appeared in: Articial Intelligence, 84, 1996, pp. 209{255.
Limited Reasoning in First-Order
Knowledge Bases with Full Introspection
Gerhard Lakemeyer
Institute of Computer Science III
University of Bonn
Romerstrae 164
D-53117 Bonn, Germany
[email protected]
Abstract
A fundamental problem in knowledge representation is that reasoning, if based on
classical logic, is inherently intractable or even undecidable. A principled approach
to specifying more ecient inference mechanisms is to use weaker logics with nonstandard model theories.
In this paper we extend earlier work on a logic of belief for decidable deductive
rst-order reasoning by adding the ability to introspect. The new logic allows us to
formally specify the beliefs of deductively limited yet fully introspective rst-order
knowledge bases. The complexity of reasoning in this framework reduces to the
complexity of a special validity problem of the logic and we obtain various tractability/decidability results. To demonstrate the usefulness of the logic for knowledge
representation purposes we show how it can be applied to the specication of routines to query and update a knowledge base. Other interesting aspects of the logic
include distinctions between knowing that and knowing who by way of quantifying-in
and a very tight coupling between the limited beliefs of a knowledge base and those
that follow under the assumption of perfect logical reasoning.
1 Introduction
Knowledge representation systems usually need to represent incomplete and thus highly
complex information. As a result, reasoning about this kind of information is computationally intractable if based on classical logic. The following slight variation of an example by
Moore [Moo82] illustrates a major source of the complexity of reasoning about incomplete
knowledge.
1
Suppose there are three blocks A, B, and C such that B is next to A and C is
next to B. Furthermore let the color of A be green and the color of C be blue.
Block B is hidden under a blanket and hence its color is not visible.
While not immediately apparent, it follows that there must be a green block next to a
non-green block. The reason why it takes (a little) thinking to arrive at this conclusion is
because it takes reasoning by cases in order to arrive at it. In particular, a reasoner has to
consider two cases, one where block B is green and another where B is not green.
Reasoning mechanisms that allow for unrestricted use of case analysis are bound to
be intractable or even undecidable. In domains like knowledge-based robots operating
in a dynamic environment we are therefore faced with a serious dilemma. On the one
hand, a robot needs to access vital information contained in its KB. On the other hand, a
robot has to act in real time and cannot wait indenitely for the KB to produce answers
to queries. As a compromise, one could imagine that the KB gives up on computing all
the logical consequences of its stored information and provides instead a limited inference
service, which guarantees timely responses which are correct but not necessarily complete
compared to full inference. For example, suppose a robot is faced with the above blocksworld scene and is asked to make sure that there is a green block next to a non-green
one. One way for the robot to satisfy this request is, of course, to use a theorem prover to
determine that no action needs to be taken. Another way is to take a quick but shallow
look at the information available to determine if the request has already been satised. If
that is inconclusive, the robot might simply decide to move A next to C or to remove the
blanket to determine B's color.
What should a limited inference mechanism look like? The obvious solution would be
to use a complete inference engine such as a theorem prover and simply use a time-out
or some other ad-hoc method to constrain it. While this can certainly produce a correct
and ecient reasoner, there is a serious drawback in that we have virtually no control over
what kinds of answers the reasoner might produce. In eect, we cannot tell by looking at
the contents of the KB what the system actually knows. Such a knowledge representation
system, whose functionality is largely unknown, seems highly undesirable.
In contrast to these ad-hoc methods, there have been a number of approaches that
address the problem in a more principled way [Lev84b, Fri86, Pat87, Lak94]. The main idea
is to employ nonstandard logics which specify weaker yet computationally more ecient
deductive inference systems. The clear advantage of these approaches is that the inference
mechanisms are well understood, that is, there are sound and complete algorithms with
respect to the underlying nonstandard semantics.
It has long been recognized that agents should be able to introspect on their own
ignorance. In the above blocks-world scenario, the robot who uses only shallow reasoning
needs to realize that he does not know that there is a green block next to a non-green one
in order to initiate the appropriate actions. A more subtle use of introspection is revealed
if we make the robot's task slightly more complicated and ask him to ensure not only that
a green block is next to a non-green block but also that he knows which green block it is.
In this case, even a perfect logical reasoner will need to take action. For he knows only
that there is some green block adjacent to a non-green one but not which one it is (it could
be either A or B). Again introspection is necessary for the robot to realize that.
2
Previous approaches to limited reasoning have neglected introspection. A goal of this
paper is to demonstrate that introspection can be added to a formal model of limited
reasoning in a principled way without losing the computational advantages of the nonintrospective model. We approach the problem by proposing a logic of belief based on
earlier work [Lak94], which allows us to formally specify the beliefs of deductively limited
yet fully introspective rst-order knowledge bases. The complexity of reasoning in this
framework reduces to the complexity of a special validity problem of the logic and we
obtain various tractability/decidability results. To demonstrate the usefulness of the logic
for knowledge representation purposes we show how it can be applied to the specication
of routines to query and update a knowledge base. Other interesting aspects of the logic
include distinctions between knowing that and knowing who and a very tight coupling
between the limited beliefs of a knowledge base and those that follow under the assumption
of perfect logical reasoning.
2 The Approach
Our point of departure is a limited deductive reasoner as presented in [Lak94]. There we
make explicit use of the idea that reasoning can be viewed as the problem of computing
the beliefs of a knowledge base. Belief is modeled using a rst-order modal logic called
BLq and the beliefs of a KB are simply those that follow logically in BLq from believing
the sentences in the KB. Formally, if B denotes that is believed, the beliefs of a KB
are f j j=BKB Bg. The complexity of reasoning then reduces to the complexity of
computing belief implications, that is, the validity of sentences of the form BKB B. It
was shown that in our logic belief implication is decidable and in many cases tractable. The
result was obtained by showing that belief implication is equivalent to a decidable form of
rst-order entailment proposed by Patel-Schneider [Pat87]. In a nutshell, decidability is
obtained by ruling out certain forms of reasoning by cases. For example, neither B[(p _
:q) ^ (q _ r)] B(p _ r) nor B[P(a) _ P(b)] B9xP(x) are valid in BLq .
2.1 Introspection
The above results were obtained under the assumption that beliefs are not nested. Thus we
had no way in BLq to even talk about introspection within the logic. The goal of this paper
is to extend the logic in such a way that we are able to specify introspective knowledge
bases with good computational properties.
At rst, one might think that considering nested beliefs (with an appropriate semantics)
is all we need to do. To show that this is not quite sucient, assume, for the sake of
argument, that we already have a belief logic with arbitrarily nested beliefs and a semantics
that yields perfect introspection, that is, every belief is itself believed (j=B BB)
and for every sentence an agent does not believe, he believes that he does not believe it
(j=:B B:B). Now consider KB = p _ q. Certainly, an agent with that KB should not
believe p and, hence, by introspection, believe that he does not believe p. Note, however,
that B:Bp does not follow from BKB, that is, belief implications are not able to capture
introspective knowledge bases. The problem is that the assumption BKB is much too weak
3
for the desired conclusion because BKB only asserts that at least KB is believed, thus not
ruling out that p may be believed as well. What we need is an assumption that KB is, in a
sense, all that is believed, which would then justify the conclusion that p is not believed.
Following Levesque [Lev90], we therefore introduce a new operator O, where O is read
as is all that is believed or is only-believed. Given an appropriate semantics of O, we
are then able to prove that O(p _ q) :Bp and hence O(p _ q) B:Bp are valid in our
extended logic.
Reasoning in this framework can be thought of as determining which beliefs follow from
only-believing a given KB. The complexity of reasoning thus reduces to the complexity of
determining the validity of sentences of the form OKB B.
In our complexity analysis we will focus on knowledge bases which do not themselves
contain modal operators. What makes these KB's particularly appealing from a knowledge
representation point of view is the fact that they determine complete and fully introspective
epistemic states. For example, for every sentence , either B or :B follows from O(p_q).
2.2 Quantifying-in, Standard Names, and Equality
While the initial logic BLq does not address nested beliefs, it does address quantifying-in,
which refers to the ability to use variables within a belief that are bound outside. This allows us to distinguish between knowing that and knowing who. For example, assume a detective knows that the murderer is the driver of the red car. While the detective then certainly
knows that somebody is the murderer [j=BMurderer(driver(red car)) B9xMurderer(x)],
he will not necessarily know who the murderer is [j6=BMurderer(driver(red car)) 9xBMurderer(x)].
Semantically, this distinction is made possible in BLq by oering both rigid and non-rigid
designators. In a nutshell, non-rigid designators are the usual terms of rst-order logic like
driver(red car), whose denotation may change with dierent interpretations. Rigid designators, on the other hand, which we also call standard names, have xed interpretations.
In other words, they denote the same individual everywhere. Knowing who someone is
then modeled as knowing the individual's standard name.
With introspection and only-believing, the detective himself will now be able to reect on these distinctions and know that the murderer, whoever it is, is not known to
him because the driver of the red car is a non-rigid designator. Formally, let KB =
Murderer(driver(red car)). Then
j=OKB B(9xMurderer(x) ^ :BMurderer(x)):
Quantifying-in is only of limited use unless we also have a distinguished equality predicate
at our disposal. This way we are able to directly relate non-rigid and rigid designators.
Suppose Eddie Nogood is a standard name and assume our detective has determined that
Eddie is the murderer. Using equality, we are easily able to incorporate this information
into the detective's KB. Furthermore, it then follows that the detective knows who the
murderer is. Formally, let KB0 = KB ^ driver(red car) = Eddie Nogood. Then
j=OKB0 B(9xMurderer(x) ^ BMurderer(x)):
(1)
As will be discussed in detail later on, equality is also very useful when evaluating queries
with nested beliefs. In essence, it allows us to replace belief expressions by equality ex4
pressions, thereby reducing queries with nested beliefs to those without. For example, in
order to evaluate the validity of (1), BMurderer(x) is rst replaced by an equality expression that characterizes all the individuals of whom the detective knows that they are the
murderers, which in this case is simply x = Eddie Nogood. Hence (1) reduces to
j=OKB0 B(9xMurderer(x) ^ x = Eddie Nogood):
(2)
2.3 Explicit versus Implicit Belief
In [Lev84b], Levesque introduced the notions of explicit and implicit belief. From a KR
point of view, the explicit beliefs are those that follow from an agent's KB when we take
into account his resource limitations. The explicit beliefs are therefore those which we have
been talking about all along and which will be the main focus of this paper. The implicit
beliefs, on the other hand, can be thought of as those that follow from the KB without
any resource limitations. For example, implicit beliefs are closed under (classical) logical
implication.
To account for implicit belief, we add a new modal operator L to our logic, which
is essentially a weak S5 operator. We make this addition mainly for two reasons. For
one, it allows a direct comparison between our new model of belief and a more traditional
account. For another, our semantics of explicit and implicit belief exhibits a much tighter
connection between the two notions than other approaches such as [Lev84b, FH85]. In
particular, we will show that for a given interpretation all implicit beliefs follow logically
from the explicit beliefs. This narrow view seems justied from a KR point of view, where
all the beliefs of the agent should be grounded in his KB.
It seems plausible to allow an arbitrary mixing of L with the other modal operators
B and O. For example, consider the sentence BLp, where the agent is said to explicitly
believe that he implicitly believes p. One way to make sense of explicitly believing implicit
beliefs is to treat Ls inside Bs as if they were themselves Bs. That way, one could model
agents that reason about implicit beliefs the same way as they reason about explicit beliefs.
However, in favor of a simpler semantics we ignore this issue altogether by restricting the
language so that the L operator may occur only outside the scope of a B or O.
