two rules for fillers of role compositions and inverse roles:
o1 :: r1 :o2; o2 :: r2 :o3
o1 :: r:o2
*
*
o1 :: r1:r2 :o3
o2 :: r :o1
(167)
(168)
Obviously these rules are not complete for object recognition. They are incomplete
with respect to case reasoning, for example, which can be easily checked by
considering the examples presented in [Donini et al. 92] and [Royer, Quantz 94].
In order to capture the case reasoning required in these examples we would have to
introduce disjunction of descriptions. The following rule, for example, introduces
such disjunctions:
o :: 8r:fo1 ; : : :; on g_ ; o :: 9r:c
*
o1 :: c _ : : :
_ on :: c
which could then be used by a rule as
o :: r:fo1 ; : : :; on g; o1 :: c _ : : :
_ on :: c *
o :: 9r:c
Since case reasoning is notoriously complex, it has been suggested to
weaken the semantics in order to avoid these complexities [Donini et al. 92,
Royer, Quantz 94].
7.3 Flexible Inference Strategies
To conclude our investigation on inference rules for DL we will now sketch their
usefulness for the implementation of DL systems providing flexible inference
strategies.
As mentioned in Section 2 most existing DL systems follow the normalizecompare approach. In order to determine whether t1 v t2 both terms are first
normalized and then structurally compared. One advantage of this approach is
that the normalforms of terms can be cached and reused in later comparisons.
Thus information is precompiled in order to speed up query answering.
Given a set of inference rules, most rules can be used for normalization as well
as for comparison. Consider, for example, the rule
c1 u c2
v c3; r2 v r1; r2 v
r3
* 8r1 :c1 u 9r2:c2 v 9r3 :c3
7 CONCLUSION AND FUTURE WORK
84
This rule can be immediately used for comparison between two concepts—if
a concept contains the term 9r3:c3 , we have to check whether the other concept
contains the terms 8r1 :c1 and 9r2:c2 and then test the subsumption relations between
the respective roles and concepts. For normalization, however, we would rather
use the following format:
N
7! 9r1:c1; 8r2:c2 7!
9r1:c1 u c2
Intuitively this rule says that the term 9r1 :c1 u c2 should be included into a normalform whenever 9r1 :c1 and 8r2 :c2 are contained in the normalform and r1 v r2
r1
v
r2
holds.
In [Royer, Quantz 92, Sect. 5] we have shown that a particular partitioning of
the rules into normalization and comparison rules yields vivid normal forms and
is therefore complete. Due to the wide range of applications in which DL systems
can be used, however, a fixed inference strategy seems problematic. Whereas a
precompilation of information might be advantageous in Information Technology
in order to guarantee fast retrievals, in Natural Language Processing it seems more
adequate to perform only limited normalizations.
The basic idea of flexible inference strategies is thus to explicitly represent
most of the inference rules in a normalization and a comparison format, and then
to allow the choice of different combinations of normalization and comparison
rules. This would also allow to switch off rules completely if they are not required
in a particular application.24 The main theoretical problem is then to prove completeness and termination of the different possible combinations of normalization
and comparison rules.
Acknowledgements
During our work on inference rules we have benefitted from discussions with Franz
Baader, Alex Borgida, Martin Fischer, Bob MacGregor, Albrecht Schmiedel, and
Philippe Volle.
24
Obviously the choice of a particular inference strategy for a given application is supposed to
be made by the developer of the DL system and not by the knowledge engineer using the system.
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[Gabbay, Guenthner 83], 133–188
[Winston 75] P.H. Winston (Ed.), The Psychology of Computer Vision, New York:
McGraw-Hill, 1975
[Woods 75] W.A. Woods, “What’s in A Link: Foundations for Semantic Networks” in [Bobrow, Collins 75], 35–82
89
A
Proofs for Numerical Operators
In the following we assume that all the numerical restrictions participate effectively
in the proof (the formulae obtained by eliminating some of the numerical restriction
terms are assumed to be not provable). We also asume that all the quantifier
elimination steps have already been performed. As explained at the beginning of
Section 5, we will only consider all, atleast, and atmost in the following, since
some is just a special case of atleast.
A.1 Formulae with Only One Creative Term
As shown in Section 5.2, no inessential cuts need to be considered during the
proof search. By the property of homogeneous lifting, no proof possibility arises
between oneof terms and role restrictions (except by propositional proofs). Thus
the possibly provable subsumption formulae consist of a left atleast together with
one or several right atleast, left atmost, or left all. Note that the right atleast can
be rewritten as a left atmost, which leaves only two cases to check.
Negation duality between atmost and atleast
We first examine the provability conditions of formulae of the form
p r1:c1 u q r2:c2 v ?
By Proposition 3, we know that the sequent is provable iff its instantiation closure
is provable. This gives a first provability condition: q + 1 p. (Otherwise, there
is no injective instantiation function for atmost, thus the formula is provable only
if p r1 :c1 is inconsistent, which we suppose not to be the case).
We get to prove a sequent like:
^ii==1n (r1(a; bi) ^ c1(bi)); dif (~b); 8~x(^jj==n1 (r1(a; xj ) ^ c1(xj )) ! equ(~x) ) ;
n
where ^jj =
x) is an abbreviation denoting the complete
=1 (r2 (a; xj ) ^ c2 (xj )) ! equ(~
q:r2:c2 )(~x) all the injective instantiation functions ~x upon ~b.
list of terms (
Necessarily, assuming normal proofs, the original sequent is provable iff
^ii==n1 (r1(a; bi) ^ c1(bi)); dif (~b); (^jj==1n (r2(a; xj ) ^ c2(xj )) ! equ(~x) ) ;
atmost
The proof continuation is necessarily propositional. Besides the standard steps for
eliminating the ^-connectors, which are assumed to be already done, the next step
can only be an elimination of !.
Let us assume that the term on which !-left is applied is the particular instantiation of the (
::)(x~0). One gets a first sequent to prove:
atmost
^ii n1 (r1(a; bi) ^ c1(bi)); dif (~b); equ(~x) ) ;
=
=
A PROOFS FOR NUMERICAL OPERATORS
90
which is trivially valid by the choice of ~x itself. Then, the second sequent to prove
is:
^ii
n r
a; bi) ^ c1 (bi)); dif (~b) ) ^jj
=
=1 ( 1 (
n r
a; xj ) ^ c2 (xj ))
=
=1 ( 2 (
It is straightforward to deduce the proof conditions c1 v c2 and r1 v r2. If several
instantiated terms are unfolded during the proof, with different sets ~x, then by an
easy induction, one easily checks that the same proof conditions arise.
With about the same kind of manipulations, one obtaines easily the proof of
the formula:
q < p; c2 v c1 ; r2 v
* > v p r1:c1 t q r2 :c2
r1
(39)
Similarly, the inconsistency schema between one atleast term and and one all term
is derived very easily.
c1 u c2
v ?; r1 v
* n r1:c1 u 8r2 :c2 v ?
r2
(41)
The preceding inference rules establish not only the inconsistency condition between atmost and atleast, but also the negation duality between atmost and
atleast. One important outcome is that in the following we can restrict to inconsistency formulae i.e. formulae with inconsistent succedent ? (alternatively,
sequents with empty succedents). Indeed, the former result together with the negation principles for the other role restrictions all and some induce the following
lemma.
Lemma 5 Every subsumption formula in TL= can be rewritten into an equivalent
inconsistency formula, provided that its succedent is homogeneously quantified.
Proof: The upper-level quantified terms in subsumption formulae correspond to
role restrictions. By the Negation and De Morgan rules, one can rewrite any formula having a role restriction in the succedent into an equivalent formula having
the corresponding dual term in the antecedent.
2
Below are the rules capturing inconsistency between atleast and atmost (38),
and between atleast and all(41, partition between between atleast and atmost(39),
the negation duality between atmost and atleast (35,36), and monotonicity for
atleast (37) and for atmost(40):
q < p; c1 v c2 ; r1 v
c2 u c1 v ?; r1 v
q + 1 p; c2 v c1 ; r2 v
r2
r2
r1
* q r2 :c2 u p r1:c1 v ?
* 8r2 :c2 u p r1:c1 v ?
* > v q r2:c2 t p r1 :c1
(38)
(41)
(39)
A.1 Formulae with Only One Creative Term
m=n 1
m=n+1
q p; c1 v c2 ; r1 v r2
p q; c2 v c1 ; r2 v r1
*
*
*
*
91
:n r:c =: m r:c
: n r:c =: m r:c
p r1:c1 v q r2:c2
p r1:c1 v q r2:c2
(35)
(36)
(37)
(40)
Modus Ponens
Now we examine inconsistency formula of the form
atmost(: : : ) u atleast(: : : ) u all(: : : ) v ?
by assuming that no strict subformula is provable. By Proposition 3 we consider
the instantiation closure of the sequent corresponding to the above formula. We
then get a sequent of the form:
a; bi) ^ c1 (bi)); dif (~b)
j
=n
^j=1 (r2(a; xj ) ^ c2(xj )) ! equ(~x); r3(a; y) ! c(y)
^ii
n r
=
=1 ( 1 (
) ;
where by convention the terms instantiated by ~x and y represent the list of all
possible corresponding instantiations, with ~x an injective substitution into ~b and y
an element of ~b . This gives the first condition q + 1 p (otherwise, ~x cannot be
injective and the proof is independent of q r2 :c2). We will use in the following
p
to denote the formula ^ii=
=1 (r1 (a; bi )).
Now the proof search proceeds by choosing a (correctly) instantiated atmost
term and eliminating the implication connector.25 One gets one sequent of the
form
; r3 (a; y) ! c3(y); equ(~x) ) ;
which is trivially valid because of the choice of ~x (every equality is of the form
bi = bj and thus is provable with the corresponding inequality in dif (~b)). One
gets another sequent of the form:
; r3 (a; y) ! c3 (y) ) ^jj
n r
a; xj ) ^ c2(xj ))
=
=1 ( 2 (
Elimination of all the conjunction connectors provides the sequents
; r3 (a; y) ! c3 (y) )
; r3 (a; y) ! c3 (y) )
r2 (a; xj )
c2 (xj )
If there exists one xj such that both sequents are provable without one left expression (r3 (a; y ) ! c3 (y )), then the proof reduces to a simple inconsistency schema
between p r1 :c1 and q r2 :c2 . We thus assume the contrary; in particular c1 6v c2 .
25
Note that the other terms remain in the formula—for the sake of brevity, we omit to write them
explicitly.
A PROOFS FOR NUMERICAL OPERATORS
92
Then for all the xj , one needs to unfold some expression of the list (r3(a; y ) !
c3 (y )). Let us call yj the particular instantiation chosen for y in the second sequent.
One gets to prove the sequents: ; c3 (y ) ) c2 (xj ) and ) r3(xj ); c2 (xj ). To
prove both sequents under the assumption c1 6v c2, one must choose y = xj . One
obtains the condition c1 u c3 v c2 from the first sequent and the condition r1 v r3
from the latter one26 .
Finally, the first sequent gives the two sequents ; c3(y ) ) r2 (a; xj ) and
) r3 (a; y); r2 (a; xj ). From the first sequent, by separation between role and
concept sequents, one gets the condition r1 v r2 (unless ; c3 (y ) ) ; is provable,
but in this case the proof reduces to the inconsistency between p r1:c1 and 8r3 :c3 ).