1
2
Related Work
As already mentioned, this work builds on previous work on limited deductive reasoning [Lak94], which also discusses related work on limited deduction at length. Introspection, which is the focus of this paper, has received a lot of attention in KR, especially with
regards to Moore's autoepistemic logic (e.g. [Moo85, Kon88, MT89, Lev90]). All of these
approaches work with implicit belief only, thus do not address resource limitations.
3
See [HM92] for an introduction to the dierent modal logics such as weak S5.
See [FH85] for a logic that deals with implicitly believing explicit beliefs.
A notable exception are Donini et al. [DLNNS92], who characterize a decidable form of rst-order
introspective reasoning. The price they pay, however, is an underlying language with rather limited
expressiveness.
1
2
3
5
Konolige [Kon85] seems to be the rst to address the issue of modeling introspection
under resource limitations. However, rather than proposing an actual instance of a computationally attractive reasoner, he presents a general framework in which one can be formalized. Elgot-Drapkin and Perlis [EP90, Elg91] attack the issue of reasoning under resource
limitations including introspection by explicitly modeling proof-steps. In [Kon85], Konolige also questions whether quantifying-in is compatible with nice computational behavior.
His argument is that, while introspection in monadic predicate calculus is decidable, it becomes undecidable if we add quantifying-in. In this paper, we are able to at least partially
absolve quantifying-in from being the computational villain.
Except for the notion of limited belief, our logic shares many features with Levesque's
logic presented in [Lev90], which in turn is based on [Lev84a]. In particular, we extend
Levesque's notion of implicitly only-believing to one of explicitly only-believing. Our treatment of implicit belief and quantifying-in are essentially like Levesque's. Finally, this paper
is based on and expands on the author's Ph.D. thesis [Lak92c]. Various results of this paper
have also appeared, in a preliminary form, in [LL88, Lak90, Lak91a].
Paper Outline
The rest of the paper is organized as follows. In Section 3, we introduce the logic OBLIQUE ,
which denes the formal semantics of explicit belief, implicit belief and only-believing. In
Section 4, we take a closer look at the connection between explicit and implicit belief in
this logic. Next we concentrate on a class of sentences which, when only-believed, can be
said to uniquely represent an epistemic state of an agent, which is of particular interest
from a knowledge representation point of view. In Section 6, we focus on the problem
of computing the explicit beliefs represented by a knowledge base. Finally, we apply our
results to the specication of routines for querying and updating a knowledge base and
provide an extended example.
3 The Logic OBLIQUE 4
3.1 The Language
Denition 1 The Language OL
Primitives
The primitives of the language consist of countably innite sets of variables, standard
names, function and predicate symbols (of every arity) including the equality predicate
=. We denote the standard names by N . The function symbols are partitioned
into two disjoint sets FREG and FSK, each of which contains innitely many function
symbols of every arity.
Thanks to Hector Levesque, who suggested that name to me. It may be read as \Only Belief Logic
with Quantiers and Equality."
4
6
Terms
A term is either a variable, a standard name, or a function symbol whose arguments
are themselves terms.
Closed Terms
A closed term is a term not containing any variables.
Primitive Terms
A primitive term is either a constant or a function symbol whose arguments are
standard names.
F-Terms: An f-term is a term other than a variable followed by the symbol 4 . (The
meaning of f-terms and that of extended terms below will become clear once we discuss
the semantics of the logic in Section 3.2 .)
Extended Terms: An extended term is either a term, an f-term, or of the form f (t ; : : : ; tk),
1
where f is a k-ary function symbol and the ti are extended terms. (In other words,
f-terms may not occur nested within an extended term.)5
Atomic Formulas: Atomic formulas (or atoms) are predicate symbols with extended terms
as arguments.
Primitive Formulas
A primitive formula is an atom whose arguments are standard names.
Formulas A formula is either an atom or can be obtained by the usual rules for :, _, 9,
L, B, and O, provided that no L occurs within the scope of a B or O. For technical
reasons, we require that no variable is bound more than once within the scope of a
modal operator.
Ordinary Formulas
Ordinary formulas are formulas without any occurrences of f-terms.
Basic Formulas
Basic formulas are formulas that do not contain any occurrences of L or O.
Objective Formulas
Objective formulas are formulas that contain no modal operators.
Sentences
Sentences are formulas without free variables.
Notation: Other logical connectives like ^, _, , , and 8 are dened in the usual
way. In order to distinguish standard names from ordinary constants syntactically, we
denote them as #1; #2; #3; : : : We use inx notation for the equality predicate and write
t 6= t instead of :(t = t ). true and false are used as shorthand for (#1 = #1)
1
2
1
2
For example, ( )4 and ( 4 ( 4)) are f-terms. ( 4 ), where is a variable, and ( 4 ( 4)4)
are not.
5
f a
g a; b
;h a
f x
7
x
g a; b
;h a
and (#1 6= #1), respectively, with the obvious intended meaning. Sequences of terms or
variables are sometimes written in vector notation. E.g., a sequence of variables hx ; : : :; xk i
is abbreviated as ~x. Also, 9~x stands for 9x : : : 9xk . If a formula contains the free
variables x ; : : :; xk , [x =t ; : : :; xk =tk ] denotes with every occurrence of xi replaced by
ti. If it is clear from the context which variables are being replaced, we sometimes simply
write [t ; : : :; tk ].
1
1
1
1
1
1
3.2 The Semantics of Basic Sentences
In our model-theoretic characterization of OBLIQUE , we rst turn to the semantics of basic
sentences only, that is, sentences without occurrences of O's and L's. Since the semantics of
basic sentences is a direct extension of the semantics of the logic BLq of [Lak94], we begin
by describing informally the main ideas underlying BLq and then discuss the necessary
modications to account for nested beliefs and equality.
The semantics of BLq combines ideas from relevance logic [AB75], in particular, a
fragment of relevance logic called tautological entailment [Dun76], with a possible-world
style interpretation of belief [Kri63, Hin62, Hin71].
From possible-world semantics BLq borrows the idea that an agent in a given state of
aairs is able to imagine (or access) other states of aairs, which determine his epistemic
state. Except for a special treatment of existential quantication described below, the
agent's beliefs are simply those sentences that hold in all those accessible states of aairs.
In classical possible-world semantics, the states of aairs are called worlds and each
world is essentially a rst-order structure as we know it from classical logic. In contrast,
the states of aairs in BLq , which are the same in OBLIQUE , are more general rstorder structures, which we call situations. Instead of assigning either true or false to a
sentence, situations assign independent true- and false-support, an idea borrowed from
tautological entailment of relevance logic. In other words, situations are four-valued in
that a sentence at a situation may have true-support, false-support, both, or neither. As
was already shown in [Lev84b], belief based on four-valued situations is no longer closed
under modus ponens. While this presents already a substantial departure from classical
logics of belief, substituting worlds by situations alone does not result in a decidable form
of belief implication, which was one of the goals of BLq . For that purpose, existential
quantication within belief is restricted in an intuitionistic fashion. In ordinary possibleworld semantics, a sentence 9xP(x) is believed just in case in every accessible world there is
an individual such that P holds for that individual. In BLq we require the agent in addition
to know a term t such that P(t) holds in all accessible situations. Note that in both cases
the individuals that satisfy P may vary among the accessible situations. However, in BLq
the agent also needs to know a common description like the driver of the red car. As a
result of this modication, believing P(a) _ P(b) no longer implies believing 9xP(x) in BLq .
If it were not for quantifying-in, the extension from non-nested beliefs in BLq to nested
beliefs in OBLIQUE would be straight-forward. In fact, without changing the semantics
of BLq at all, we obtain a model of nested beliefs with perfect introspection simply because
the accessible situations are the same for all situations in a given model. To understand
6
6
This corresponds to the logic weak S5 of classical possible-world semantics(see, for example, [HM92]).
8
the complication introduced by quantifying-in, let us go back to the detective example from
Section 2.2. In the semantics for non-nested beliefs [Lak94], the detective's belief that someone is the murderer [ B9xMurderer(x) ] is satised just in case the detective imagines a set
of situations such that there is a term like driver(red car) and Murderer(driver(red car))
holds in all situations. Now we want to express, in addition, that the detective does not
know who the murderer is, expressed as
B(9xMurderer(x) ^ :BMurderer(x)):
(3)
If we just substitute driver(red car) for x as before, we obtain that
Murderer(driver(red car)) ^ :BMurderer(driver(red car))
holds in all situations the detective thinks possible, which leads to a contradiction because
of perfect introspection. In order to x this problem, we need a better characterization
of knowing or not knowing who someone is. In our example, it simply means that the
detective does not know the identity of the driver of the red car. We can formalize this idea
by saying that the detective does not know of the denotation of the term driver(red car)
that he or she is the murderer. In order to convey this distinction between a term and its
denotation we use f-terms, which were already introduced syntactically on page 7. Thus,
when substituting driver(red car) for x in (3), we use the f-term driver(red car)4 inside
the modal context and obtain that
Murderer(driver(red car)) ^ :BMurderer(driver(red car)4 )
holds in all situations. More specically, in every situation s which the detective imagines Murderer(driver(red car)) holds, yet it is not known of the individual denoted by
driver(red car) at s whether he or she is the murderer.
We begin our formal investigations with the denition of situations, which form the
core of the model theory. Situations are rst-order structures with a universe of discourse,
a denotation function mapping terms into the universe of discourse and two sets T and F
of primitive formulas. T contains primitive formulas which are considered to have truesupport and F contains those considered to have false-support. The true- and false-support
of arbitrary objective sentences is then dened in the usual recursive fashion, keeping in
mind that there need to be separate rules for true- and false-support. If the sets T and F
partition the primitive formulas, we obtain the equivalent of a classical two-valued world.
However, we allow the two sets to overlap arbitrarily and not every primitive formula needs
to be in either T or F , thus giving us a four-valued semantics in general. The universe
of discourse is xed for all situations and is simply the set of all standard names. This
choice allows a proper treatment of quantifying-in and also simplies the formalism, since
quantication can be understood substitutionally and situations only need to deal with
primitive formulas in the denition of true- and false-support.
Denition 2 Denotation Function
A denotation function d is a mapping from closed extended terms into the standard
7
A more detailed discussion of using standard names as the universe of discourse can be found
in [Lev84a].
7
9
names such that the following three conditions hold:
1: d(n) = n if n 2 N
2: d(f (t ; : : : ; tk)) = d(f (d(t ); : : : ; d(tk ))) for terms f (t ; : : :; tk )
3: d(t4 ) = d(t) for f-terms t4.
Note that denotation functions do not treat f-terms in any special way. As will be seen
later on, f-terms receive their special status only in connection with the interpretation of
the modal operators.
Denition 3 First-Order Situations
A situation s is a triple s = < T; F; d >,
where T and F are subsets of the set of primitive formulas and d is a denotation function.
T and F can be arbitrary sets of primitive formulas except for equality, which has a
xed interpretation.
1: (n = m) 2 T i n and m are identical standard names.
2: (n = m) 2 F i n and m are distinct standard names.
Worlds are situations with a classical 2-valued interpretation of primitive formulas.
Denition 4 Worlds
A situation w = < Tw ; Fw ; dw > is called a world, i
1
1
1
(~n) 2 Tw () P(~n) 62 Fw for all primitive formulas P(~n).
P
The following denitions (5{11) all have to do with the interpretation of B. The reader
may want to skip them on a rst reading and consult them only when necessary.
Denition 5 Objective Level of a Formula
A primitive or formula contained in a formula occurs at the objective level of if it
does not occur within the scope of a modal operator.
Denition 6 Existentially Quantied Variables
Let be a formula in OL. Let x be a variable that is bound at the objective level of some
formula such that either = or B , O , or L is a subformula of .
x is said to be existentially (universally) quantied in i x is bound within in the
scope of an even (odd) number of :-operators in .
In other words, a variable is existentially quantied if it occurs in an even number of
negation signs relative to the nearest enclosing B, O, or L (if any). For example, in
9x:B9yP(x; y), both x and y are considered existentially quantied.