From the second sequent one gets no new necessary condition (if role disjunction
is available, the condition r1 v r3 t r2 could be obtained for the choice y = xj , but
this condition is redundant with previous conditions r1 v r2 and r1 v r3).
To summarize, we obtain Modus Ponens in inconsistency form (42), as well
as Modus Ponens for atleast (43) and for atmost(44):
q < p; r
q p; r
q p; r
v
1 v
1 v
1
;
r ;r
r ;r
v
1 v
1 v
r2 r1
2
2
;
r ;c
r ;c
r3 c1 u c3
3
3
v
1 u c3 v
1 u c3 v
c2
c2
c2
* q r :c
* p r :c
* q r :c
2
1
2
u p r1:c1 u 8r3:c3 v ? (42)
(43)
1 u 8r3 :c3 v q r2 :c2
(44)
2 u 8r3 :c3 v p r1 :c1
2
Additivity Rules wrt one single atleast term
Now let us examine the cases where several left atmost coexist in the antecedent
with one single atleast. We basically consider formulae of the form:
n r:c u p r1:c1 u q r2:c2 v ?
Because of Proposition 3 one can restrict to instantiation-closed sequents. This
gives the first conditions: p<n and q<n. We call ) ; the instantiation-closed
sequent obtained from the precedent subsumption formula. If the proof appeals
only to instantiations of one unique subterm atmost, we are in the same situation
as before.
Let us suppose that the proof uses instantiations of both atmost terms.
p:r1 :c1)(~x) and one term
We get a sequent containing one term (
(
q:r2:c2 )(~y). After application of !-left, we get four sequents to prove:
atmost
; equ(~x); equ(~y)
; equ(~x)
; equ(~y)
atmost
)
)
)
)
;
^ii
^jj
^ii
a; xi) ^ c1 (xi))
a; yj ) ^ c2 (yj ))
1
j
(r1 (a; xi) ^ c1 (xi )); ^j
p
q
= +1
( 1(
=1
= +1
( 2(
=1
p
= +
=1
r
r
q
a; yj ) ^ c2(yj ))
= +1
( 2(
=1
r
The choice of y is thus dependent of xj . If y were chosen globally, independently of the
particular xj , then we would have obtained only simple inconsistency schemata.
26
A.1 Formulae with Only One Creative Term
93
As the instantiation functions are injective into the set B of eigen-parameters
created by the atleast term in , the three first sequents are trivially valid (because
dif (B~ ); equ(~x) ) ; and dif (B~ ); equ(~y) ) ;). We thereby get the conditions
p + 1 n and q + 1 n.
The other proof conditions can thus come only from the fourth sequent (we will
call this sequent 4 in the following). One gets conditions like ) c1(x); c2 (y )
i.e. sequents with right disjunctions to prove. If the proof proceeds without _-left
rules, then by the Constant Separation lemma for disjunctions, one gets to prove
either ) c1(x) or ) c2(y ) (by choosing x 6= y , which one can always assume
because of p; q 1). Then the proof reduces to a simple inconsistency schema
between c, c1 , and c2 . Consequently, possible new proof conditions come only by
considering proofs with application of _-left rules or ^-right rules. It means that
either c is disjunctive (then _-left rules are applied for eliminating the disjunction)
or c1 or c2 are conjunctive (then ^-right rules are applied for eliminating the
conjunction).
Case where c contains one disjunction. Now, let us first assume that c is in
disjunctive form: c = d1 t d2 , where d1 and d2 are disjunction free. Let us assume
that c1 and c2 are conjunction free. By exhaustive applications of _-left rules, we
get to prove (up to variable renaming) a collection of sequents m , for m ranging
from 0 to n:
m; (atmost p:r1 :c1 )(~x); (atmost q:r2 :c2)(~y) ) ;
n
i=m
i=n
where m = ^ii=
=1 (r(a; bi )); ^i=1 (d1 (bi )); ^i=m+1 (d2 (bi )). m counts the numbers
of elements affected to d1 in the partition of c between d1 and d2 . In the following,
for some given m we write D1 for the set of eigen-parameters for d1, D1 = fbi :
1 i mg, and D2 for the set of eigen-parameters for d2, D2 = fbj : m + 1 j ng. For each value of m, the proof relies as before upon some sequent:
m ) ^ii
p
a; xi) ^ c1 (xi)); ^jj
= +1
( 1(
=1
r
q
a; yj ) ^ c2 (xj ))
= +1
( 2(
=1
r
for some particular choices of ~x and ~y depending locally on m.
1. For the limit cases m = 0 and m = n, whatever the choices of ~x and ~y are,
there is always one couple (x; y ); x 6= y . The corresponding sequent (after
the elimination of the conjunctions) is provable by Constant Separation (by
assumption, m is disjunction free). So we get at least conditions ri v r and
dj v ci , for j = 1; 2.
In the following we assume that neither dj v c1, for j = 1; 2, nor dj v c2 ,
for j = 1; 2, prove uniformely all the sequents (otherwise, we obtain the
simple inconsistency schema).
A PROOFS FOR NUMERICAL OPERATORS
94
2. For m p, ~x takes necessarily one element x2 in D2, then the sequents
obtained from 4 by fixing c1 (x2 ) cannot be proved by ) c1 (x2). By
Constant Separation, for all y different from x2 we get to prove ) c2(y )
and ) r2(y ). This gives r v r2 and arbitrary conditions dj v c2 , for j = 1
or j = 2 or both. When both conditions arise a simple inconsistency is
obtained between q r2 :c2 and n r:c. We assume the contrary.
3. The case where d1 v c1, d1 v c2, d2 6v c1, and d2 6v c2 is excluded by
considering m = n (the sequents with distinct x and y are unprovable!).
Hence we get d1 v c1, d2 v c2, d1 6v c2 , and d2 6v c1 .
4. To respect these conditions, the existence of one couple (x; y ) such that
x 6= y; x 2 ~x \ D2; y 2 ~y \ D1, is impossible (because again an unprovable
sequent would be obtained). Hence, for all m, ~x and ~y must be locally chosen
such that either ~x is included in D1 or ~y is included in D2. For m p,
~y must be necessarily included in D2. We get the necessary condition
q + 1 n m, i.e. p + q + 1 n. Then all the sequents m, for m p, are
proved with the terms r2 and c2 wrt the current conditions.
5. Finally, for n m = q , ~y cannot be chosen to be included in D2; there is
one y 2 D1 and ~x D1. Then the sequents m ) r1(x); c2 (y ), x 6= y , are
provable iff m ) r1 (x) is provable. This gives the final condition r v r1.
Then it is straightforward to check that all sequents m are indeed proved!
We get the Additivity Rule:
p + q < n; r v r1; r v r2 ; c v c1 t c2
*
n r:c u q r1:c1 u p r2:c2 v ?
By the negation duality between atleast and atmost, we get as corollary the
additivity principles of atmost wrt disjunction.
r v r1; r v r2
* p r1 :c1 u q r2:c2 v p + q r:c1 t c2
Case where c1 contains one conjunction. Let us assume that c1 = c1a u c1b. By
^-right 4 gives:
) ^ii
q
a; xi) ^ (c1a u c1b)(xi)); ^jj
= +1
( 1(
=1
r
p
a; yj ) ^ c2(xj ))
= +1
( 2(
=1
r
which can be further transformed as follows
) ^ii 1q 1 (r1 (a; xi) ^ c1a(xi) ^ c1b(xi)); ^jj p1 1 (r2 (a; yj ) ^ c2 (xj ))
) (^ii q1 1 (r1 (a; xi) ^ c1a(xi))) ^ (^ii 1q 1 (r1 (a; xi) ^ c1b(xi)));
= +
=
= +
=
= +
=
= +
=
A.1 Formulae with Only One Creative Term
^jj
) ^ii
^jj
) ^ii
) ^ii
95
a; yj ) ^ c2 (xj ))
r a; xi) ^ c1a(xi )) ^ ^ii q1 1 (r1(a; xi) ^ c1b (xi));
r a; yj ) ^ c2 (xj ))
1
j p 1
(r1 (a; xi) ^ c1a(xi )); ^j 1 (r2 (a; yj ) ^ c2 (xj ))
1
j p 1
(r1 (a; xi) ^ c1b (xi)); ^j 1 (r2 (a; yj ) ^ c2 (xj ))
p
= +1
( 2(
=1
q
p
r
= +1
( 1(
=1
= +1
( 2(
=1
q
= +
=
= +
=1
q
= +
=
= +
=1
= +
=
This reduces to proving both formulae, with conjunction free terms in the upper
bound numerical restrictions:
n r:c u q r1:c1a u p r2:c2 v ?
n r:c u q r1:c1b u p r2:c2 v ?
From the results in the previous paragraph we get for each formula conditions c v
c1a t c2 and c v c1b t c2. By summing up these conditions, again one derives the
proof condition c v c1 t c2 . No new rule comes out but the precedent ones.
Complete proof by structural induction. Formally, the complete proof for
arbitrary formulae will work by structural induction on the roles and concept
terms, by considering that the terms quantified by lower bounds (resp. by upper
bounds) numerical restrictions atleast (resp. atmost) are in Disjunctive Normal
Form (resp. Conjunctive Normal Form).
The induction steps proceeds, exactly as in the two cases above, by reducing
to proofs of simpler formulae either obtained by disjunction elimination in c (then
one needs to consider all the sequents m where the antecedent describes a possible
partition of B among the disjuncts of c) or conjunction elimination of c1 or c2 (then
one needs to consider proving the formula for all the conjuncts). In each case,
local proof conditions of the form d v c1a t c2 are obtained, d being some disjunct
of c, c1a being some conjunct of c1. The combination of the local proof conditions
gives always the general condition c v c1 t c2 .
General Additivity Rules. We now examine the generalization to more than
two atmost terms. Let us consider formulae with k atmost terms:
n r:c u p1 r1:c1 u p2 r2:c2 u : : : u pk rk :ck v ?
P
k
We prove by induction on n+ ii=
=1 (pi ) that the formula is provable iff cis subsumed
by a disjunction of the ci such that the corresponding restriction numbers sum up
to strictly less than n. The reasoning proceeds again by assuming c in disjunctive
form and combining the different conditions obtained in the different partition of
B among the disjuncts of c. (The details are left to the reader—the reasoning
exploits the same principles as those explained before).
A PROOFS FOR NUMERICAL OPERATORS
96
To summarize, we obtain the general additivity rule wrt a single atleast term
Pi k (pi) < n; ri v r; c v c t c t : : : t ck
i
=
=1
*
1
2
(46)
n r:c u p1 r1:c1 u p2 r2:c2 u : : : u pk rk :ck v ?
and the additivity for atmost wrt disjunction:
r v r1; r v r2
* p r1 :c1 u q r2:c2 v p + q r:c1 t c2
(47)
Formulae with several all terms
If several all terms coexist, they also contribute by left implicative terms, more
precisely they contribute by right role terms and left concept terms. So assuming
a total of h unfolding steps of instantiated all terms, we get to prove sequents
generalizing the four next sequents to an arbitrary number of role/concept terms:
; c3 (y)
; c3 (y)
)
)
)
)
r2(a; xj )
r2(a; xj ); r3 (a; y )
c2 (xj )
c2 (xj ); r3 (a; y )
The conditions are similar to simple Modus Ponens: the choice of the y ’s must
depend on the x’s for realizing exact matching between the parameters. The
introduction of one term 8r:c modifies the proof of the additivity rules by possibly
weakening the subsumption relation dj v ci by c u dj v ci . Typically, one gets
additive Modus Ponens like:
p + q < n; r v r1; r v r2 ; r3 v r1; r4 v r2 ; c3 u d1 v c1 ; c4 u d2 v
*
n r:d1 t d2 u q r1:c1 u p r2:c2 u 8r3:c3 u 8r4:c4 v ?
c2
(48)
A.2 Formulae with exactly two eigen-parameter-creating terms
By duality between atleast and atmost, we only consider formulae of the form:
c u q r1 :c1 u p r2 :c2 v ?