Denition 7 Admissible Terms
Let be a formula and x existentially quantied in . A term t is said to be an admissible
substitution for x with respect to i every variable y in t is universally quantied in and x occurs within the scope of y.
If the context is clear, we often say t is admissible for x or t is admissible.
10
Note that an admissible term for x may contain at most those variables that would appear
if we were to Skolemize x in the usual way.
Example 3.1 Admissible Terms
Let = 9x8y9zP(x;y; z) ^ BQ(x; z). Then only closed terms are admissible terms for x.
The admissible terms for z are precisely those terms that contain at most y as a variable.
For example, f (a), where a is a constant, and h(y; y) are admissible. x,z, and g(u), where
u is a new variable, are not admissible for z.
8
Denition 8 Let be a sentence and let ~x = hx ; : : : ; xk i be a sequence of the existentially
quantied variables bound at the objective level of . @ denotes with all 9xi removed.
Example 3.2 Let = 9w8x9yP(w;x; y) ^ B9zQ(z).
Then @ = 8xP(w; x; y) ^ B9zQ(z).
1
Note that existential quantiers within modalities are left untouched.
Denition 9 A formula is existential-free i contains no existential (universal)
quantiers at the objective level within the scope of an even (odd) number of :-operators.
Example 3.3 Existential-free
Examples of existential-free formulas are :9xP(x) and 8xP(x) ^ B9yQ(x; y). On the other
hand, 8xP(x) ^ 9yQ(x; y) is not existential-free. Note that only the objective level of a
formula matters. Note also that @ is existential-free for every .
Denition 10 Let be a formula with free variables ~x = hx ; : : :; xk i. ( may contain
other free variables as well.) Let ~t = ht ; : : : ; tk i be a sequence of terms.
1
1
[~x=~t] is with every occurrence of xi at the objective level replaced by ti and every occurrence of xi inside the scope of a modal operator replaced by ti 4 if ti is not a variable and
by ti otherwise.
9
Example 3.4
Let a and b be constants and let = P(x ) ^ BQ(x ; x ; x ).
Then [ x =a; x =b; x =#27]] = P(a) ^ BQ(a4; b4; #27).
1
1
2
1
2
3
3
Note the dierence to [x =a; x =b; x =#27] = P(a) ^ BQ(a; b; #27), that is, [: : :] indi1
2
3
cates regular substitutions, while [ : : :] indicates that substitutions within modalities use
f-terms.
Denition 11 Let be a sentence and s = hT; F; di a situation.
s is obtained from by replacing every occurrence of t4 by the standard name d(t), if t
is closed, and by t otherwise.
As a matter of fact, these restrictions on admissible terms are more convenient than essential. What
really matters about admissible terms is that they do not introduce new free variables.
9 Variables are exceptions because they cannot be made into f-terms, i.e., 4 is not an f-term.
8
xi
11
Example 3.5 Let = P(a4) ^ 8xB(Q(f (x)4) _ R(a4)) and s =< T; F; d > such that
d(a) = #1. Then
s = P(#1) ^ 8xB(Q(f (x)) _ R(#1))
The true- and false-support for basic sentences can now be dened. Let s = hTs; Fs; ds i be
a situation and M a set of situations. Let P(~t) be an atomic sentence and let and be
sentences except in rule 4. where may contain the free variable x.
1. M; s j=T P(~t) () P(ds (~t)) 2 Ts
M; s j=F P(~t) () P(ds(~t)) 2 Fs
2. M; s j=T : () M; s j=F M; s j=F : () M; s j=T 3. M; s j=T _ () M; s j=T or M; s j=T M; s j=F _ () M; s j=F and M; s j=F 4. M; s j=T 9x () for some n 2 N; M; s j=T xn
M; s j=F 9x () for all n 2 N; M; s j=F xn
Let ~x = hx ; : : :; xk i be a sequence of the existentially quantied variables bound at
the objective level of .
5. M; s j=T B () there are admissible ~t such that for all s0 2 M; M; s0 j=T s @[~x=~t]
M; s j=F B () M; s j6=T B
Notation: Since the semantics of an ordinary subjective sentences only depends on the
set of situations in question, we usually write M j=T (M j=F ) instead of M; s j=T (M; s j=F ). Similarly, for objective we simply write s j=T or s j=F , since M only
plays a role if a sentence mentions a belief.
To get a better understanding of the interpretation of B, let us look at an example:
Example 3.6 Let M = fs j s j=T P(a)g for some constant a.
Claim: M j=T B(9xP(x) ^ :BP(x)).
Proof: Let = 9xP(x) ^ :BP(x).
By denition, M j=T B i 8s 2 M; M; s j=T s@[ x=t] for some admissible t. Note
that, in this case, s = because does not contain any f-terms.
Note that s @[ x=a] = P(a) ^ :BP(a4). It suces to show that
for all s 2 M; M; s j=T P(a) ^ :BP(a4):
Let s = hTs ; Fs; dsi be any situation in M . M; s j=T P(a) follows immediately from
the denition of M .
1
12
To show that M; s j=T :BP(a4), assume that ds (a) = n for some standard name
n. By the denition of M , there must be a situation s0 2 M with a dierent denotation function such that s0 j6=T P(n). Therefore, M; s j=T :BP(n) which implies
M; s j=T :BP(a4).
3.3 Maximal Sets
Before turning to the denitions of implicit belief and only-believing, we need to address a
complication, which has to do with the fact that there are far more sets of situations than
there are basic belief sets, that is sets of all basic sentences that are believed at some set of
situations. For example, let M = fs j s j=T pg for some atomic sentence p and let M be
M with an arbitrary situation s removed. One can show that M and M still believe the
same basic sentences. It turns out to be very convenient to restrict the semantics of belief
to certain canonical sets of situations, which we call maximal sets. In a nutshell, maximal
sets are those which cannot be extended without changing some of their basic beliefs. (In
the above example, M is maximal while M is not.) As far as basic beliefs are concerned,
using only maximal sets or not does not make any dierence, which is reassuring. The
benet of maximal sets will become clear once we dene the meaning of O and L.
We start out with the denition of basic belief sets. For simplicity, we only consider
basic beliefs without f-terms.
Denition 12 Basic Belief Sets
A set of ordinary basic sentences is called a basic belief set if there is a set of situations
M such that = f j M j=T B; where is an ordinary basic sentenceg.
10
Theorem 1 Basic belief sets are uniquely determined by their existential-free objective
sentences.
Proof : This theorem was proven in [Lak92a] for a variant of our notion of explicit belief.
The proof carries over with only minor modications. This theorem is actually surprising since Levesque has shown that in the case of implicit belief (which we dene below) basic belief sets are not uniquely determined by their objective
sentences [Lev90].
Denition 13 Equivalence of Sets of Situations
Two sets of situations M and M are equivalent (M t M ) i their basic belief sets
are the same.
1
2
1
2
Denition 14 Maximal Sets
A set of situations M is maximal i no proper superset of M is equivalent to M .
Note that the empty set is a maximal set. This is easy to see because fg j=T Bfalse and
for all non-empty sets M , M j6=T Bfalse.
Denition 15 For any set of situations M , let M = fs j M t M [ fsg g.
+
10
Levesque introduces a similar notion in his paper on implicit belief and only-believing [Lev90].
13
Since M [fsg = M for all s 2 M , we obtain immediately that M M . In the following,
we will show that M is maximal, that is, M is the unique largest superset equivalent to
M.
Theorem 2 For any set of situations M , M t M and for all sets of situations M 0 such
that M t M 0 , M 0 M .
+
+
+
+
+
Proof :
1. To prove M t M , it suces to show that M and M agree on all ordinary, objective, and existential-free sentences (Theorem 1.) Assume to the contrary that there
is an ordinary, objective, and existential-free sentence on which M and M disagree,
that is, since M M , M j=T B but M j6=T B. In other words, there is an s 2 M
such that s j6=T (note that is existential-free). But then M 6t M [fsg, a contradiction.
+
+
+
+
+
+
2. Now we need to show that for every set of situations M 0 such that M 0 t M , M 0 M .
Let s 2 M 0. In order to show that s 2 M , it suces to show that M t M [ fsg.
In particular, it suces to show that M and M [ fsg agree on all ordinary, objective,
and existential-free sentences. Since M 0 t M by assumption, for all ordinary, objective,
and existential-free sentences , if M j=T B then s j=T . Therefore, if M j=T B then
M [ fsg j=T B as well. Thus M [ fsg has at least the objective beliefs M has. On the
other hand, since M M [ fsg, M [ fsg has certainly no more objective beliefs than M .
Thus M and M [ fsg have the same objective beliefs and hence the same basic belief sets.
+
+
The following two corollaries are immediate consequences of Theorem 2.
Corollary 3.1 M is unique and maximal.
+
Corollary 3.2 For any maximal sets of situations M and M 0, if M t M 0, then M = M 0.
The following corollary gives us an alternative characterization of M .
Corollary 3.3 For any set of situations M ,
+
M
+
= fs j for all ordinary, objective, and existential-free sentence ,
if M j=T B then M; s j=T .g:
Proof : Let us denote the right hand side by M .
M M : Let s 2 M . Then, since for all ordinary, objective, and existential-free sentences , if M j=T B then M; s j=T , M and M [ fsg agree on all ordinary, objec+
tive, and existential-free beliefs. Thus they agree on all ordinary basic beliefs and,
hence, M t M [ fsg. By the denition of M , s 2 M follows.
M M : Let s 2 M . By the denition of M , M t M [ fsg. Therefore, M and
M [ fsg agree on all ordinary, objective, and existential-free sentences. Thus, for
any such sentence , if M j=T B, then s j=T . Hence, s 2 M . +
+
+
+
14
+
Corollary 3.4 For all basic sentences , sets of situations M , and situations s,
M; s j=T i M ; s j=T and M; s j=F i M ; s j=F .
Proof : A simple induction on the structure of (omitted). +
+
The main result of this section has been that maximal sets can be taken as unique representatives of basic belief sets. In particular, as pointed out in Corollary 3.4, as far as
basic sentences are concerned, it does not matter whether one considers arbitrary sets of
situations or just maximal ones.
3.4 The Semantics of Implicit Belief and Only-Believing
The semantics of implicit belief is a slight modication of classical possible-world semantics.
The only dierence is that f-terms need to be taken care of as well. Their treatment is the
same as for explicit belief.
M; s j=T L () for all worlds w 2 M; M; w j=T s
M; s j=F L () M; s j6=T L
Note that implicit beliefs are interpreted only with respect to classical worlds contained
in the set of situations M . Furthermore, in contrast to explicit belief, there is no special
treatment of existential quantiers. In fact, their treatment is the same as in classical
possible-world semantics. For example, M; s j=T L9xP(x) i for every world w in M there
is a standard name n such that w j=T P(n).
As for only-believing, the basic intuition is that a set of situations M only-believes a
sentence i M is the largest set that believes . There is, however, one complication
that needs to addressed. Consider the sentence = 9xP(x). What should it mean for M
to only-believe ? Since a necessary requirement is that M believes , there must be some
closed term a such that M j=T BP(a). It may be tempting to let M = fs j s j=T P(a)g
for some closed term a. But that seems too strong. For example, to say that all the
detective believes is that someone is the murderer conveys a lot less information than all
the detective believes is that the driver of the red car is the murderer.
One way around this problem is to require the terms that are used when only-believing
an existentially quantied sentence to convey no information about the world. In other
words, the terms should behave like Skolem functions or internal identiers. For that
reason, the function symbols of OL have been partitioned into two sets, the regular function symbols FREG and the set FSK, whose members we also call sk-functions. Then
M j=T O9xP(x) just in case M = fs j s j=T P(a)g for some constant a 2 FSK. Finally, the
idea that function symbols in FSK do not convey information about the world can be dealt
with pragmatically when using the logic to specify interaction routines for querying and
updating a KB. In particular, a user is restricted to a language without function symbols
from FSK. Thus the KB is told information about the world using only sentences mentioning functions in FREG . Those in FSK are reserved for internal use by the KB. (See Section 7
for more details on this.)
Before turning to the formal denition of only-believing, it is worth pointing out that
the above problem does not arise in the case of implicitly only-believing as studied by
15
Levesque [Lev90]. There, the sentence 9xP(x) is implicitly only-believed just in case
M = fw j w is a world and w j=T 9xP(x)g. It is easy to see that M is the largest set of
worlds such that M j=T L9xP(x). Note that, in contrast, 9xP(x) is not even explicitly
believed at fs j s is a situation and s j=T 9xP(x)g.
Denition 16 Sk-terms
Let be a sentence and x an existentially quantied variable bound at the objective level
of . Let U (x) be a sequence of the universally quantied variables in whose scope x is
bound. Let f 2 FSK be a function symbol of arity jU (x)j occurring nowhere else in . Then
f (U (x)) is called an sk-term (for x).
Note that an sk-term is also an admissible substitution.
Let be a sentence, where ~x = hx ; : : :; xk i is a sequence of the existentially quantied variables at the objective level.
1
M; s j=T O () there is a sequence of sk-terms ~tSK for ~x such that for all s0;
s0 2 M i M; s0 j=T s@[~x=~tSK]
M; s j=F O () M; s j6=T O
Truth, Logical Consequence, Validity, and Satisability in OBLIQUE
The notions of truth, logical consequence, validity, and satisability are dened with respect
to worlds and maximal sets of situations only.
A formula is true at a maximal set of situations M and a world w if M; w j=T . is
false if M; w j6=T . A sentence is logically implied by a set of sentences ( j=) i for
all maximal sets M and worlds w, if M; w j=T for all 2 , then M; w j=T . A formula
is valid (j=) i is logically implied by the empty set. is satisable i : is not valid.
3.5 Properties of the Logic
In this section, we discuss properties of the logic with an emphasis on the various forms of
belief. The purpose is not to provide an exhaustive analysis but to concentrate on those
properties that relate the various notions of belief to the classical ones. In subsequent
sections, we will study in detail the usefulness of the logic for KR purposes.
Since we care mostly about ordinary sentences, we restrict our attention to that class
only.
11
12
An exhaustive analysis would include a complete axiomatization of the logic, which may be dicult
to nd. Even in the case of implicit belief and implicitly only-believing, Halpern and Lakemeyer [HL95]
showed recently that Levesque's axiomatization [Lev90] is incomplete in the rst-order case. Furthermore,
it is easily seen that any complete axiomatization of Levesque's logic is nonrecursive.
12With few exceptions, the properties discussed in this section carry over directly to sentences with
f-terms.
11
16
It is not hard to see that OBLIQUE subsumes the logic BLq of [Lak94]. In particular,
the semantic rules of OBLIQUE , restricted to the language Lq of BLq , reduce to those of
BLq . In addition, as far as basic sentences are concerned, it does not matter whether or
not maximal sets are used (Corollary 3.4).
3.5.1 Implicit Belief
Ordinary implicit belief satises all the properties of weak S5, that is, beliefs are closed
under classical logical consequence and full introspection. In fact, apart from the additional
treatment of f-terms, L is nothing but Levesque's notion of implicit belief as described
in [Lev90]. We therefore will not go into any more details regarding implicit belief and
turn right to the properties of explicit belief.
3.5.2 Explicit Belief and Only-Believing
Let ; and be ordinary sentences not containing L. Let and , in addition, be
subjective and let n be a standard name. (Proofs can be found in the Appendix.)
Perfect Introspection
1. j=B BB
2. j=:B B:B
Quantifying-in
3. j=Bxn 9xB
4. j=9xB B9x
5. j=Bxa B9x
6. j=8xB B8x provided is existential-free
7. j=B8x 8xB
Logical Omniscience for Subjective Sentences
8. j=(B( ) ^ B) B
Accuracy and (Partial) Completeness of Subjective Beliefs
9. j=(B ^ :Bfalse) 10. j= B provided is existential-free.
Relating Only-Believing and Explicit Belief
11. j=O B.
Relating Explicit Belief and Implicit Belief
17
12. j=B L.
Note that explicit and implicit belief of ordinary sentences have the same properties regarding introspection and quantifying-in. Also, similar to the propositional case, an agent's
explicit beliefs lack logical omniscience only about the world, not about what is believed.
In fact, explicit belief of subjective sentences is very similar to weak S5 except for the
proviso in property 10. The connection between explicit belief and implicit belief on the
one hand (11.) and between only-believing and explicit belief on the other (12.) are as
expected.
4 The Connection Between Explicit and Implicit Belief
In the last section, we observed that believing a sentence explicitly logically implies that
the sentence is also believed implicitly. While this is not surprising, it turns out that there
is a much stronger connection between the explicit and implicit beliefs at a given set of
situations. The following theorem says that if we are given only the basic beliefs of an agent
(and, by Theorem 1, we only need to consider objective ones), then we can reconstruct all
beliefs including what the agent only-believes and what he implicitly believes.
Theorem 3 Let = f j M j=T B; where is basic and ordinaryg be the basic belief
set for a maximal set of situations M . Let be any ordinary sentence (except in part 1,
where may not contain any Ls). Then
1: M j=T O
2: M j=T L
3: M j=T L
Proof :
i
i
i
fB j 2 g [ f:B j 62 g j= O:
fB j 2 g [ f:B j 62 g j= L:
j=:
1. To prove the only-if direction, let M j=T O and let M 0 be a maximal set such that
M 0 j=T fB j 2 g [ f:B j 62 g. Thus M and M 0 agree on their basic belief
sets, i.e., M t M 0. By Corollary 3.2, M = M 0 and, therefore, M 0 j=T O .
To prove the if direction, let fB j 2 g[f:B j 62 gj=O . By the denition of
, we obtain immediately that M j=T fB j 2 g [ f:B j 62 g and, therefore,
M j=T O .
2. The same argument proves part 2 with O replaced by L .
3. To prove the if direction, assume j= , that is, for all maximal sets of situations
M 0 and worlds w, if M 0; w j=T , then M 0; w j=T . Since, by the denition of ,
M j=T B for all 2 , M j=T L for all 2 holds as well (proved later as
property 12).
Next, let w be a world in M . (If M is empty, then M j=T L holds vacuously.) Then
M; w j=T for all 2 . Thus M; w j=T and, therefore, M; w j=T . Since this
holds for all w 2 M , M j=T L follows.
18
Conversely, assume that M j=T L . Let M 0 be a maximal set of situations and w a
world such that M 0; w j=T . We need to show that M 0; w j=T .
Note that, if 2 , then B 2 . Also, if 62 , then :B 2 .
Therefore, M and M 0 agree on all basic beliefs, that is, M t M 0. Since both M and
M 0 are assumed to be maximal, by Corollary 3.2, M = M 0.
Since, by assumption, M 0; w j=T , we obtain, in particular, that for all ordinary,
objective, and existential-free 2 , if M 0 j=T B then M 0; w j=T . Thus, by
Corollary 3.3, w 2 M 0. Since M = M 0 and M j=T L by assumption, M 0 j=T L .
Finally, since w 2 M 0, M 0; w j=T . If we had used non-maximal sets, neither what is only-believed nor what is implicitly
believed would follow from the explicit beliefs and non-beliefs:
Example 4.1 Consider the following sets of situations M and M 0, where p and q are
distinct atomic sentences.
M = fs j s j=T pg
M 0 = fs j s j=T pg fw j w is a world and w j6=T qg
The only dierence between M and M 0 is that q is true at all the worlds contained in M 0.
It is easy to see that M and M 0 agree on all explicit beliefs. (Note that M 0 is not maximal
and M 0 = M .) However, M 0 implicitly believes q while M does not!
The question may arise why one should not allow epistemic states as modeled by M 0.
The answer is that, from a KR point of view, it is hard to imagine that an agent whose
beliefs are represented by a KB has implicit beliefs that cannot be inferred from his KB or,
for that matter, from his explicit beliefs. Since the purpose of the logic OBLIQUE is to
form a formal basis of a KR system, it seems appropriate to aim for a model theory that
matches our intuitions as tightly as possible. Incidentally, none of the other approaches in
the literature that deal with both explicit and implicit belief, such as [Lev84b] and [FH85],
addresses this issue. They all allow epistemic states, where agents may have implicit
beliefs independent of their explicit beliefs. A more detailed study of this issue can be
found in [Lak91b].
+
5 Determinate Sentences
So far our discussion of OBLIQUE has been quite independent from any representational
aspect, that is, nothing depended on whether or not the beliefs of an agent are in any way
represented symbolically. In the rest of the paper we will study OBLIQUE specically
from this knowledge representation point of view.
We begin with a closer look at how and to what extent the logic allows us to draw a
connection between a knowledge base and a corresponding epistemic state (= set of beliefs).
The range of possible epistemic states in OBLIQUE is dened by the dierent maximal
sets of situations. Only-knowing denes a mapping from knowledge bases into epistemic
states for many sentences. For example, given the primitive formula P(n), fs j s j=T P(n)g
19
is the only set which satises OP(n). Hence we obtain that for every sentence , either B
or :B is logically implied by OP(n). For sentences with existential quantiers such a nice
mapping generally does not exist. For example, O9xP(x) is satised by fs j s j=T P(ask )g
for every constant in FSK. Note, however, that these sets of situations are very similar. In
particular, it can be shown that they agree on all beliefs that do not mention sk-functions.
Thus if we restrict epistemic states to consist of beliefs without sk-functions, O9xP(x)
again represents a unique epistemic state. As was mentioned before, we do not really
limit ourselves by disregarding beliefs with sk-functions. In a sense, all we do is reserve
sk-functions to be used only as internal identiers of the system. A user who asks queries
and updates the KB is unaware of them and uses function symbols only from the set FREG .
With these observations, we call a sentence determinate if O is uniquely satisable
after replacing all existential quantiers at the objective level of by sk-terms.
Not all sentences are determinate. This is because O is essentially an autoepistemic
operator. For example OBP(n) is not satisable at all and O(BP(n) P(n)) is satised by
exactly two sets of situations, namely fs j s j=T P(n)g and the set of all situations. Having
multiple epistemic states in cases like this one is completely analogous to the multiple stable
expansion phenomenon of Moore's autoepistemic logic [Moo85]. We will not discuss the
autoepistemic aspect of OBLIQUE here. The interested reader is referred [LL88] instead.
See also [Lev90] in the case of only-believing and implicit belief.
We now investigate determinate sentences in more detail. The rest of this section is
organized as follows. After formally dening determinate sentences in OBLIQUE , we show
that all objective sentences are determinate. We then prove that determinate sentences
uniquely determine epistemic states without sk-functions (Theorem 4). The main result
of this section is that, if a determinate sentence is only-believed, the question whether an
ordinary basic sentence is explicitly believed reduces to the question whether an ordinary
objective sentence is believed (Theorem 5).
Although the results in this section hold for non-basic as well as for basic sentences,
we by and large ignore the non-basic part because it will not be used later on. Restricting
ourselves to basic sentences also simplies the presentation.
First we introduce some convenient notation.