The need for homogeneous quantifier elimination induces that c is made of role
restrictions. So c consists only of atmost or all terms. Given the results of
Section 5.2 one can restrict to equality-complete sequents (i.e. sequents in which
b = d is decidable for every couple of parameters). The point being that all
equality complete extensions must be checked.
A.2 Formulae with exactly two eigen-parameter-creating terms
97
It is straightforward to check that no inessential cut wrt an equality inside
B or D (typically, bi = bj or bi = bi) is relevant (we get on the one hand a
sequent with trivially inconsistent antecedent, on the other hand the same sequent
as the one before cut). Similarly, if bi = di belongs already to the sequent,
then any inessential cut wrt bi = dj , for j 6= i, produces one sequent with
bi = dj ; bi 6= bj and another trivially provable by inconsistency of the antecedent
(containing bi = dj and bi = di , reducing to di = dj by substitution).
Up to some renaming, one can always assume that the inessential cuts involved
in the proof introduce a set of equalities/inequalities such that bi = dj is decidable
for every index i and j . We write (B D)k in the resulting formula: k denotes
the number of equalities of (B D)k , all the other terms are inequalities. We also
assume that the equalities (B D)k are bi = di ; i k .
We get to prove sequents of the form ; (B D)r ) ; where r ranges from 0 to
min(p; q). Furthermore, because of Proposition 3, one has to consider instantiation
closed sequents only. We therefore introduce the following notations:
B
D
1
2
B D)k
Br
Dr
k
c
(
=
fb1; b2; :::; bpg
fd1; :::; dqg
=
((r1 (
=
=
=
=
=
=
a; b1); :::; r1(a; bp)); (c1 (b1); :::; c1(bp)); dif (B~ ))
produced by creative quantifier elimination applied to p r1 :c1
~ ))
((r1 (a; d1 ); :::; r1 (a; dq )); (c1 (d1 ); :::; c1 (dq )); dif (D
2 is the left expression produced by q r2:c2
^ii==k1 (bi = di ) ^k6=l (bk 6= dl )
fbi : 8i rg
fdi : 8i rg
1 ^ 2 ^ (B D)k
denotes the instantiation closure of , the sequent expression for c.
In conclusion of all these preliminary remarks, proving the initial formula reduces
min(p;q) :
to proving the collection of sequents (k )kk=
=0
k
k
=
=
k ; c ) ;
1 ; 2; (B D)k
Then the proof proceeds by case analysis on the possible forms for c. We will
always assume that 1; 2 ) ; is not provable. Thus all the forthcoming proofs
will necessarily decompose some expression in c .
Proving formulae p r1:c1 u q r2:c2 u
We get to prove all sequents of the form
k r3:c3 v ?
r ; (atmost k:r3 :c3 )(~x) ) ;
98
A PROOFS FOR NUMERICAL OPERATORS
Let us consider one particular instantiation of the atmost term, wrt ~x, used during
the proof (recall that ~x is injective over B [ C , cf. Proposition 3). The two
sequents to prove are:
r ; equ(~x) ) ;
r ) ^ii
k
a; xi); c3 (xi))
= +1
( 3(
=1
r
(We will not consider the trivial cases where c1 or c2 is subsumed by ?.) For
r = 0, the elements of B and D are made distinct, so the first sequent is proved
by the subsequent dif (B [ D); equ(~x ) ;. When r 6= 0, some elements of B
and D are made equal. There are two cases depending on whether or not the proof
exploits the terms c1 and c2 or just the equality/inequality terms of the sequent.
In the first case the sequent can be proved by the subsequent
b1 = d1 ; c1(b1 ); c2(d1 ) ) ;
This gives one sufficient condition: c1 u c2 v ?. Then, for r 6= 0, the second
sequent is proved by its antecedent being inconsistent. The case r = 0 gives the
remaining conditions: r1 v r3 , c1 v c3 and similar conditions for r2 and c2 .
For the second case assume that c1 u c2 v ? does not hold. Then the first
sequent can only be proved by its subsequent:
dif (B ); dif (C ); (B D)r ; equ(~x) ) ;
If ~x is totally included in B (or D), then the proof works by dif (B ); equ(~x) ) ;.
Then the second sequent is easily seen to be proved by r1 v r3 and c1 v c3. We
end up with the classical negation duality between atmost and atleast.
New conditions are obtained only from ~x taking elements both from B and D.
Take x1 = bi and x2 = dj . Let us try to prove the sequent obtained from the first
sequent above:
dif (B ); dif (C ); (B D)r ; bi = dj ) ;
By completeness of (B D)r , it is easy to see that the proof only fails if i = j r.
Hence to be provable for every r ~x must be chosen such that it contains no pair
fbi; dig, hence the condition k + 1 p + q r, and more generally k + 1 max(p; q).
Let us assume that p = min(p; q ) and q = max(p; q ). Let us consider the case
where p < k + 1 q . Hence the proof does not reduce to a simple inconsistency
schema wrt the single term p r1c1 . For the case r = p, when B is “embedded”
into D, ~x takes necessarily one element d 2 D n Br . One gets to prove, from the
second sequent
r )
r )
r3 (d)
c3 (d)
A.2 Formulae with exactly two eigen-parameter-creating terms
Assuming no use of right disjunction for c3and r3 , one gets the conditions r2
and c2 v c3 , hence a simple inconsistency schema wrt q r2 c2.
99
v r3
If some right disjunction occurs, say for c3, meaning that c3 = c1a t c1b is
decomposed, one may get the weaker condition c2 v c1a instead of c2 v c3. Then
the inference results from the simple inconsistency schema between q r2 :c2 and
k r3c1b and the monotonicity schema for atmost.
We conclude these considerations with the following Lemma:
Lemma 6 The subsumption formula
p r1:c1 u q r2:c2 u k r3:c3 v ?
is provable iff one of the following conditions holds:
k < p, r1 v r3, c1 v c3
k < q, r2 v r3, c2 v c3
k < p + q, r1 v r3, r2 v r3, c1 v c3, c2 v c3
By the way we have proved the following lemma:
Lemma 7 dif (B ); dif (D); (B D)r ; equ(~x)
pair (bi; di ); i r.
) ; is provable iff ~x contains no
Proving formulae p r1:c1 u q r2:c2 u k r3 :c3 u n r4:c4 v ?
We have to search for provability conditions of sets of sequents r , with r min(p; q), like:
1 ; 2; (B D)r ; (atmost k:r3 :c3)(~x); (atmost n:r4 :c4 )(~y) ) ;
In the following, we assume that the proof does not reduce to the precedent case.
Thus at least two atmost expressions are unfolded, with two different instantiation
vectors ~x and ~y. If ~x and ~y are equal, or ~x ~y, then k n; by contraction
everything happens as if one single atmost were unfolded; the situation is the
same as with one single atmost. Thus in all the following we assume that ~x and ~y
are not included in each other.
We have to prove four sequents (1,: : : ,4):
1 r ; equ(~x); equ(~y)
2
r ; equ(~y)
3
r ; equ(~x)
4
r
)
)
)
)
;
^ii
^jj
^ii
^jj
k
n
k
n
r3(a; xi); c3 (xi ))
r4 (a; yj ); c4 (yj ))
r3(a; xi); c3 (xi ));
r4 (a; yj ); c4 (yj ))
= +1
(
=1
= +1
(
=1
= +1
(
=1
= +1
(
=1
A PROOFS FOR NUMERICAL OPERATORS
100
The proof for 1 requires more clever analysis because the presence of double
equalities in the antecedent may open new proof possibilities via substitutions. But
by the assumption about equality-complete sequents things remain quite simple.
Let us assume that neither r ; equ(~x) ) ; nor r ; equ(~y) ) ; is provable.
There is a pair (bk ; dk ) 2 ~x such that k r and a pair (bu ; du ) 2 ~y such that u r.
Then the subssequent r ; bk = dk ; bu = du ) ; from 1 is unprovable! Thus at
least one of ~x and ~y is provable alone wrt (B D)r . Again, k + 1 p + p r.
Now let us assume that r ; equ(~x) ) ; is provable but not r ; equ(~y) ) ;.
~y contains some (bu ; du); u r. 1 and 3 are trivially proved but not 2. 2
gives all the sequents r ; equ(~y) ) c3(x) and r ; equ(~y) ) r3(a; x), 8x 2 ~x.
All the sequents obtained by eliminating the _ connector in equ(~y) are trivially
proved (antecedent inconsistent) except from the (bu ; du ). Thus the only significant
sequents 2 are of the form: r ; bu = du ) c3 (x); 8x (and similarly for the role
sequents). But r ; bu = du ) c3(x) is (by contraction) equivalent to r ) c3 (x).
Moreover, if r ; equ(~y) ) ; is not provable, then 2 reduces to the collection of
sequents r ) c3(x) and r ) r3(x); then 4 is proved by 2.
The proof for 4 requires elimination of the right conjunctions. 4 splits into
four new types of sequents, for all the possible pairs (i; j ) i k + 1; j n + 1:
4a
4 b
4 c
4 d
r
r
r
r
)
)
)
)
r3 (a; xi); r4 (a; yj )
r3 (a; xi); c4 (yj )
c3(xi ); r4(a; yj )
c3(xi ); c4 (yj )
Due to the absence of any role forming operators in the first sequent, the only proof
possibilities are by assumptions ri v rj 1 i 2; 3 j 4. By the Separation
Principle for role/concept sequents, if for some j neither r1 v rj nor r2 v rj , then it
means that the proof must be independent from m rj :cj ; contrarily to the initial
assumption. Then we have ri v r3 and ri v r4 for 1 i 2.
In the following, like in Section A.1 we distinguish the cases where the sequents
contain left disjunction and right conjunction or not. This is because the terminal
steps in the proofs usually apply the Constant Separation Principle (cf. Lemma 4),
which works with sequents free of left disjunction or right conjunction. Again, the
general proof will proceed by structural induction on the disjunctive structure of
the terms appearing in lower bounds numerical restrictions and on the conjunctive
structure of the terms appearing in upper bounds numerical restrictions. We begin
with proofs without _-left and ^-right rules, then consider proofs with _-left rules,
and finally investigate proofs with ^-right rules.
Proofs without _-left and ^-right rules. We first assume that r1 , c1, r2, and c2
are free of disjunction and that r3, c3 , r4 , and c4 are free of conjunction. Therefore,
no _-left rule and no ^-right rule are applied and the Constant Separation principle
A.2 Formulae with exactly two eigen-parameter-creating terms
101
is safely applicable. Furthermore we always assume that p and q are greater than
2. (Otherwise there is no proof with atmost terms—~x and ~y are identical and take
all the parameters b; d; in the case where b = d is assumed, 1 is unprovable!)