Denition 17 Let M be a set of situations, an ordinary basic sentence and let ~x be the
existentially quantied variables at the objective level of . Let ~t be admissible terms (or
sk-terms) for ~x. If B or O hold for ~t at M , we use the abbreviation B~t and O~t,
respectively. Formally,
M j=T B~t i for all situations s, if s 2 M , then M; s j=T @[~x=~t]
M j=T O~t i for all situations s, s 2 M i M; s j=T @[~x=~t]
Denition 18 Determinate Sentences
Let be an ordinary, basic sentence and ~x a sequence of the existentially quantied
variables at the objective level of .
is determinate i O is satisable and for all admissible sk-term substitutions ~t of ~x
and for all sets of situations M and M ,
if M j=T O~t and M j=T O~t, then M = M :
1
1
2
1
2
20
2
Informally, Denition 18 says that for any sk-term substitution of the existentially quantied variables of a determinate sentence the corresponding epistemic state is unique.
Lemma 5.1 Every ordinary objective sentence is determinate.
Proof : Let be an ordinary and objective sentence. First, if is existential-free and
M j=T O for a set of situations M , then s 2 M i s j=T , which follows straight from
the denition of O. Thus M is unique in case is existential-free.
Now let be an arbitrary ordinary objective sentence. Let M j=T O and M 0 j=T O
such that M j=T O~t and M 0 j=T O~t for sk-terms ~t. Since is objective, this is the same
as M j=T O@[~x=~t] and M 0 j=T O@[~x=~t]. Therefore, since @[~x=~t] is an existential-free
objective sentence and existential-free objective sentences are only-believed at a unique set
of situations, M = M 0. The following lemma is useful for the main results of this section.
Lemma 5.2
Let : F ! F be a bijection over function symbols such that f and (f ) are of the same
arity for all f , and (f ) 2 FREG ((f ) 2 FSK ) i f 2 FREG (f 2 FSK ).
For a given situation s = hT; F; di dene s = hT; F; d i such that
d (f (~n)) = d((f )(~n)) for all primitive terms f (~n).
For any (not necessarily maximal) set of situations M , let M = fs j s 2 M g. For any
formula (term t), let () ((t)) denote (t) with every function symbol f replaced by
(f ).
If is basic, then
1: M; s j=T M; s j=F i
i
M; s j=T ()
M; s j=F ()
2: If M is maximal, then M is maximal.
Proof : 1. is proved by induction on the structure of .
First, a simple induction argument on the structure of terms shows that for any denotation
function d and term t, d (t) = d((t)) (omitted).
Base case: M; s j=T P(~t) i P(ds (~t)) 2 Ts i P(ds((~t))) 2 Ts i M; s j=T (P(~t)).
The argument for j=F is completely symmetric.
Induction Step The cases for :, _, and 9 follow easily by induction.
21
M; s j=T B
M; s0 j=T (s )@[~t]
for some admissible ~t and for all s0 2 M
M; s0 j=T ((s )@[~t] ) for all s0 2 M by induction
M; s0 j=T ()s @[ (~t)]] for all s0 2 M (note that (s ) = ()s
because ds (t) = ds((t)) by def. of ds )
then M; s j=T (B)
because (~t) is admissible.
Conversely,
Let M; s j=T (B)
then M; s0 j=T ()s @[~t0]
for some admissible ~t0 and all s0 2 M
then M; s0 j=T ()s @[ (~t)]] for some admissible ~t, which must exist
because is bijective
then M; s j=T B
by the same argument as above
Let
i
i
i
2. Let M be maximal. To prove that M is maximal, it suces to show that M M.
Let s 2 M . Then, since is bijective, s = s0 for some s0.
Let be an ordinary, objective, and existential-free sentence. By Corollary 3.3, if
M j=T B, then M; s j=T . Therefore, using part 1 of the lemma, if M j=T B(),
then M; s0 j=T (). Note that, since is bijective, is a bijection over ordinary, basic
and existential-free formulas as well. Thus we obtain that for all ordinary, basic, and
existential-free , if M j=T B, then M; s0 j=T . Hence s0 2 M , which implies s0 = s 2
M .
We can now show that determinate sentences as dened above in fact deserve their name.
Theorem 4 Let be a determinate sentence. For any ordinary basic without skfunctions, exactly one of j=O B and j=O :B holds.
Proof : Let M and M 0 be two maximal sets of situations such that M j=T O and
M 0 j=T O. Then M j=T O~t and M 0 j=T O~u for admissible sk-terms ~t and ~u, respectively.
Now dene a bijection : F ! F such that (~t) = ~u and all f 2 FREG are mapped onto
themselves as well as those f 2 FSK that occur in . Let M0 = fs j s 2 M 0g as dened in
Lemma 5.2.
First, we show that M0 j=T O~t. By assumption, M 0 j=T O~u , i.e., for all situations
s, s 2 M 0 i M 0; s j=T @ [~u] . By the denition of , @[~u] = (@[~t] ). Therefore, by
Lemma 5.2, for all s, s 2 M0 i M0 ; s j=T @[~t] , that is, M0 j=T O~t.
Let be basic and without sk-functions. By Lemma 5.2, M 0 j=T B i M0 j=T B
because ( ) = . Also, since is determinate and both M j=T O~t and M0 j=T O~t,
M = M0 . Thus M j=T B i M 0 j=T B .
Since this holds for arbitrary maximal M and M 0 such that M j=T O and M 0 j=T O,
the theorem follows. The rest of this section is concerned with proving that, if a determinate sentence is onlybelieved, the question whether an arbitrary ordinary sentence is explicitly believed reduces
to the question whether an ordinary objective sentence is believed. Since the proof is fairly
involved, we begin by sketching the main ideas rst.
+
+
22
Denitions 19 and 20 describe the transformation from believing an arbitrary sentence
to believing an equivalent objective sentence. The idea is as follows. Suppose a determinate sentence is only-believed. The question whether a sentence is believed can be
transformed into the equivalent question whether a certain objective sentence is believed
by performing the following transformations on . For any subsentence B in , where is objective and does not contain free variables, replace B by true or false depending
on whether the belief holds at M or not. In case contains free variables, replace B by
an equality expression as described in Denition 19. This expression describes precisely for
which standard name substitutions of the free variables the belief holds at M . Most importantly, although there may be innitely many standard names that can be substituted,
these can always be captured with a nite equality expression. It is not hard to see that, if
these substitutions are performed according to the denitions, that is from the innermost
beliefs to the outermost ones, the result will be an objective sentence.
Lemma 5.4 and 5.7 provide the justication why these transformations are legitimate.
Other lemmas are needed for technical reasons. Finally, the main result appears in Theorem 5.
Denition 19 Let be a determinate sentence, M a set of situations such that M j=T O ,
and an ordinary formula possibly containing free variables. Let n ; : : : ; nm be all the
standard names occurring in or , and let n be a standard name not occurring in or
.
8
>> true
if is closed and M j=T B
>< false
if is closed and M j=F B
RESB [ M; ] =
>> W((x = n ) ^ RES [ M; x ] )_ otherwise, where x is free in i
B
ni
>: V
( (x 6= ni ) ^ RESB[ M; xn ] nx )
13
1
Lemma 5.3 Consider a bijection : N ! N . For a given situation s = hT; F; di
dene s = hT ; F ; d i s.t. for any primitive formula P(n1; : : :; nk ) and primitive term
f (n1; : : : ; nk ),
P( (n1); : : : ; (nk )) 2 T
i P(n1; : : :; nk ) 2 T
P( (n1); : : : ; (nk )) 2 F
i P(n1; : : :; nk ) 2 F
d (f ( (n1 ); : : :; (nk )) = (d(f (n1 ; : : :; nk )))
For any set of situations M let M = fs j s 2 M g. is extended to apply to terms
and formulas in the usual way. Let M be a set of situations, s a situation, and a basic
sentence. Then
1: M ; s j=T ()
M ; s j=F ()
i
i
M; s j=T M; s j=F 2: If M is maximal, then M is maximal.
Proof :
13
By the way, the same process applies in the case of implicit belief [Lev90].
23
1. Induction on the structure of .
(First, a simple induction shows that for any denotation function d and closed term
t,
d ( (t)) = (d(t)) (omitted).)
If is atomic, then there are two cases:
a) = P(~t) for an arbitrary predicate symbol P other than =.
b) = (t = t0) for closed terms t and t0.
Let s = hT; F; di with s = hT ; F ; d i as dened above.
a) s j=T P(~t) i P(d(~t)) 2 T i P( (d(~t)) 2 T by denition of T
i P(d ( (~t))) 2 T by denition of d
i s j=T P( (t)).
(The proof for j=F is analogous.)
b) s j=T (t = t0) i d(t) = d(t0) i (d(t)) = (d(t0)) (because is a bijection)
i d ( (t)) = d ( (t0)) i s j=T (t = t0).
The cases for :, _, and 9 follow easily by induction.
Let M ; s j=T (B) and let ~x be the existentially quantied variables at the objective level of .
Then there are admissible ~t for ~x such that for all s0 2 M ; M ; s0 j=T ()s @[~x=~t] .
Since is bijective, there are terms ~t0 such that ~t = (~t0) and the ~t0 are admissible as
well. Also, since ds ( (t)) = (ds (t)), ()s = (s ).
Therefore, for all s0 2 M ; M ; s0 j=T (s )@[~x= (~t0)]] and, by induction,
for all s 2 M; M; s j=T s @[~x=~t0] , which implies M j=T B.
Conversely, let M; s j=T B. Then there are admissible ~t such that for all s 2 M ,
M; s j=T s @[~t]. By induction, for all s0 2 M ; M ; s0 j=T (s @[~t]).
Since the (~t) are admissible and ()s = (s), we obtain that for all s0 2 M ;
M ; s0 j=T ()s @[~x= (~t)]], which implies M ; s j=T (B).
2. Assume that M is maximal. In order to prove that M is maximal, let s be any situation and assume that for all ordinary, objective, and existential-free , if M j=T B,
then s j=T (). By Corollary 3.3, all we need to show is that s is in M .
First, since is bijective, s = s0 for some s0. By assumption (), for all ordinary,
objective, and existential-free , if M j=T B (), then s0 j=T (). Thus, by part 1
of the lemma, if M j=T B, then s0 j=T .
Finally, by Corollary 3.3, s0 2 M and by the denition of M , s0 (= s) 2 M . Lemma 5.4 Let be a determinate sentence and an objective formula with free variables
x ; : : : ; xk . Let M be a set of situations such that M j=T O . Then for any s 2 M and
1
any substitutions of closed terms t1; : : :; tk for x1; : : :; xk ,
M; s j=T RESB [ M; ] [t1; : : :; tk ] i M; s j=T (B)[[t1; : : :; tk ]
(Note the use of [ :] on the right hand side, making sure that ti4 is substituted for xi . See
also the example below.)
24
Proof : By induction on the number of variables in .
Let have no free variables. Then M; s j=T RESB[ M; ] i RESB [ M; ] = true i M; s j=T B.
Assume the lemma holds for k variables. Let have k + 1 variables. Let t be a closed
term intended as a substitution for the variable x with ds (t) = n.
Case: n occurs in or . M; s j=T RESB[ M; ] [t ; : : :; tk ; t] i M; s j=T ((x = n) ^
RESB [ M; xn )]][t ; : : :; tk ; t] (all other candidates are not satised because ds (t) = n.)
i M; s j=T RESB[ M; xn] [t ; : : :; tk ] i M; s j=T (Bxn)[[t ; : : : ; tk] (by induction) i
M; s j=T (B)[[t ; : : :; tk ; t] .
Case: n does not occur in or .
M; s j=T RESB [ M; ] [t ; : : :; tk ; t] i M; s j=T RESB[ M; xn] nx[t ; : : : ; tk; t]
i M ; s j=T RESB[ M; xn] nx[ (t ); : : : ; (tk ); (t)] (by Lemma 5.3, where is a bijection over standard names that swaps n and n and that is the identity function
otherwise)
i M ; s j=T RESB [ M; xn ] [ (t ); : : : ; (tk )] (since ds ( (t)) = n)
i M ; s j=T (Bxn )[[ (t ); : : :; (tk )]] (by induction)
i M; s j=T (Bxn)[[t ; : : :; tk ] (by Lemma 5.3) i M; s j=T (B)[[t ; : : : ; tk ; t] . 1
1
1
1
1
1
1
1
1
1
1
1
To illustrate the previous lemma, let us consider an example.