For each sequent r one of ~x , ~y must be provable: k + 1orn + 1 p + q r.
Let us assume that q = min(p; q ).
In the sequent 0 , i.e. when no equation bi = di is assumed, both ~x and ~y
are provable. Assuming, there are two elements x 6= y , one gets to prove from
4d: 0 ) c3 (x); c4(y). The proof proceeds necessarily by Constant Separation,
giving conditions cj v ci , i, j = 1 or 2.
When there are elements common to ~x and ~y, the 4d sequents 0 )
c3 (x); c4(x) give also conditions like ci v c3 t c4 , i= 1 or 2. From sequents
r ) c3 (x); c4(x) one can also get (by substitution wrt an equality of (B D)r )
the condition c1 u c2 v c3 t c4.
In all the following we assume that the proof does not proceed by a simple
inconsistency schema wrt one atleast term; this means we do have neither c1 v c3 ,
nor c1 v c4 , nor c2 v c3 , nor c2 v c4 . c1 v c or c1 v c uniformely over all sequents
r , for c = c3 or c4 . Then, by summing up the local conditions found for all the
sequents r , one must get maximal conditions which are (boolean) combinations
of:
c1 u c2 v c3, c1 u c2 v c4, c1 t c2 v c3, c1 t c2 v c4
c1 v c3 t c4, c2 v c3 t c4, c1 v c3 u c4, c2 v c3 u c4
Case 1: Both c3 and c4 have maximal conditions c1 u c2 v cj . This means
that we do not have ci v cj (1 i 2; 3 j 4) in a sequent r , because then the condition c1 u c2 v cj will be subsumed by ci v cj . But, for the
sequent 0, we have no equality assumption at all ; the sequent 0 ) c3 (x); c4 (y ),
with x =
6 y, is necessarily proved by Constant Separation and gives conditions
ci v cj . Consequently for at least one of c1 or c2 c1 u c2 v cj is not a maximal
condition.
Case 2: c3 has maximal condition c1 u c2 v c3 . In order to avoid a proof by
c1 v c3 or c2 v c3 , one must insure that there is no sequent r in which ~x alone is
provable with some x not belonging to Br [ Dr . For r k , the above situation
where ~x is provable and included in Br [ Dr is not possible. If ~x were provable,
but not ~y, then the sequent 2 should be proved, with at least one x outside Br [ Dr ;
a condition ci v c3 would arise, contrarily to the assumption. Two cases arise.
1. Let us first assume that ~x is not provable but ~y is; then 3 must be proved.
This gives the new condition n + 1 p + q k . One gets conditions
c1 v c4 or c2 v c4 (c1 u c2 v c4 is excluded by taking r = 0). One gets both
conditions if ~y is large enough to take elements from B n Dr and D n Br , i.e.
A PROOFS FOR NUMERICAL OPERATORS
102
when n + 1 > p. Otherwise, c1 v c4 or c2 v c4 alone makes the inference
reducing to a simple inconsistency schema wrt one of the atleast terms.
Finally, the role conditions for r4 are similar to the conditions for c4; both r1
v r4 and r2 v r4 can be derived unless reduction to a simple inconsistency
schema is possible.
2. Now let us assume that both ~x and ~y are provable wrt r (r k ), then 4 alone
is to be proved. Again, one gets n + 1 p + q k . One element x of ~x is
necessarily outside Br [ Dr ; the corresponding sequents r ) c3 (x); c4(y )
cannot be proved by r ) c3(x) (ci v c3 is forbidden). Thus r ) c4 (y ),
for all y 6= x, r ) r4 (y ) for all y , and r ) c3(x); c4 (x) if x 2 ~x \ ~y,
have to be proved. If r = 0 in particular, because Br [ Dr is empty, all the
sequents 0 ) c3 (x) are unprovable, then the proof reduces to a proof by
c4 and r4 with 3. One gets the conditions c1 v c4 and c2 v c4 (excluding
the conjunctive condition c1 u c2 v c4 by Case 1) unless ~y is included in B
or D (then a simple inconsistency arises). The roles conditions follow.
We end up with Left Additivity of atleast wrt disjunction:
(r1
v
r3 or r2
v
r3 or r1
u r2 v
r3 ); r1
*
v r4; r2 v r4; k + n < p + q
1
p r1:c1 u q r2:c2 u k r3:c1 u c2 u n r4:c1 t c2 v ?
Case 3: Both c1 and c2 have maximal conditions ci v c3 t c4. To obtain these
maximal conditions, no condition ci v c3 or ci v c4 must arise in any r : For r = 0,
both ~x and ~y are provable. Assuming ~x and ~y not included in each other, one can
choose two elements x 6= y , x 2 ~x n ~y and y 2 ~y n ~x. The corresponding sequent
from 4d must be necessarily proved by Constant Separation, giving the forbidden
conditions ci v cj for j = 3 and j = 4. Thus no proof arises in this case, except
proofs independent from c3 or c4 , and thus reducible to a simple inconsistency
schema.
Case 4: c2 has maximal condition c2 v c3 t c4 . In order to obtain the maximal
condition c2 v c3 t c4 , but neither c2 v c3 nor c2 v c4 , the only possibility, in
absence of any left disjunction, is to prove sequents like r ) c3 (x); c4 (x) with
x 2 D.
1. We first assume that there is some element d 2 D \ ~x \ ~y and r = 0. d is
the unique element of D \ ~x \ ~y. If there were two of them, d1 ; d2, then
the sequent 0 ) c3 (d1 ); c4(d2 ) should be proved, giving necessarily one
forbidden condition. For x 6= d the sequent 0 ) c3(x); c4 (d) is necessarily
proved by Constant Separation and c3 . This means that ~x n fdg B . The
same holds for ~y. Then k and n p, c1 v c3 and c1 v c4 .
A.2 Formulae with exactly two eigen-parameter-creating terms
103
The role conditions are obtained from the sequents 4b and 4c by choosing
d as argument of cj and x or y different from d, one gets r1 v r3, r1 v r4 .
It remains to prove the sequents 0 ) c3 (d); r4(d) and 0 ) r3(d); c4(d).
One gets r2 v r3 and r2 v r4.
For the other sequents r , if p < q , the same kind of reasoning may still hold
with D n Dr instead of D. But if q p, then for r = q , necessarily one of
~x and ~y is provable in q , hence k + 1 p or n + 1 p. Then the proof
reduces to a simple inconsistency schema between c1 and c3 or c4 .
2. Now assuming ~x \ ~y \ D = ; and r = 0, then we cannot have sequents like
0 ) c3 (x); c4(y) where x and y both belong to D and are distinct, otherwise
some forbidden condition c2 v cj would arise by Constant Separation. Then
one of ~x or ~y must be included in B , say ~x, then the conditions c1 v c3 , r1
v r3, k + 1 p are derived. The proof reduces to a simple inconsistency
schema.
We finally obtain Weak Right Additivity:
p < q; k p; n p; c1 v c3 u c4; c2 v c3 t c4 ;
r1 v r3; r1 v r4 ; r2 v r3 ; r2 v r4
*
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
NB: The rule is called “weak” because the assumptions about the number restrictions are not maximal. The general Right Additivity cannot be derived without
assuming proofs with left disjunctions (see proofs with disjunctions below).
Case 5: c1 has maximal condition c1 v c3 u c4 . The case where both c1 and c2
have maximal conditions ci v c3 u c4 will be considered in Case 7. The case where
neither c2 v c4 nor c2 v c3 corresponds to the precedent case. In the following,
we assume that c2 v c4 holds but not c2 v c3 . (The situation also corresponds to
c1 t c2 v c4 and not c1 t c2 v c3 (cf Case 6).)
To avoid the forbidden condition c2 v c3 one must insure that there is no
sequent r in which ~x is provable alone and takes elements from D n Dr . For
r k, if ~x is provable then it takes necessarily one element from D n Dr and then
~y must be provable also. Otherwise ~y is provable alone. Two cases arise.
1. Assume that both ~x and ~y are provable in r and x 2 ~x \ (D n Dr ). As usual
the condition k + n < p + q is obtained. Then the sequent r ) c3(x); c4 (y )
and r ) c3 (x); r4 (y ) have to be proved independently of c3 (x). One gets
arbitrary conditions for c4 and r4 .
A PROOFS FOR NUMERICAL OPERATORS
104
2. Only ~y is provable in r . This is the same as above. For r > k , ~y alone
provable means that n + 1 max(p; q ) and the proof reduces to a simple
inconsistency schema. Otherwise, ~y is not provable (n + 1 > max(p; q ));
then ~x is provable, takes no element outside from D n Dr and the proof
proceeds by 2. All role conditions for r1 may come (including r2 v r3 ).
We obtain Partial Right Additivity
k + n + 1 p + q; (r1 v r3 or r2 v r3 or r1 u r2 v r3 );
r1 v r4; r2 v r4; c1 v c3 ; c1 v c4 ; c2 v c4;
*
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
Case 6: c3 has maximal condition c1 t c2 v c3 . If c4 has also the same maximal condition, this is Case 7. Otherwise, as already mentioned, the situation
reduces to Case 5.
Case 7: Both c3 and c4 have maximal conditions c1 t c2 v cj . Then all the
conditions ci v cj are satisfied, which means that all the concept parts of the
sequents are trivially verified. Then the proofs relies only upon role conditions.
By pure logical symmetry between role terms and concept terms, one can repeat
the same reasonings as in the precedeing Cases 1-5 and derive the same conditions
up to the inversion between role and concept terms.
For example, then both maximal conditions r1 u r2 v r3 and r1 u r2 v r4 are
impossible (cf. Case 1). If r3 has the maximal condition r1 u r2 v r3 , then an
inference schema similar to those derived in Case 2 are obtained. One gets Left
Role Additivity:
r1 u r2
v r3; r1 v r4; r2 v r4; c1 t c2 v c3; c1 t c2 v c4;
k+n+1p+q k
*
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
We would get also the Weak Right Role Additivity schema for roles and Partial
Right Role Additivity if role disjunction were allowed.
Case 8: Both c1 and c2 have maximal conditions ci v c3 u c4. c1 v c3 u c4
and c2 v c3 u c4 is equivalent to the list of conditions c1 v c3 , c1 v c4, c2 v
c3 , c2 v c4 . This is equivalent to c1 t c2 v c3 and c1 t c2 v c4 . This is Case 7.
Proofs with _-left Rules. We now consider the more general case where the
concept and role terms are possibly in disjunctive forms. This does not reduce to
the precedent case, because the _-left rules give a collection of sequents of the
form r to prove, instead of a single one, therefore offering more proof combination
A.2 Formulae with exactly two eigen-parameter-creating terms
105
possibilities. The only possible change from the disjunction-free proofs comes
from the application of the Constant Separation Principle to concept terms or
role terms (if there is role disjunction). But, by the symmetries between concept
and role terms one can restrict at first to the case where one only has concept
disjunction. (Then rules for roles with disjunction will be obtained by applying
the logical symmetry as already shown above).
We here consider only the case where c1 = c1at c1b . There is no loss of
generality in doing so (by induction wrt to the number of disjunctive connectors
in the antecedents). In fact one shows here that the rules for formulae with
disjunctions are necessarily obtained by some algebraic combinations of the rules
for disjunction-free formulae. The combination operation can be run as often
as one “fixed point” is not obtained, i.e., as long as the set of proof conditions
is not stable. Due to the symmetry between roles and concepts, the algebraic
manipulations must also operate by inversion of role and concept conditions.
Elimination of the disjunctions yields a collection of sequents ( e ), quite similar
to , except from the antecedents which are now of the form (up to variable
renaming): e = 1; 2 , where
a; b1); :::; r2(a; bp); (c1a(b1); :::; c1a(be); c1b(be 1 ; :::; c1b(bp)); dif (B~ ))
and 0 e p. Then, for each sequent one can search for proof conditions in a
1
=
(r1(
+
similar way as in the disjunction-free case.
We define (E; F ) as being the partition of B defining the breakdown of c1
among its disjuncts c1a and c1b in a given sequent e : B = E [ F; E \ F = ;.
We will first consider the special cases in which e = 0 or e = p, and then the
inconsistency schemata for c1a and c1b .
Limit cases e = 0 or e = p. For these limit values of e, the sequents reduce
to the disjunction free case, with c1 = c1a or c1= c1b . We get locally conditions
coming from the previous cases, one for c1a in place of c1 , the other for c1b in place
of c1. This means that the global proof conditions are at least obtained by some
“algebraic”, consistent combinations of the local conditions for 0 and p. In some
sense, the global conditions belong to the algebraic closure of the set of conditions
for left disjunction free formulae. Let us examine what algebraic conditions are
possibly obtained. It is easy to see that the algebraic closure is strictly bigger than
the initial set.
1. exchanging the roles of c3 and c4 , gives the new proof conditions, called
Union/Intersection Schema:
c1a u c2 v c3 , c1a t c2 v c4 , c1b u c2 v c4 , c1b t c2 v c3
By summing up, one gets the following conditions:
A PROOFS FOR NUMERICAL OPERATORS
106
k + n + 1 p + q, c1 u c2 v c3 u c4 , c1 t c2 v c3 t c4
Similarly, applying Left Additivity with both c1 = c1a t c1b and r1 = r1a t r1b
gives the symmetrical proof conditions for Union/Intersection:
r1 u r2 v r3 u r4 , r1
t r2 v r3 t r4, c1 u c2 v c3 u c4, c1 t c2 v c3 t c4
2. The combination of two Partial Right Additivity rules, by exchanging the
roles of c3 and c4 , gives as new rule the Right Additivity Schema:
r1
v r3; r2 v r4; r1 v r4; r2 v r4; c1 v
c3 u c4 ; c2
v
c3 t c4;
k+n<p+q
*
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
This is obtained by adding
the concept conditions when the Left Additivity Schema is applied for
(c1 = c1a , c2): c1a 6v c3 , c1a v c4, c2 v c3 , c2 v c4 ;
the concept conditions when the Left additivity schema is applied for
(c1 = c1b , c2 ) and the roles of c3 and c4 are inverted: c1b 6v c4 , c1b v c3,
c2 v c3, c2 v c4
3. Assuming that r1 is also disjunctive, r1= r1a t r1b , the same reasoning as
in (1) applies simultaneously for c1 and r1 yielding the full Right Additivity
Schema with completely symmetrical conditions for roles and concepts
r1
v r3 u r4; r2 v r3 t r4; c1 v
c3 u c4; c2
k+n<p+q+1
v
c3 t c4 ;
*
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
4. If the same schemata are used, without exchanging the roles of c3 and c4,
then by left additivity of concept disjunction, one gets the same schema for
c1a t c1b as well.
5. Combination of Left Additivity and Inconsistency gives Left Additivity or
Union/Intersection. For Example:
n min(p; q); c1b u c2 v c4; c1b t c2 v c3 ; c1a v c3 ; k + 1 p
*
c1 u c2 v c3 u c4; c1 t c2 v c3 t c4
A.2 Formulae with exactly two eigen-parameter-creating terms
107
It is easy to see that all the combinations give one of the above rules except
for the case where two inconsistency schemata are combined for c1a and c1b .
Indeed, in all other cases, complete conditions for the concept and role terms are
obtained, i.e. complete classification information about all the role and concept
terms involved in the formula. It is then easy to check the satisfaction of the
remaining sequents for other values of e, 0 < e < p. This step is not worked out
here; it causes no particular problem and one checks easily that Right Additivity
and Union/Intersection are indeed valid inference rules.
But the combination of two different inconsistency schemata gives only conditions about c1. We exclude the case where similar inconsistency schemata are
obtained for c1a and c1b (if the inconsistency is between (c1a , c) and (c1b , c) for
the same c, then we get a simple inconsistency schema between c1 and c). If 0
is proved by inconsistency between c1a and c3 and p is proved by inconsistency
between c1b and c4, one gets the conditions: k+1 , n+1 p , r1 v r3 , r1 v r4 , c1a
v r3 , c1b v r4 but c1a6v c4, c1b6v c3. The resulting conditions do not characterize
c2 nor r2 . Here more reasoning is needed to obtain complete proof conditions. In
the following, we focus on that case. We will show that no new inference schema
but the precedent ones will be derived.
Proofs by Inconsistency Schemata for c1a and c1b. The initial conditions are
like: c1a v c3 , c1b v c4 , c1a 6v c4 , c1b 6v c3 , k+1 p , n+1 p. Then c1 v c3 t c4
is obtained as maximal condition for c1 (c1 6v c3 , c1 6v c4). We now consider the
sequents e , for arbitrary values of e, 0 < e < p.
We first begin by checking the provability conditions for the pairs (x; y ) in
B B . By the assumptions about c1a and c1b it is easy to see that in all the
sequents re where both ~x and ~y are provable, then there cannot be any couple
(xf ; ye ) ~
xx~y, xf 6= ye , xf 2 F n Dr , xe 2 F n Dr (otherwise the sequent
er ) c3(xf ); c4(ye ) would give one forbidden condition c1b v c3 or c1a v c4 ).
This means that either ~x D [ E [ Fr or ~y D [ F [ Er . (Er , Fr denote the
subsets of E and F whose elements are equated to some elements of D in re ).
For r = 0, one gets k + 1 q + e = p + q f or n + 1 q + f = p + q f .
Consequently, with an adequate choice of e = n or f = k , the necessary condition
k + n < p + q is always obtained.
Now let us examine what conditions come for c2 . They result from the sequents
of 4 with parameters (x; y ) such that one of x and y belongs to D.
1. We first prove that there must be necessarily conditions for c2 otherwise the
proof reduces to a simple inconsistency schema wrt p r1 :c1 . No necessary
condition about c2 will be derived iff in all the sequents re , one of ~x or ~y
is included in B [ Dr . Let us assume this is the case. Let us take r = 0
and k = e: in 0e, either ~x or ~y is included in B ; if it is ~x then ~x E
(because c1b 6v c3 ) but then k + 1 e; so ~y F (because c1a 6v c3) and
108
A PROOFS FOR NUMERICAL OPERATORS
n + 1 f = p e = p k. One gets k + n + 1 p and one inconsistency
schema between c1 and the two atmost terms. The same holds for role
conditions.
2. Let us assume that we do not get both conditions c2 v c3 and c2 v c4 . Let
us assume that c2 6v c3. Then both c2 6v c3 and c1b 6v c3 . This means that
there is no sequent re in which ~x is provable alone and ~x takes an element
of (D [ F ) n Er (i.e. whenever there is an element x in D [ F , it must be
equated to some element of E to keep consistent with the assumptions).
(a) If one gets the conjunctive condition c1 u c2 v c3 , then for r = 0
the proof must proceed with c4 (same reasoning as in Case 2); for
e = k; r = 0, to respect c4 6v c1a then ~y must be included in F [ D,
then n + 1 p k and one gets a simple inconsistency schema between
c1 and c3 t c4.
(b) Let us assume that c2 6v c3 and q n. If q n p 1; n = q +e0; e0 p q 1. Let us consider qp f , where f = n, D is embedded in F
(Dr = D = Fr F ), but there is no equality assumption between
E and D. Then if ~y is provable, it takes necessarily one element ye
outside Fr [ Dr . By c1a 6v c4, necessarily ~x must be provable and
necessarily included in E (to avoid the forbidden conditions about c3 ).
Then k + 1 p f; k + n < p. One gets again an inconsistency
schema between c1 and c3 t c4 .
(c) Let us assume that c2 6v c3 and n < q . One gets an inconsistency
schema between c3 and c2 unless r2 6v r4 . In that case, by symmetry
between roles and concepts in b2, one gets that r2 6v r4 is possible only
if k < q .
Then, by choosing e = k and making E included in D by suitable
equality assumptions, then ~x takes at least one element from F , hence
it cannot be provable alone. The proof necessarily proceeds by ~y in
2 or in 4 . If 2 is to be proved, to satisfy the concept condition in
ee ) c4(y) ~y must be included in D [ F ; to satisfy the role conditions
ee ) r4 (y), ~y must be included in F . Then again n + 1 p k!
If 4 is to be proved with ~x is provable in ee , then k + 1 = e + 1 q + f ,
i.e. p + 1 q . Then there is an element xf in F such that er ) c3 (xf )
is unprovable). The same sequents as before must be proved for all y ,
except possibly for xf if xf is the unique element of ~x \ ~y; As xf 2 F
once again one gets n + 1 f = p k .
To conlude this case, we always get both conditions c2 v c3 and c2 v c4.
By symmetry, the same reasoning holds for the role conditions. One finally
gets Right Additivity or Inconsistency.
A.2 Formulae with exactly two eigen-parameter-creating terms
109
Proofs with ^-right rules. The final case is when proofs require ^-right rules.
This means that the roles and concepts coming from the atmost terms can possibly be in conjunctive form. One can proceed with structural induction like in
Section A.1. Then the rules for conjunctive c3 or c4 will again be obtained by
combining the local proofs conditions obtained in above for each conjunct. One
can easily check that the combinations will not produce new rules (the set of
conditions already obtained is closed).
For example, if c3 = c1a t c1b, the combination of conditions given by Right
Additivity, one for (c1a, c4 ) and the other for (c1b , c4 ), by inversing the roles of c1
and c2, gives again the Union/Intersection rule.
Summary of the inference rules for general terms
We have proved one important constructive property, which states intuitively that
all the proof conditions for
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
are necessarily combinations of the proof conditions given by the three inference
rules obtained for the disjunction/conjunction free terms.
Proposition 11 The set of proof conditions for the subsumption formula,
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
contains the set of proof conditions obtained for terms without left disjunction and
right conjunction and is closed under the following operations:
1. combination of arbitrary proof conditions for the disjuncts of the terms
encapsulated in atleast operators;
2. combination of arbitrary proof conditions for the conjuncts of the terms
encapsulated in atmost operators;
3. inversion of role/ concept conditions.