Example 5.1 Let = P(#1) ^ P(#2), = P(x), and M = fs j s j=T g.
It is easy to see that M j=T O . For any situation s = hT; F; di and closed term t we obtain
M; s j=T RESB(M; )[t]
i M; s j=T (t = #1) _ (t = #2)
i d(t) = #1 or d(t) = #2
i M; s j=T BP(t4)
i M; s j=T BP(x)[[t]:
The following denition describes how to convert any ordinary basic sentence into an
objective sentence. Lemma 5.7 shows that this transformation is indeed equivalent to the
original sentence.
Denition 20 Let be a determinate sentence and M a set of situations such that
M j=T O . Let be an ordinary basic formula.
kkM
= ; for objective k:kM = :kkM
k _ kM = kkM _ k kM
k9xkM = 9xkkM
kBkM = RESB [ M; kkM]
Lemma 5.5 kkM is objective.
Proof : A simple induction on the structure of . Omitted. 25
Lemma 5.6 k@kM = kkM@.
Proof : A simple induction on the structure of . Note that @ only removes existential
quantiers at the objective level and k : : : kM does not alter the objective level of a formula.
Lemma 5.7 Let be a determinate sentence and M a set of situations such that M j=T O .
Let be an ordinary basic formula with free variables ~x = hx ; : : :; xk i and let ~t =
ht ; : : :; tk i be a sequence of closed terms.
M; s j=T [~x=~t] i M; s j=T kkM[~x=~t]
M; s j=F [~x=~t] i M; s j=F kkM [~x=~t]
Proof : By induction on the structure of .
Let ~t4 = ht 4; : : :; tk 4 i. If is objective, the lemma holds trivially because [~t] = kkM[~t].
Given the denition of kkM, the cases for :, _, and 9 follow easily by induction.
Finally,
M; s j=T (B)[[~t]
i M; s j=T B([~t4 ])
by denition of [~t]
0
4
s
0
@
i M; s j=T [~t ] [~t ]
for some admissible ~t0 and for all s0 2 M
i M; s0 j=T [~n]@[~t0]
for ds (~t) = ~n
0
0
@
i M; s j=T [~n; ~t ]
i M; s0 j=T k@kM [~n; ~t0]
by induction
0
0
@
i M; s j=T kkM [~n; ~t ]
by Lemma 5.6
0
0
4
s
@
i M; s j=T kkM[~t ] [~t ]
i M; s j=T B(kkM[~t4 ])
i M; s j=T (BkkM)[[~t]
i M; s j=T RESB[ M; kkM] [~t] by Lemma 5.4
i M; s j=T kBkM[~t]
by denition of k : : : kM 1
1
1
Lemma 5.8 Let be an ordinary determinate sentence. Let M and M 0 be maximal sets
of situations such that M j=T O and M 0 j=T O. Then for any ordinary basic formula without sk-functions, k kM = k kM .
Proof : The proof is by induction on the structure of .
If is objective, the lemma holds trivially because k kM = = k kM .
The cases for the logical connectives :, _, and 9 follow easily by induction using the
denition of k : : : k.
Let us now consider kB kM = RESB[ M; k kM] and kB kM = RESB [ M; k kM ] . Thus we
need to show that RESB[ M; k kM ] = RESB[ M; k kM ] . By induction, k kM = k kM . By
Theorem 4, M and M 0 agree on all explicit beliefs that do not mention function symbols
in FSK. With that, a simple inductive argument on the number of free variables in , using
the denition of RESB , establishes that RESB[ M; ] = RESB[ M 0; ] . 0
0
0
0
0
0
Now we can nally prove that determining whether an arbitrary basic belief follows from
only-believing a determinate sentence reduces to determining whether a certain objective
belief follows.
26
Theorem 5 Let be an ordinary determinate sentence and an ordinary basic sentence,
which does not contain function symbols in FSK . Given Lemma 5.8, kkM is the same for
all M such that M j=T O . Hence let us write kk instead of kkM for such M . Then
j=O B i j=O Bkk:
Proof :
Let M j=T O . Then M j=T B i M j=TkBkM (by Lemma 5.7) i M j=T RESB[ M; kk]
(by denition of k : : : k) i M j=T Bkk (by Lemma 5.4). Thus, whenever M j=T O ,
M j=T B i M j=T Bkk and we are done. Note that this result rests squarely on the denition of kkM (Denition 19 and 20). In this
regard, = plays a crucial role in turning a basic sentence into an objective one. It is mainly
for this reason that = was introduced as a two-valued, or classical, equality predicate rather
than a four-valued one as all the other predicates.
In summary, this section introduced determinate sentences, which determine complete
epistemic states as long as these do not mention sk-functions. An important result was that
basic beliefs without sk-functions that follow from only-believing a determinate sentence
reduce to objective beliefs. While interesting in its own right, this result will play an
important role in the next section on decidability results in OBLIQUE .
6 Decidability Results
In this section, we are concerned with the problem of deciding whether a basic belief follows
from only-believing an objective sentence. We restrict our attention to ordinary sentences
only. So far we have obtained complexity results only for subclasses of sentences. Note
that, while Theorem 7 is subsumed by Theorem 8, the former has been included as a special
case because it has a much simpler proof than the more general case. We begin by noting
that, if we restrict ourselves to a propositional language without equality, reasoning is in
fact tractable.
Denition 21 A basic quantier-free formula is in conjunctive normal form, if is a conjunction of disjunctions, where every disjunct is either a literal or of the form B
or :B such that itself is in conjunctive normal form.
Theorem 6 Let and be basic quantier-free sentences without equality in conjunctive
normal form. Furthermore let be objective.
Then the validity of O B can be determined in O(jj j j).
Proof : This theorem is a slight variant of a result in [LL88]. In order to address the complexity of reasoning in the quanticational case, we rst need to
establish some results regarding various validity preserving transformations on sentences
of the form O B and B B .
Lemma 6.1 Skolemization
Let be an ordinary objective sentence with existentially quantied variables ~x, an
ordinary basic sentence whose function symbols are in FREG or contained in . Then
j=O B i j=O@[~x=~t] B for sk-terms ~t:
27
Proof : Assume j=O B and let M j=T O@[~x=~t]. Thus s 2 M i s j=T @[~x=~t] for
sk-terms ~t. In other words, M j=T O which implies, by assumption, M j=T B .
Conversely, assume j=O@[~x=~t] B for admissible ~t and let M j=T O.
Then M = fs j s j=T @[~u] for some sk-terms ~u g.
Dene a bijection : F ! F such that (~t) = ~u, (f ) = f for all f 2 FREG, and (f ) = f
for all f 2 FSK that occur in . Then (@[~x=~t]) = @ [~x=~u]. Since M j=T O@[~x=~u] =
(@[~x=~t]), by Lemma 5.2, M j=T O@[~x=~t] and thus, by assumption, M j=T B . Since
( ) = , by Lemma 5.2, M j=T B . Lemma 6.2 Let be an ordinary, objective, and existential-free sentence. Let PCNF be in prenex conjunctive normal form. Then
j=B BPCNF and j=O OPCNF .
Proof : A similar result was proven for the logic BLq of [Lak92c, Lak94]. Since the proof
generalizes in a straightforward way, it is omitted here. Denition 22 For any ordinary objective sentence , let <[] = fs j s j=T g.
Given the denition of <[], it is easy to see that
Lemma 6.3
For any ordinary, objective, and existential-free , <[] is maximal and <[] j=T O.
Proof : The lemma follows immediately from the denition of <[] and the fact that is
existential-free. Note that the lemma does not hold for arbitrary objective sentences containing existentially
quantied variables. For example, if = 9xP(x), then <[] j6=T O. In fact, is not even
explicitly believed at <[].
Lemma 6.4 Let be an objective, ordinary, and existential-free sentence. Then for any
ordinary basic sentence ,
j=O B i j=O Bk k< :
[ ]
Proof : Since is objective and existential-free, there is a unique <[] such that
<[] j=T O. What is left to show is that <[] j=T B i <[] j=T Bk k< .
<[] j=T B i there are admissible ~t such that for all s 2 <[], <[]; s j=T @[~t]
i there are admissible ~t such that for all s 2 <[], <[]; s j=T k k< @[~t] by Lemma 5.6
and 5.7 i <[] j=T Bk k< . [ ]
[ ]
[ ]
Lemma 6.5 Let and be ordinary objective sentences, where is existential-free.
j=O B i j=B B
28
Proof : Assume j=B B and let M j=T O. Then M j=T B and thus, by assumption, M j=T B .
Conversely, assume j=O B and let M j=T B. Then M <[]. Since <[] j=T O,
<[] j=T B by assumption and thus M j=T B . Lemma 6.6 Let and be ordinary objective sentences, where is existential-free. Let
~x = hx ; : : : ; xki be a sequence of the existentially quantied variables in .
j=B B i j=B B @[~x=~t]
for some admissible terms ~t for ~x.
1
Proof :
only-if direction: Assume j=B B . We show that we can choose an admissible ~t
for the existentially quantied variables in independent of the set of situations in
question.
For that reason, let us consider Mmax = fs j s j=T g. Since is existential-free, i.e.,
= @, Mmax j=T B. Then, by assumption, Mmax j=T B , i.e., for some admissible
~t and for all s 2 Mmax, s j=T @[~t].
Now consider any set of situations M such that M j=T B. Since is existential-free,
s j=T for all s 2 M and, therefore, M Mmax. Thus for all s 2 M , s j=T @[~t],
where ~t are the same admissible terms as in the case of Mmax. Finally, since @[~t] is
existential-free, M j=T B @[~t].
if direction: Conversely, assume that j=B B @[~t] for some admissible ~t
and let M j=T B. Then M j=T B @[~t], i.e., for all s 2 M , s j=T @[~t] (since
@[~t] is existential-free), from which M j=T B follows immediately. Theorem 7 Let be an ordinary objective sentence without equality. Let be an ordinary
basic sentence without equality or quantifying-in, i.e., no subformula B of contains free
variables.
Then the validity of O B is decidable.
Proof :
j=O B
i j=O0 B
where 0 is with all existentials replaced by sk-terms
i
i
j=O0 Bk k<[0]
j=B0 Bk k<[0]
(Lemma 6.1)
by Lemma 6.4
by Lemma 6.5 because k k< 0 is objective
[
]
Since k k< 0 is objective, it suces to show the decidability of the validity problem for
B0 B 0 for an arbitrary objective and existential-free 0 and any objective 0 which is
the result of k:k< 0 . A simple induction argument then shows the decidability of k k< 0 ,
where is basic and without equality and quantifying-in. This, together with the above
[
]
[
]
[
29
]
equivalences, establishes the decidability of j=O B .
Given the restrictions on , it is easy to see that we only need to consider objective 0
whose only occurrences of equality expressions are of the form (#1 = #1) and (#1 6= #1),
which we denoted as true and false (see Denition 19 on page 23). Let 00 be 0 with
occurrences of true and false removed in the usual manner (e.g. ( _ false) reduces to
). Thus either 00 = true or 00 = false or 00 has no occurrences of true or false.
Then
j=B0 B 00
i 00 = true or
00 has no occurrences of true or false and B0 B 0 is valid in BLq ,
which was shown to be decidable in [Lak92c, Lak94]. In [Lak92c, Lak94] it was shown that belief implication, that is, the validity problem for
sentences of the form B B , where both and are ordinary objective sentences
without equality, is equivalent to Patel-Schneider's t-entailment [Pat87]. Patel-Schneider
shows that, while t-entailment is intractable in general, it is ecient for a large class of
sentences that are relevant from a practical point of view (e.g., queries can be assumed to
be at most logarithmic in length with respect to the size of the knowledge base). From the
proof of Theorem 7 it is obvious that the complexity of introspective reasoning is dominated
by the complexity of belief implication. Thus in the case where belief implication is ecient,
the extension to introspective reasoning is ecient as well. Hence, at least in cases without
quantifying-in and equality, there is really no penalty to be paid when moving from a
purely deductive reasoner to an introspective one.