We give below the most general rules, with symmetrical conditions for roles and
concepts. One easily sees Left Additivity and Right Additivity are subsumed by
Union/Intersection (49).
r1 u r2
v r3; r1 t r2 v r4; c1 u c2 v c3; c1 t c2 v c4;
k+n<p+q
*
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
A PROOFS FOR NUMERICAL OPERATORS
110
r1
r1 u r2
v r3 t r4; r2 v r3 u r4; c1 v
c3 t c4; c2
v
c3 u c4 ;
k+n<p+q
*
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
v r3 u r4; r1 t r2 v r3 t r4; c1 u c2 v
c3 u c4 ; c1 t c2
k+n<p+q
*
p r1:c1 u q r2:c2 u k r3:c3 u n r4:c4 v ?
v
c3 t c4 ;
(49)
Generalization to an arbitrary number of numerical restrictions
With Left Additivity and Right Additivity we get exactly the terminological approximations (lower and upper bounds) of the exact cardinality equations of Set
Theory:
card(X [ Y )
card(X ) + card(Y ) card(X \ Y )
When X is intuitively assimilated to the set fb : r(a; b) ^ c1(b)g and Y
fb : r(a; b) ^ c2(b)g:
=
to the set
1. Left Additivity gives the lower bound approximation:
card(X ) + card(Y ) card(X \ Y ) card(X [ Y )
p r:X u q r:Y u k r:X u Y v p + q k r:X t Y
2. Right Additivity gives the upper bound approximation:
card(X ) + card(Y ) card(X \ Y ) card(X [ Y )
p r:X u q r:Y u k r:X u Y v p + q k r:X t Y
By combining the Righ and Left Additivity rules, one can prove the general
subsumption formulae which represent the lower/upper bound approximations of
the general cardinality laws of Set Theory:
card([ii 1n Xi )
=
=
=
X
(
j 1;j 2;:::;jk
k
1)k+1 (card(\ii=
=1 (Xji )))
We illustrate this by showing the derivation of the law for a three-set union.
card(X1 [ X2 [ X3 )
card(X1 ) + card(X2 ) + card(X3 )
card(X1 \ X2 ) card(X2 \ X3 ) card(X3 \ X1 )
+card(X1 \ X2 \ X3 )
With X = c1 t c2 , Y = c3, and n = p + q k , one gets:
=
111
p r:c1 t c2 u q r:c3 u k r:c1 t c2 u c3 v n r:c1 t c2 t c3
One can replace the subterm p r:c1 t c2 , and p = u + v
w, by the subterm:
u r:c1 u v r:c2 u w r:c1 u c2
One can replace the subterm k r:c1 t c2 u c3 , and k = f + g h and the fact
that (c1 t c2 ) u c3 is equivalent to (c1 u c3) t (c2 u c3), by the subterm:
f r:c1 u c3 u g r:c2 u c3 u h r:c1 u c2 u c3
Finally we get the formula:
u r:c1 u v r:c2 u q r:c3 u w r:c1 u c2u
f r:c1 u c3 u g r:c2 u c3 u h r:c1 u c2 u c3 v n r:c1 t c2 t c3
with the numerical constraints: n = (u + v + q ) + h (w + f + g )
NB: lower bound estimates for odd number unions are counted positively and
upper bound estimates for even number unions are counted negatively. This is
quite consistent with the lower bound approximation of the exact set theoretical
law.
The same kind of reasoning would yield the upper bound approximation,
by inversing the roles of atmost and atleast, but keeping the same numerical
constraints. The generalization to the general rules for arbitrary unions is left to
the reader (easy induction).
Generalization to all terms
This is quite similar to the section with one single atleast term. 8r:c may only
contribute by weakening the subsumption relations c1 v c2 by c1 u c v c2 . Basically, any one of the previous rules can be weakened by the addition of all terms,
changing the basic identity axiom c1 t c2 v c1 t c2 into (c3 u c1a) t (c4 u c1b) v
c1 t c2 with c3 u c1a v c1 and c4 u c1b v c2.
B Survey of Inference Rules
In this section we list for each term-forming operator the inference rules in which
it occurs. Rules for atomic terms are listed under atomic concepts and atomic
roles.
B SURVEY OF INFERENCE RULES
112
all
>v c
c1 v c2 ; r2 v r1
c2 u c1 v ?; r1 v r2
> v c2 t c1; r1 v r2
c1 u c2 v c3 ; r3 v r1 ; r3 v r2
c1 u c2 v c3 ; r2 v r1 ; r2 v r3
c3 v c1 t c2 ; r2 v r1 ; r2 v r3
q
c2 u c1 v ?; r1 v r2
p; r1 v r2; r1 v r3; c1 u c3 v c2
q p; r1 v r2; r1 v r3;
c1 u c3 v c2
r1 v r2 ; , v r1r3 ; c1 v c2
n jS j; r2 v r1
c1 v c2
c1 t c2
v c3; r3 v r1 t r2
o1 :: 8r:c; o1 :: r:o2
o1 :: 8r:c; o2 :: : c
*
)
*
*
*
*
*
*
*
*
*
*
*
*
> v 8r:c
(10)
8r1:c1 v 8r2:c2
(12)
8 r2:c2 u 9 r1:c1 v ?
(14)
> v 9r2:c2 t 8r1:c1
(15)
8r1:c1 u 8r2:c2 v 8r3:c3 (16)
8r1:c1 u 9r2:c2 v 9r3:c3 (18)
8r3:c3 v 9r1:c1 t 8r2:c2 (19)
:9r:c =: 8r:: c
(27)
:
:8r:c = 9r:: c
(28)
:
1 r:c = :8r:: c
(32)
:
0 r:c = 8r:: c
(33)
n r1:c1 u 8r2:c2 v ?
(41)
p r1:c1 u 8r3:c3 v q r2:c2 (43)
*
*
*
*
*
*
*
*
*
*
q r2:c2 u 8r3:c3 v p r1:c1 (44)
n r1:c1 u n r2:> v 8r3:c2(45)
8r1:S _ v n r2:>
(55)
8>r:c1 v c2
(131)
r
> v 8?:c
(132)
8r1:c1 u 8r2:c2 v 8r3:c3 (133)
> v 8rjc:c
(134)
:
8r1.r2:c = 8r1:(8r2:c)
(135)
8r.r :c v c t 0 r:>
(136)
m r1:> u 8r1:(n r2:c)
v m n r1.r2:c
(147)
*
*
o2 :: c
o1 :: : (r:o2 )
(162)
(163)
and (Concepts)
c1
v
c1
c2 ; c1
v
v
c2
c3
*
*
*
)
c1 u c2 v c1 t c3
c1 u c3 v c2
c1 v c2 u c3
(2)
(3)
(5)
113
c1 v c2 ; c2 u c3 v
c2 u c1 v ?; r1 v
c1 u c2 v c3 ; r3 v r1; r3 v
c1 u c2 v c3 ; r2 v r1; r2 v
c4
r2
r2
r3
c1 u c2 v c3
c1 v c3 t c2
q < p; c1 v c2; r1 v r2
c2 u c1 v ?; r1 v r2
c2 u c1 v ?; r1 v r2
q < p; r1 v r2 ; r1 v r3 ;
c1 u c3 v c2
q p; r1 v r2 ; r1 v r3 ;
c1 u c3 v c2
q p; r1 v r2 ; r1 v r3 ;
c1 u c3 v c2
r1 v r2; , v r1 r3 ; c1 v c2
r v r1 ; r v r2
*
*
*
*
*
*
)
*
)
*
*
*
*
*
c1 u c3 v c4
8r2:c2 u 9r1:c1 v ?
8r1:c1 u 8r2:c2 v 8r3:c3
8r1:c1 u 9r2:c2 v 9r3:c3
cu:cv ?
c1 v c3 t : c2
c1 u : c2 v c3
:(c1 u c2) =: : c1 t : c2
:(c1 t c2) =: : c1 u : c2
q r2:c2 u p r1:c1 v ?
n r1:c1 u 8r2:c2 v ?
n r1:c1 u 8r2:c2 v ?
(7)
(14)
(16)
(18)
(20)
(22)
(23)
(25)
(26)
(38)
(41)
(41)
* q r2 :c2 u p r1 :c1 u 8r3 :c3 v ?
(42)
* p r1:c1 u 8r3 :c3 v q r2:c2
(43)
*
*
*
*
S1 \ S2 6= ; *
o :: c1 ; o :: c2 *
)
q r2:c2 u 8r3:c3 v p r1:c1
(44)
n r1:c1 u n r2:> v 8r3:c2
(45)
p r1:c1 u q r2:c2 v p + q r:c1 t c(47)
2
:
_
_
_
S u T = (S \ T )
(52)
r:S1 u :r:S2 v ?
(152)
o :: c1 u c2
(159)
and (Roles)
v r2; r1 v
r1 v
r1 v r2 ; r2 u r3 v
r3
r2
r4
v r3; r2 v
r4
v r2; r3 v r4; c1 v
c2
r1
r1
r1
*
) r1 v r2 u r3
* r1 u r3 v r2
* r1 u r3 v r4
* r u : r =: ?r
* r1 u r2 v r3 u r4
* r1 u r2 v r1 t r3
* : r1 u : r2 =: : (r1 t r2 )
* : r1 t : r2 =: : (r1 u r2 )
* c1 jr1 u r3 v c2 j(r2 u r4)
(58)
(64)
(65)
(66)
(67)
(68)
(69)
(72)
(87)
B SURVEY OF INFERENCE RULES
114
r1 v
r1 v
r1 v
r1 v
r2 ; r3 v
r2 ; r3 v
r2 ; r3 v
r2; c1 v
v c2
v c2
v c2
v c4
r1 v r2; c1 v c2 ; c3 v c4
r1 v r2; c1 v c2 ; c3 v c4
r1 v r2; c1 v c2 ; c3 v c4
r1 v r2 ; r3 v r4
r1 v r2 ; r3 v r4
c1 u c2 v ?
S3 = S1 \ S2
* r1 jc1 u r3 v (r2 u r4)jc2
* c1 j(r1 u r3 ) v c2 jr2 u r4
* (r1 u r3 )jc1 v r2jc2 u r4
* c3 j(c1 jr1 ) v (c2 u c4) jr2
* (c1 u c3 )jr1 v c4 j(c2 jr2)
* (r1 jc1 )jc3 v r2j(c2 u c4)
* r1 j(c1 u c3 ) v (r2 jc2 )jc4
* r1 u r3 v (r2 u r4 )
* (r1 u r3 ) v r2 u r4
* (>rjc1 ).(c2 j>r) =: ?r
* r1 :S1 u r2 :S2 v r1 u r2:S3
r4 ; c1
r4 ; c1
r4 ; c1
c2 ; c3
(88)
(89)
(90)
(96)
(97)
(99)
(100)
(107)
(108)
(121)
(150)
anyrole
> v c; >r v r
> v c; >r v r
>r v r
>r v r1; >r v r2
c1 v c2
*
*
*
*
*
*
*
*
*
*
*
*
*
*
rv
>r
>r =: r t : r
>r =: : ?r
: >r =: ?r
>r v c jr
>r v rjc
>r v r
>r v r1.r2
8>r:c1 v c2
> v 9>r:c
9?r:c v ?
> v n >r:c
0 >r:c v ?