In a generalization of Theorem 7, we now allow some form of quantifying-in and show
that reasoning is still decidable.
Theorem 8 Let be an ordinary objective sentence without equality. Let be an ordinary
basic sentence without equality such that any free variable x in a subformula B in is a
universally quantied variable in . In other words, only universally quantied variables
may participate in quantifying-in.
Then the validity of O B is decidable.
Proof : See Appendix. The theorem shows that, in the case of limited reasoning, quantifying-in does not necessarily push us over the decidability edge, thus at least partially absolving the poor computational record quantifying-in has in classical modal logic as noted in [Kon85]. While
Theorem 7 only allows quantifying-in for universally quantied variables, we conjecture
that the theorem can in fact be strengthened to allow arbitrary forms of quantifying-in,
but we have not yet found a proof one way or the other.
So far in our complexity analysis, we have not allowed any explicit use of equality
expressions. In the proof of Theorem 8, some equalities are introduced when resolving
nested B's with quantifying-in inside . Of course, allowing equalities of that type within
explicitly does not aect decidability, since the proof would still go through. However,
unrestricted use of equality expressions certainly destroys decidability because equality
has its classical two-valued meaning. We are then left with two options. For one we
30
can investigate restricted uses of equality, which do not lead to undecidability. One such
case obtains when the KB is of the form KB0 ^ E , where KB0 contains no equality and E
is a conjunction of ground atomic equality expressions and their negations. Rather than
allowing restricted uses of classical equality, one could also introduce an additional equality
predicate with a weaker semantics. Exactly what such a semantics should look like is a
challenging open problem.
14
7 A KR Service: ASK and TELL
After a fairly lengthy excursion into nonstandard modal logic, we now apply our results to
the specication of routines to interact with a knowledge base.
To that end, we use the methodology proposed in [Lev84a] which treats a KB as an
abstract data type. The idea is that a user is given a set of KR service routines that allow
her to interact with a KB in a meaningful way. In particular, these routines are specied
in terms of what the KB knows (the knowledge level ) rather than how this knowledge is
represented (the symbol level ). Of course, it is just as important to show exactly how
the knowledge level specication can be realized at the symbol level, and it is at this level,
where it makes sense to talk about the computational complexity of the KR service.
We focus on two core operations, ASK and TELL, which allow a user to query a KB and
to add new information to it.
15
7.1 ASK
takes two arguments, an epistemic state represented by a set of situations M and a
sentence in an interaction language IL, and it returns one of four answers: YES if is
believed to be true, NO if is believed to be false, UNKnown if is neither believed to be
true nor believed to be false, and INConsistent if is both believed to be true and false.
The interaction language is restricted to the following subset of OL.
Denition 23 The Interaction Language IL
IL = f 2 OL j is an ordinary basic sentence without sk-functions.g
As we noted earlier, not being able to refer to FSK as a user is not really a restriction,
since there is still an innite supply of other function symbols available. Not being able to
mention O within a query is a restriction, but it seems of minor importance, since ASKing
whether something is all that is known has little practical value.
Denition 24 ASK at the Knowledge Level
Let S denote the set of all situations.
ASK : 2S IL
! fYES, NO, UNK, INCg
ASK
16
The obvious solution of a four-valued equality predicate without connection to the denotation function
would work but would also be very uninteresting.
15The terms knowledge and symbol level are due to [New81].
16It would be much more interesting if we had operators that allowed us to talk about more focussed
versions of only-believing such as: is this all you know about the driver of the red car? See [Lak92b] for a
proposal in this regard.
14
31
8 YES if M j= B ^ :B:
>>
< NO if M j=TT :B ^ B:
ASK[ M; ] =
>> UNK if M j=T :B ^ :B:
: INC otherwise:
Given this abstract characterization of ASK that makes no commitment as to how the
epistemic state of the system comes about, we now turn to the case where epistemic states
are represented by objective knowledge bases.
First note that ASK is the only means by which a user is able to probe what the system
believes. Thus, as far as the user is concerned, the epistemic state consists only of sentences
in IL. With this in mind, we call a set of sentences an external belief set i there is
a set of situations M such that
= f 2 IL j M j=T Bg:
From the discussion in Section 5, it follows that objective sentences uniquely determine
external belief sets in the sense that any two sets of situations that only-believe the same
objective sentence agree on the corresponding external belief set (a direct consequence of
Theorem 4 on page 22). Therefore, objective sentences can be said to represent external
belief sets.
Given this notion of representation, it is not hard to give a symbol level account of
ASK for objective existential-free KBs. In the next section, where we discuss TELL, we
will see that the restriction to objective existential-free representations suces for the
given interaction language. For the following theorem, recall that <[KB] is dened as
fs j s j=T KBg.
Theorem 9 ASK at the Symbol Level
Let KB be an ordinary, objective, and existential-free sentence in OL and let be in IL.
8 YES if j=OKB (B ^ :B:)
>><
NO if j=OKB (:B ^ B:)
ASK[ <[KB]; ] =
>> UNK if j=OKB (:B ^ :B:)
: INC otherwise:
Proof : It suces to show the following.
1: <[KB] j=T B i j=OKB B:
2: <[KB] j=T :B i j=OKB :B:
To prove the if-direction of 1:, let j=OKB B. If <[KB] is empty, then <[KB] j=T B
holds vacuously. Otherwise, since <[KB] j=T OKB, <[KB] j=T B follows by assumption.
Conversely, let <[KB] j=T B. Since KB is objective, KB is determinate. Therefore, by
Theorem 4, exactly one of j=OKB B and j=OKB :B holds. Since <[KB] j=T B by
assumption, j=OKB B follows.
The proof of 2: is analogous. While, in general, there are four possible answers that may be returned by ASK, there
are cases where we can narrow the range of answers. For example, if <[KB] is empty,
32
then ASK[ KB; ] = INC for all sentences in IL. This can only occur if the KB contains
an inconsistency that is due to equalities such as 8x(x 6= x). Note that j=O8x(x 6=
x) (B ^ B:) for all sentences in IL, since the only set of situations that supports
the truth of O8x(x 6= x) is the empty set.
If <[KB] is not empty and the query is a subjective existential-free sentence, the answer
is either YES or NO because the KB has complete and accurate knowledge of subjective
existential-free sentences (see page 17).
Finally, if the query is an existential-free sentence without function symbols using
equality as the only predicate, the answer is YES or NO as well. This is no coincidence
since, given a subjective existential-free sentence , <[KB] j=T i <[KB] j=T kk< KB
(Lemma 5.7), and kk< KB is an existential-free sentence without function symbols using
equality as the only predicate.
17
[
[
]
]
7.1.1 Open Queries
Before we turn to the TELL operation, let us briey consider one important extension to
the ASK operation. So far, ASK handles only so-called Yes/No-questions, that is queries in
the form of closed formulas. Often, however, one is interested in the individuals that satisfy
a parametrized query, that is, a formula with free variables. For example, an appropriate
answer to the query Teacher(x) consists of all the teachers that are known to the KB. Such
queries are often referred to as Wh-questions.
It is not hard to dene a ASKWH operation using the existing machinery that we developed. Let KB be an ordinary, objective, and existential-free sentence in OL and let be
a formula in IL with free variables ~x = hx ; : : :; xk i. Then let
ASKWH [ KB; ] = f~
n = hn ; : : :; nk i j j=OKB B[~x=~n]g:
Note that the set returned by ASKWH may be innite.
For example
ASKWH [ 8xP(x); P(x)]] = N . Nevertheless, it is possible to give a nite description of the
answer set using the k:k mechanism on page 25. Thus, an alternative denition of ASKWH
would be
ASKWH [ KB; ] = kk< KB :
In the above example, ASKWH [ 8xP(x); P(x)]] = (x = x), from which the set of satisfying
standard names can be read o.
1
1
[
]
7.2 TELL
ing the system a sentence means that the current epistemic state is transformed into
a new state which reects the changes that result from acquiring the belief . Capturing
this idea at the knowledge-level is somewhat more complicated than in the case of ASK.
The details are explained after the formal denitions are given. First, in contrast to ASK,
it is necessary that the current state be representable in the following sense:
TELL
An inconsistency due to predicates other than equality does not result in the empty set of situations.
For example, <[P( ) ^ :P( )] consists of all situations that support both the truth and falsity of P( ), of
which there are innitely many.
17
a
a
a
33
Denition 25 Representability
A set of situations M is representable i there is an objective existential-free sentence
KB such that M = <[KB].
Next, we need a way to uniquely pick new sk-functions relative to a given sentence.
Denition 26 Let sk choose be a function that, when given any two formulas and in OL, returns a sequence ~tsk of sk-terms for the existentially quantied variables at the
objective level of such that none of the functions symbols in ~tsk occurs in .
Note that such a function sk choose can easily be obtained by imposing an ordering on the
sk-functions and always picking the least available sk-functions that do not appear in .
The details are not important. For our purposes, all that matters is that such a function
exists and is easily computable.
Denition 27 TELL at the Knowledge Level
Let M be representable. Then M j=T OKB for some objective existential-free KB in OL.
Let be in IL with existentially quantied variables ~x at the objective level. Let ~tsk =
sk choose (KB; ), i.e. ~tsk is a sequence of sk-terms for ~x such that none of the sk-functions
in ~tsk appears in KB.
TELL[ M; ] = M \ fs j M; s j=T @ [ ~
x=~tsk ] g
To better understand the denition, let us rst look at the simpler case, where does not
contain existentially quantied variables at the objective level. In this case
[ M; ] = M \ fs j M; s j=T g;
TELL
which is of the same form as in the denition of TELL in [Lev84a]. The intuition behind
the intersection of M with another set is that an agent who acquires new knowledge
does so by ruling out certain possibilities (situations). If is objective, this amounts to
ruling out all those situations that do not support the truth of . In case mentions
beliefs, i.e., contains subformulas of the form B , these are interpreted with respect to
the current epistemic state M . For example, if the only known teacher at M is John,
TELL[ M; 8x(BTeacher(x) married(x))]] results in a new state that now believes that
John is married.
Note that the result of a TELL[ M; ] may be the empty set if is an inconsistent
subjective sentence such as BP(a) ^ :BP(a) or an inconsistent sentence about equality
such as 8x(x 6= x).
The occurrence of existentially quantied variables at the objective level of complicates the denition of TELL somewhat because existentials can only be known according
to our logic if the system has a name (an admissible term) for the existential. Also these
names must not reveal any information about the world. Thus, they are chosen to be
sk-terms (~tsk ), which do not belong to the interaction language.
Finally, it is necessary to choose sk-terms that are not already in use at the current
state M . For that reason, we restrict M to be representable, which allows us to determine
which sk-terms have been used so far (as given by KB).
34
As noted in [Lev84a], there are other, more subtle reasons, why one should restrict
to representable sets. Roughly, if we allowed M to be representable by an innite
set of sentences, which Levesque calls !-representable, it may happen that the result of a
TELL is no longer ! -representable.
Theorem 10 TELL at the Symbol Level
Let KB be an ordinary, objective, and existential-free sentence and a sentence in IL.
Let ~x be the existentially quantied variables at the objective level of and let ~tsk =
sk choose (KB; ) be sk-terms for ~x which do not already occur in KB. Then TELL[ <[KB]; ]
is representable and
TELL[ <[KB]; ] = <[KB ^ kk< KB@[~
x=~tsk ]]:
TELL
[
]
Proof :
Let TELL[ <[KB]; ] = M . By Denition 27, M = <[KB] \ fs j <[KB]; s j=T @[~x=~tsk ] g.