> v >r :S
(57)
(61)
(62)
(73)
(78)
(79)
(105)
(118)
(131)
(137)
(138)
(141)
(145)
(148)
anything
>v c
> v c2 t c1; r1 v r2
* cv >
*
) > v 8r:c
* > v 9r2 :c2 t 8r1 :c1
* >v ct:c
(9)
(10)
(15)
(21)
115
q < p; c1 v c2 ; r1 v r2
r1 v r2 ; , v r1r3 ; c1 v c2
jS j n
n jS j; r2 v r1
> v c; r1 v r2
> v c; r1 v r2
> v c; >r v r
> v c; >r v r
*
*
*
*
*
*
*
*
cv ? *
*
*
*
*
*
*
*
> v q r2:c2 t p r1:c1
n r1:c1 u n r2:> v 8r3:c2
> v n r:S _
8r1:S _ v n r2:>
r1 v c jr2
r1 v r2 jc
>r v c jr
>r v rjc
> v n r:c
> v 8?r:c
> v 8rjc:c
> v 9>r:c
> v n >r:c
> v n ?r:c
> v >r :S
o :: >
(39)
(45)
(54)
(55)
(74)
(75)
(78)
(79)
(29)
(132)
(134)
(137)
(141)
(146)
(148)
(158)
atleast
cv
? * n r:c v ?
* 1 r:c =: 9r:c
* 1 r:c =: :8r:: c
m = n 1 * :n r:c =: m r:c
m = n + 1 * : n r:c =: m r:c
q p; c1 v c2; r1 v r2 * p r1:c1 v q r2 :c2
q < p; c1 v c2; r1 v r2 * q r2:c2 u p r1 :c1 v ?
q < p; c1 v c2; r1 v r2 * > v q r2 :c2 t p r1 :c1
q < p; r1 v r2 ; r1 v r3 ;
c1 u c3 v c2 * q r2:c2 u p r1 :c1 u 8r3 :c3 v ?
q p; r1 v r2 ; r1 v r3 ;
c1 u c3 v c2 * p r1:c1 u 8r3 :c3 v q r2 :c2
r1 v r2 ; , v r1 r3; c1 v c2 * n r1 :c1 u n r2:> v 8r3 :c2
n = jS j * r:S =: n r:S _
jS j < n * n r:S _ v ?
* > v n >r:c
(30)
(31)
(32)
(35)
(36)
(37)
(38)
(39)
(42)
(43)
(45)
(50)
(53)
(141)
B SURVEY OF INFERENCE RULES
116
o1 :: c; : : :; on :: c
* n ?r:c v ?
* n cjr:> v c
* m r1:(n r2:c) v n r1.r2 :c
* o :: n r:c
(142)
(143)
(144)
(165)
atmost
cv
? * > v n r:c
(29)
:
* 0 r:c = 8r:: c
(33)
:
* 0 r:c = :9r:c
(34)
:
m = n 1 * :n r:c = m r:c
(35)
:
m = n + 1 * : n r:c = m r:c
(36)
q < p; c1 v c2 ; r1 v r2 * q r2 :c2 u p r1:c1 v ?
(38)
q < p; c1 v c2 ; r1 v r2 * > v q r2:c2 t p r1 :c1
(39)
p q; c2 v c1 ; r2 v r1 * p r1 :c1 v q r2 :c2
(40)
q < p; r1 v r2; r1 v r3;
c1 u c3 v c2 * q r2 :c2 u p r1:c1 u 8r3 :c3 v ?
(42)
q p; r1 v r2; r1 v r3;
c1 u c3 v c2 * q r2 :c2 u 8r3:c3 v p r1 :c1
(44)
(45)
r1 v r2 ; , v r1r3 ; c1 v c2 * n r1:c1 u n r2 :> v 8r3 :c2
r v r1; r v r2 * p r1 :c1 u q r2:c2 v p + q r:c1 t c2 (47)
jS j n * > v n r:S _
(54)
n jS j; r2 v r1 * 8r1 :S _ v n r2 :>
(55)
c v 0 r1 :>; r2 v r1 * c jr2 v ?r
(82)
c v 0 r1 :>; r2 v r1 * r2 jc v ?r
(85)
c v 0 r1 :>; r2 v r1 * r2 jc v ?r
(86)
* 8r.r :c v c t 0 r:>
(136)
* 0 >r:c v ?
(145)
r
* > v n ?:c
(146)
* m r1 :> u 8r1 :(n r2 :c)
v m n r1.r2:c
(147)
Atomic Concepts
*
cv c
(1)
117
c1
v
c2
*
*
*
*
*
*
cv
>
?v c
:: c =: c
8>r:c1 v c2
9cjr:> v c
n cjr:> v
c
(9)
(8)
(24)
(131)
(139)
(143)
Atomic Roles
* r =: r
* r v >r
* r =: : : r
* ?r v r
* r1 v cjr2
* r1 v r2jc
* cjr1 v r2
* r1 jc v r2
* r1 v r2
* r1 v r2
* r1 v r2
* r1 v r2
(56)
(57)
(60)
(63)
(74)
(75)
(76)
(77)
(101)
(102)
(103)
(104)
* >r v r1 .r2
* r1.r2 v ?r
* r1.r2 v ?r
* (>rjc1 ).(c2 j>r) =: ?r
(118)
(119)
(120)
(121)
c1 j(r1 .r3) v c2 jr2 .r4
c1 jr1 .r3 v c2 j(r2.r4 )
c1 j(r1 .r3) v c2 jr2 .r4
r1jc1 .r3 v (r2 .r4)jc2
(r1 .r3 ) v r4 .r2
r1 .r3 v (r4 .r2 )
(122)
(123)
(124)
(125)
(126)
(127)
> v c; r1 v r2
> v c; r1 v r2
r1 v r2
r1 v r2
r1 v r2
r1 v r2
r1 v r2
r1 v r2
comp
r2 v
r1 v
r1 v
r1 v
r1 v
>r v r1; >r v r2
r1 v ?r
r2 v ?r
c1 u c2 v ?
r
c2 j>; c1 u c2 v ?;
r2 ; r3 v r4 ; c1 v c2
r2 ; r3 v r4 ; c1 v c2
r2 ; r3 v r4 ; c1 v c2
r2 ; r3 v r4 ; c1 v c2
r1 v r2 ; r3 v r4
r1 v r2 ; r3 v r4
*
*
*
*
*
*
B SURVEY OF INFERENCE RULES
118
r1
r1
v
v
r1
r2 ; r3
r2 ; r3
v r3; r2 v
v r4; r5 v
v r4; r5 v
r4
r6
r6
o 2S
o1 :: r1:o2 ; o2 :: r2 :o3
*
*
*
*
*
*
*
*
*
*
r1 .r2 v r3 .r4
(r1 .r3).r5 v r2 .(r4.r6 )
r1 .(r3.r5 ) v (r2.r4).r6
8r1.r2:c =: 8r1:(8r2:c)
8r.r :c v c t 0 r:>
9r1.r2:c =: 9r1:(9r2:c)
m r1:(n r2:c) v n r1.r2:c
m r1:> u 8r1:(n r2:c)
v m n r1.r2:c
r1 :r2 :S v 9r1:(r2:o)
o1 :: r1 :r2 :o3
(128)
(129)
(130)
(135)
(136)
(140)
(144)
(147)
(155)
(167)
domain
> v c; r1 v r2
r1 v r2
> v c; >r v r
> v c; >r v r
r v ?r
cv ?
c v 0 r1 :>; r2 v r1
r1 v r2; r3 v r4 ; c1 v c2
r1 v r2; r3 v r4 ; c1 v c2
r1 v r2; r3 v r4 ; c1 v c2
r1 v r2 ; c1 v c2
r1 v r2 ; c1 v c2
r1 v r2; c1 v c2 ; c3 v c4
r1 v r2; c1 v c2 ; c3 v c4
r1 v r2 ; c1 v c2
r1 v r2 ; c1 v c2
r1 v r2 ; c1 v c2
r1 v r2 ; c1 v c2
r1 v r2; r3 v r4 ; c1 v c2
r1 v r2; r3 v r4 ; c1 v c2
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
v cjr2
c jr1 v r2
>r v c jr
>r v rjc
c jr v ?
c jr v ?
c jr2 v ?r
c1 jr1 u r3 v c2 j(r2 u r4 )
c1 j(r1 u r3 ) v c2 jr2 u r4
c1 j(r1 t r3 ) v c2 jr2 t r4
c1 jr1 v : (: c2 j: r2 )
c1 jr1 v c2 jr2
c3 j(c1 jr1) v (c2 u c4 ) jr2
(c1 u c3) jr1 v c4 j(c2 jr2)
c1 jr1 v (r2 jc2 )
(r1 jc1 ) v c2 jr2
r1 jc1 v (c2 jr2 )
(c1 jr1 ) v r2 jc2
c1 j(r1.r3 ) v c2 jr2.r4
c1 jr1 .r3 v c2 j(r2 .r4 )
9cjr:> v c
r1
(74)
(76)
(78)
(79)
(80)
(81)
(82)
(87)
(89)
(91)
(93)
(95)
(96)
(97)
(113)
(114)
(115)
(116)
(122)
(123)
(139)
119
* n cjr:> v c
* cjr:S v c
* c u r:o v c jr:o
(143)
(153)
(154)
fills
n = jS j *
*
*
S3 = S1 \ S2 *
*
S1 \ S2 6= ; *
*
*
o 2S *
*
o1 :: 8r:c; o1 :: r:o2 *
o1 :: 8r:c; o2 :: : c *
o1 :: r:o2 ; o2 :: c *
o1 :: c; : : :; on :: c *
o1 :: r:o2 ; o2 :: c *
o1 :: r1 :o2; o2 :: r2 :o3 *
o1 :: r:o2 *
: n r:S _
> v >r :S
?r :S v ?
r1 :S1 u r2 :S2 v r1 u r2 :S3
:
(:r):o = : (r:o)
r:S1 u :r:S2 v ?
c jr:S v c
r:S
=
c u r:o v c jr:o
r1 :r2:S v 9r1 :(r2 :o)
r :o v 9r :(9r:(r :o))
o2 :: c
o1 :: : (r:o2)
o1 :: 9r:c
o :: n r:c
o1 :: rjc :o2
o1 :: r1:r2:o3
o2 :: r :o1
(50)
(148)
(149)
(150)
(151)
(152)
(153)
(154)
(155)
(156)
(162)
(163)
(164)
(165)
(166)
(167)
(168)
inv
cv
cv
0 r1:>; r2 v r1
0 r1 :>; r2 v r1
r1 v r2
r1 v r2
r1 v r2
r1 v r2
>r v r
r v ?r
r1 v r2 ; r3 v r4
* r2jc v ?r
* r2jc v ?r
* r1 v r2
* r1 v r2
* r1 v r2
* r1 v r2
* >r v r
* r v ?r
* r1 u r3 v
(r2 u r4 )
(85)
(86)
(101)
(102)
(103)
(104)
(105)
(106)
(107)
B SURVEY OF INFERENCE RULES
120
r1 v
r1 v
r1 v
r1 v
r1 v
r1 v
r1 v
r1
r1
v
v
r2 ; r3 v
r2 ; r3 v
r2 ; r3 v
r2 ; c1 v
r2 ; c1 v
r2 ; c1 v
r2 ; c1 v
r1 v
r2 ; r3 v
r2 ; r3 v
r4
r4
r4
c2
c2
c2
c2
r2
r4
r4
o1 :: r:o2
* (r1 u r3 ) v r2 u r4
* r1 t r3 v (r2 t r4 )
* (r1 t r3 ) v r2 t r4
* c1 jr1 v (r2 jc2 )
* (r1 jc1 ) v c2 jr2
* r1 jc1 v (c2 jr2 )
* (c1 jr1 ) v r2 jc2
* r1 v r2
* (r1 .r3) v r4 .r2
* r1 .r3 v (r4 .r2)
* 8r.r :c v c t 0 r:>
* r :o v 9r :(9r:(r :o))
* o2 :: r :o1
(108)
(109)
(110)
(113)
(114)
(115)
(116)
(117)
(126)
(127)
(136)
(156)
(168)
norole
r v ?r
cv ?
c v 0 r1 :>; r2 v r1
r v ?r
cv ?
c v 0 r1 :>; r2 v r1
c v 0 r1 :>; r2 v r1
r v ?r
r1 v ?r
r2 v ?r
c1 u c2 v ?
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
>r =: : ?r
?r v r
:
r u : r = ?r
: >r =: ?r
c jr v ?r
c jr v ?r
c jr2 v ?r
rjc v ?r
rjc v ?r
r2 jc v ?r
r2 jc v ?r
r v ?r
r1 .r2 v ?r
r1 .r2 v ?r
:
(>rjc1 ).(c2 j>r) = ?r
> v 8?r:c
9?r:c v ?
n ?r:c v ?
> v n ?r:c
(62)
(63)
(66)
(73)
(80)
(81)
(82)
(83)
(84)
(85)
(86)
(106)
(119)
(120)
(121)
(132)
(138)
(142)
(146)
121
* ?r :S v ?
(149)
not (Concepts)
*
*
c1 u c2 v c3 *
)
c1 v c3 t c2 *
)
*
*
*
*
*
*
m=n 1 *
m=n+1 *
*
o 62 S *
o1 :: 8r:c; o2 :: : c *
cu:cv ?
>v ct:c
c1 v c3 t : c2
c1 u : c2 v c3
:: c =: c
:(c1 u c2) =: : c1 t : c2
:(c1 t c2) =: : c1 u : c2
:8r:c =: 9r:: c
1 r:c =: :8r:: c
0 r:c =: :9r:c
:n r:c =: m r:c
: n r:c =: m r:c
:
(:r):o = : (r:o)
o :: : S _
o1 :: : (r:o2 )
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(28)
(32)
(34)
(35)
(36)
(151)
(161)
(163)
not (Roles)
r1
r1
v r2; c1 v c2
v r2; c1 v c2
r1 v r2
r1 v r2
*
*
*
*
*
*
*
*
*
*
*
: ::r
>r =: : ?r
:
r u : r = ?r
: r1 u : r2 =: : (r1 t r2)
: r1 t : r2 =: : (r1 u r2)
: >r =: ?r
r=
c1 jr1 v : (: c2 j: r2 )
r1 jc1 v : (: r2 j: c2 )
: r1 v (: r2 )
(: r1 ) v : r2
:
(:r):o = : (r:o)
(60)
(62)
(66)
(69)
(72)
(73)
(93)
(94)
(111)
(112)
(151)
B SURVEY OF INFERENCE RULES
122
nothing
*
cv ? *
)
c2 u c1 v ?; r1 v r2 *
*
cv ? *
cv ? *
q < p; c1 v c2; r1 v r2 *
c2 u c1 v ?; r1 v r2 *
q < p; r1 v r2 ; r1 v r3 ;
c1 u c3 v c2 *
jS j < n *
*
*
*
S1 \ S2 6= ; *
?v c
9r:c v ?
8 r2:c2 u 9 r1:c1 v ?
cu:cv ?
> v n r:c
n r:c v ?
q r2:c2 u p r1:c1 v ?
n r1:c1 u 8r2:c2 v ?
q r2:c2 u p r1:c1 u 8r3:c3 v ?
n r:S _ v ?
n ?r:c v ?
0 >r:c v ?
?r :S v ?
r:S1 u :r:S2 v ?
(8)
(11)
(14)
(20)
(29)
(30)
(38)
(41)
(42)
(53)
(142)
(145)
(149)
(152)
Objects
*
)
*
o :: c1 ; o :: c2 *
)
o2S *
o2
6 S *
o1 :: 8r:c; o1 :: r:o2 *
o1 :: 8r:c; o2 :: : c *
o1 :: r:o2 ; o2 :: c *
o :: r:fo1 ; : : : ; on g; o1 :: c; : : : ; on :: c *
o1 :: r:o2 ; o2 :: c *
o1 :: r1 :o2 ; o2 :: r2 :o3 *
o1 :: r:o2 *
o :: c1 ; o :: c2
o :: c1 u c2
o :: >
o :: c1 u c2
o :: S _
o :: : S _
o2 :: c
o1 :: : (r:o2 )
o1 :: 9r:c
o :: n r:c
o1 :: rjc :o2
o1 :: r1 :r2 :o3
o2 :: r :o1
(157)
(158)
(159)
(160)
(161)
(162)
(163)
(164)
(165)
(166)
(167)
(168)
oneof
n = jS j *
r:S
: n r:S _
=
(50)
123
ST *
*
jS j < n *
jS j n *
n jS j; r2 v r1 *
o2S *
o 62 S *
S_ v T _
S _ u T _ =: (S \ T )_
n r:S _ v ?
> v n r:S _
8r1:S _ v n r2:>
o :: S _
o :: : S _
(51)
(52)
(53)
(54)
(55)
(160)
(161)
or (Concepts)
c1 v c2
c2 v c1; c3 v c1
> v c2 t c1; r1 v r2
c3 v c1 t c2 ;
r3 v r1 ; r3 v r2
c3 v c1 t c2 ;
r2 v r1 ; r2 v r3
c1 u c2 v c3
c1 v c3 t c2
q < p; c1 v c2 ; r1 v
r v r1 ; r v
r2
r2
* c1 u c2 v c1 t c3
* c1 v c2 t c3
*
) c2 t c3 v c1
* > v 9r2:c2 t 8r1 :c1
(2)
(4)
(6)
(15)
* 9r3 :c3 v 9r1:c1 t 9r2 :c2
(17)
*
*
*
)
*
)
*
*
*
*
*
*
8r3:c3 v 9r1:c1 t 8r2:c2
(19)
>v ct:c
(21)
c1 v c3 t : c2
(22)
c1 u : c2 v c3
(23)
:
:(c1 u c2) = : c1 t : c2
(25)
:
:(c1 t c2) = : c1 u : c2
(26)
:
:9r:c = 8r:: c
(27)
> v q r2:c2 t p r1:c1
(39)
p r1:c1 u q r2:c2 v p + q r:c1 t c2 (47)
8r.r :c v c t 0 r:>
(136)
or (Roles)
v
r2
v r1; r3 v
r1
r1
r2
* r1 v r2 t r3
* >r =: r t : r
* r1 u r2 v r1 t r3
* : r1 u : r2 =: : (r1 t r2 )
* r2 t r3 v r1
(59)
(61)
(68)
(69)
(70)
B SURVEY OF INFERENCE RULES
124
r1
v r3 ; r2 v
r4
v r2; r3 v r4; c1 v c2
v r2; r3 v r4; c1 v c2
r1 v r2 ; r3 v r4
r1 v r2 ; r3 v r4
c1 t c2 v c3 ; r3 v r1 t r2
r1
r1
* r1 t r2 v r3 t r4
* : r1 t : r2 =: : (r1 u r2)
* c1 j(r1 t r3) v c2 jr2 t r4
* (r1 t r3 )jc1 v r2jc2 t r4
* r1 t r3 v (r2 t r4)
* (r1 t r3 ) v r2 t r4
* 8r1:c1 u 8r2 :c2 v 8r3 :c3
(71)
(72)
(91)
(92)
(109)
(110)
(133)
range
> v c; r1 v r2
r1 v r2
r v ?r
c v 0 r1 :>; r2 v r1
c v 0 r1 :>; r2 v r1
r1 v r2; r3 v r4 ; c1 v c2
r1 v r2; r3 v r4 ; c1 v c2
r1 v r2; r3 v r4 ; c1 v c2
r1 v r2 ; c1 v c2
r1 v r2 ; c1 v c2
r1 v r2; c1 v c2 ; c3 v c4
r1 v r2; c1 v c2 ; c3 v c4
r1 v r2 ; c1 v c2
r1 v r2 ; c1 v c2
r1 v r2 ; c1 v c2
r1 v r2 ; c1 v c2
c1 u c2 v ?
r1 v r2; r3 v r4 ; c1 v c2
r1 v r2; r3 v r4 ; c1 v c2
o1 :: r:o2; o2 :: c
* r1 v r2 jc
* r1 jc v r2
* rjc v ?
* r2 jc v ?r
* r2 jc v ?r
* r1 jc1 u r3 v (r2 u r4 )jc2
* (r1 u r3 )jc1 v r2jc2 u r4
* (r1 t r3 )jc1 v r2jc2 t r4
* r1 jc1 v : (: r2 j: c2 )
* r1 jc1 v r2 jc2
* (r1 jc1 )jc3 v r2 j(c2 u c4)
* r1j(c1 u c3 ) v (r2 jc2 )jc4
* c1 jr1 v (r2 jc2 )
* (r1 jc1 ) v c2 jr2
* r1 jc1 v (c2 jr2 )
* (c1 jr1 ) v r2 jc2
* (>rjc1 ).(c2 j>r) =: ?r
* c1 j(r1.r3 ) v c2 jr2.r4
* r1 jc1 .r3 v (r2 .r4 )jc2
* > v 8rjc:c
* o1 :: rjc:o2
(75)
(77)
(83)
(85)
(86)
(88)
(90)
(92)
(94)
(98)
(99)
(100)
(113)
(114)
(115)
(116)
(121)
(124)
(125)
(134)
(166)
some
cv
? *
) 9r:c v ?
(11)
c1 v c2 ; r1
c2 u c1 v ?; r1
> v c2 t c1; r1
c3 v c1 t c2; r3 v r1 ; r3
c1 u c2 v c3; r2 v r1 ; r2
c3 v c1 t c2; r2 v r1 ; r2
v
v
v
v
v
v
*
*
*
*
*
*
*
*
*
*
*
*
*
*
o 2S *
*
o1 :: r:o2 ; o2 :: c *
r2
r2
r2
r2
r3
r3
125
9r1:c1 v 9r2:c2
8 r2:c2 u 9 r1:c1 v ?
> v 9r2:c2 t 8r1:c1
9r3:c3 v 9r1:c1 t 9r2:c2
8r1:c1 u 9r2:c2 v 9r3:c3
8r3:c3 v 9r1:c1 t 8r2:c2
:9r:c =: 8r:: c
:8r:c =: 9r:: c
1 r:c =: 9r:c
0 r:c =: :9r:c
> v 9>r:c
9?r:c v ?
9cjr:> v c
9r1.r2:c =: 9r1:(9r2:c)
r1 :r2 :S v 9r1:(r2 :o)
r :o v 9r :(9r:(r :o))
o1 :: 9r:c
(13)
(14)
(15)
(17)
(18)
(19)
(27)
(28)
(31)
(34)
(137)
(138)
(139)
(140)
(155)
(156)
(164)