By lemma 5.7 on page 26, M = <[KB] \ fs j s j=T kk< KB@[~x=~tsk ]g.
Therefore, M = fs j s j=T KB ^ kk< KB@[~x=~tsk ]g. [
[
]
]
7.3 Decidability
If we restrict ourselves to sentences in IL without quantifying-in and equality, it is easy
to see that TELL and ASK provide us with a decidable KR service. The argument goes as
follows:
1. Creating an initial KB is trivial. Simply let KB = true.
2. TELL is decidable.
(a) For any objective KB and basic sentence in IL without quantifying-in or equality,
kk< KB is computable. (See the argument in the proof of Theorem 7.)
Actually, according to the denition of TELL and the initial KB , the KB may
well contain some equalities in the guise of true and false (recall that
true = (#1 = #1)). However, such occurrences can always be easily eliminated
or we obtain the trivial case where KB equals either true or false.
(b) Choosing new sk-terms for the existentially quantied variables in kk< KB is
trivial.
3. ASK is decidable.
This follows immediately from Theorem 7.
Of course, the question is whether this KR service can be extended to handle a wider class
of sentences without losing decidability. On the basis of Theorem 8, ASK can be extended
to cover queries with quantifying-in restricted to universally quantied variables and we
conjecture that arbitrary forms of quantifying-in do not result in undecidability either. As
for TELL, the main restriction is that the new sentence that is added to the KB does not
0
[
0
]
0
[
35
]
introduce arbitrary equality expressions. As noted at the end of Section 6, it is at least
possible to allow atomic ground equality expressions and their negations.
In summary, in this section we used OBLIQUE to provide a formal specication of a
KR service consisting of routines ASK and TELL, which allow a user to query a KB and to
add information to it. If the interaction language is restricted to sentences without equality
and quantifying-in, the KR service is provably decidable. Based on results and conjectures
of the previous section, we also briey discussed possible extensions.
7.4 An Example
Let us consider the following KB about teachers, employees and courses taught. Proper
names and course numbers are assumed to be standard names.
18
9
8
>> Teacher(Jack); Teacher(Jill); Teacher(father(Jill)); >>
>=
>< Emp(Jack); Emp(Jill); Emp(Sue)
KB = > Jack 6= father(Jill); Jill 6= father(Jill);
>>
>> Teach(Jack; csc378); Teach(Jill; csc484);
: Teach(father(Jill); csc148) _ Teach(father(Jill); csc228) >;
We view the KB as a set of sentences keeping in mind that, strictly speaking, TELL and ASK require
the KB to be in the form of a single sentence.
18
36
[ KB; +] =
Answer
ASK
1. 8x(Teacher(x) Emp(x))
UNK
The answer is UNK because there may be teachers who are not known
to the KB and who are not employees.
2. 8x(BTeacher(x) Emp(x))
YES
Note the dierence to the previous query. Here the KB is only asked
about the known teachers, namely Jack and Jill.
3. 8xEmp(x) Teacher(x)
UNK
Similar to the rst query, the answer is unknown because there may or
may not be an employee who is not a teacher.
4. 8xBEmp(x) BTeacher(x)
NO
In this case the answer is clearly no because Sue, a known employee, is
not among the known teachers.
5. 9xTeacher(x) ^ :BTeacher(x)
YES
The answer is yes because we know that the father of Jill is a teacher,
but we do not know who he is.
6. 9xEmp(x) ^ :BEmp(x)
UNK
In contrast to the previous question, it is not clear whether there is an
employee who is not yet known.
Notice that without introspection none of the distinctions the previous queries exhibit
would have been possible. Also, the answers would have been no dierent if we had used
implicit instead of explicit belief. The next queries, however, demonstrate that explicit
belief is indeed weaker.
37
[ KB; +] =
ASK
7. 9xTeach(father(Jill); x)
Answer
UNK
The answer is unknown since disjunctive generalization does not hold
in OBLIQUE . The failure of an inference due to existential generalization can be overcome, however, if we replace the original query by the
following:
8. 9xTeach(father(Jill); x) _ 9yTeach(father(Jill); y) YES
Note that (8) is equivalent to (7) in classical logic. Hence we observe
that there are cases, where the answers to queries under implicit belief can be approximated by explicit belief by introducing additional
disjunctions.
9. 8x(BTeacher(x) 9yTeach(x; y))
UNK
The answer is UNK in this case because there is no xed term that can
be substituted for y which would work for all known teachers.
Since there are only two known teachers, a reformulation of the last query similar to (8.)
would also give us the anser YES. Alternatively, a YES answer would obtain as well if the
KB contained
favorite course(Jack) = csc378 ^ favorite course(Jill) = csc484:
In that case, favorite course(x) is an admissible substitution for y when evaluating the
query.
While we have seen that the inability to apply existential generalization from disjunctions can be overcome by modifying the query, such an approach will not work to overcome
the inability to apply modus ponens, which is probably the main weakness of the logic.
For example, let us consider
KB0 = KB [ f8xTeacher(x) Emp(x)g;
that is, we add the taxonomic information that every teacher is also an employee. Then
[ KB0;
ASK
(father(Jill))]] = UNK;
Emp
even though both Teacher(father(Jill)) and Teacher(father(Jill)) Emp(father(Jill))
are explicitly believed. In [LM94], Lakemeyer and Meyer address this weakness by moving
to a sorted logic, where unary predicates like Emp and Teacher are used as sort symbols.
Since sort symbols are given a classical two-valued semantics, the inference that Jill's father
19
19
This extension is based on earlier work by Frisch [Fri91]. See also [Pat87] for a dierent approach.
38
is an employee will go through in this extended framework. Most importantly, it was shown
that the decidability results of the unsorted case can be retained in the stronger sorted
logic if the information about the two-valued sort symbols is restricted in an appropriate
manner. In a nutshell, these restrictions require that there is no disjunctive information
about sort symbols. Nevertheless this still allows fairly powerful taxonomic information to
be represented. The exact limit of expressiveness, which would lead to undecidability, is
still an open question.
We conclude our discussion with a brief look at TELL. First note that TELLing a KB an
objective sentence is rather trivial since we can just add that sentence directly except that
every existentially quantied variable has to be replaced by an sk-term. To illustrate how
non-objective arguments are handled by TELL, let KB be the knowledge base introduced at
the beginning of this subsection.
1. TELL[ KB; :BTeach(Jill; csc340) :Teach(Jill; csc340)]] =
KB ^ :Teach(Jill; csc340):
Here the KB is told to assume that Jill does not teach csc340 unless it already knows
otherwise.
2. TELL[ KB; 9xEmp(x) ^ :BEmp(x)]] =
KB ^ (Teacher(ask ) ^ (ask 6= Jack ) ^ (ask 6= Jill)) ^ (ask 6= Sue)):20
This time the KB is told that there is an employee other than the known ones. Given
the current contents of the KB, this means that there is an employee other than Jack,
Jill, or Sue.
8 Summary and Future Work
In this paper, we proposed a logic of explicit and implicit belief with full introspection. We
considered a rich rst-order language with function symbols, standard names, equality and
quantifying-in, which allows distinctions such as knowing that versus knowing who. While
the semantics of implicit belief uses classical possible worlds, explicit belief is modeled
by combining possible-world semantics with ideas from relevance logic. By incorporating
the notion of only-believing into the logic, we are able to completely specify the beliefs of
introspective rst-order knowledge bases within the logic. An analysis of the complexity of
reasoning of such KB's reveals that adding introspection to an existing model of limited deductive reasoning does not necessarily lead to an increase in complexity. As an application
of our results we show how the logic can be used to formally specify interaction routines
for limited introspective knowledge bases. The example KB we considered demonstrates
that, while the underlying notion of belief is considerably weaker than the classical one, it
is still possible to extract interesting and non-trivial information.
20Occurrences of true and false introduced by RES [ <[KB] BTeacher( )]] have been removed.
B
;
39
x
There are various avenues for future research that come out of this work. For one, the
decidability results should be extended. In particular, the question whether Theorem 8
holds for sentences with unrestricted use of quantifying-in needs to be settled. We already
mentioned our recent extension of OBLIQUE to a sorted logic, thereby strengthening the
inferential power considerably. An interesting question is how far this approach can be
pushed before decidability is lost again.
Up to now we have completely neglected the fact that OBLIQUE is also an autoepistemic logic (AEL). The main dierence compared to standard AEL is, of course, that belief
is no longer closed under classical logical implication. This opens the door for interesting
complexity considerations similar to those have been undertaken in propositional default
reasoning, for example [KS89]. In regular rst-order AEL, there is little to talk about
complexity-wise, since the deductive apparatus alone is already undecidable. However,
with OBLIQUE , we have an AEL where the deductive component restricted to objective sentences is decidable (ignoring equality for the moment). One question that seems
worth investigating then is what kinds of default statements can be added to an otherwise
objective knowledge base without losing decidability, or, where is the boundary between
decidability and undecidability in terms of the various kinds of defaults in the rst-order
case?
Acknowledgements
The work reported here is a substantial revision of part of my Ph.D. thesis at the University
of Toronto. I am most grateful to my thesis advisor, Hector Levesque, who endured long
hours of discussion on this subject over many years and whose own work has greatly
inuenced the ideas presented here. I wish to thank my colleagues at the University
of Toronto for providing such a stimulating research environment and the Institute of
Computer Science of the University of Bonn for its generous support while preparing this
paper. The comments of an anonymous referee are gratefully acknowledged. The work was
nancially supported by a Government of Canada Award and the Department of Computer
Science of the University of Toronto.
Appendix
Properties of Explicit Belief
In order to prove many of the properties about explicit belief, it is useful to rst show
that, if a given situation supports the truth of a sentence with all existentials replaced by
admissible terms, then the situation also supports the original sentence, that is, when the
existentials are treated as usual. The following three lemmas establish this result.
Lemma 8.1 Let be a formula with one free variable x and M a set of situations. Then
for any situation s and closed term t such that ds (t) = n,
1: M; s j=T [ x=t] i M; s j=T xn
2: M; s j=F [ x=t] i M; s j=F xn
40
Proof : By a simple induction on the structure of (omitted). Corollary 8.2 Let , M , s, and t be as above. Then
M; s j=T [ x=t] implies M; s j=T 9x
Lemma 8.3 Let be a sentence such that every variable at the objective level is bound
exactly once. Let 9x= be with 0 or 1 occurrences of 9x at the objective level of replaced by . Let t be a term such that 9x= [ x=t] is a sentence. (In particular, any
admissible term for x satises this condition.) Then
1: M; s j=T 9x= [ x=t] implies M; s j=T if 9x occurs in an even number of :
2: M; s j=F 9x= [ x=t] implies M; s j=F if 9x occurs in an odd number of :
Proof : By induction on the structure of .
Base case: In the case of an atomic sentence, there are no occurrences of a subformula
9x and the lemma holds vacuously.
case ::
1. Let us assume that 9x occurs in an even number of :'s.
If
M; s j=T (:)9x= [ x=t] then M; s j=F 9x= [ x=t] , where 9x now occurs in
an odd number of :. Thus, by induction (2), M; s j=F and, therefore,
M; s j=T :.
2. The argument is symmetric to (1).
case _ :
1. Let M; s j=T (_ )9x= [ x=t] . Then M; s j=T 9x= [ x=t] or M; s j=T 9x= [ x=t] .
By induction, M; s j=T or M; s j=T , which immediately implies M; s j=T _
.
2. Analogous to 1.
case 9y:
1. If 9y = 9x , then the case reduces to Corollary 8.2. Otherwise, let
M; s j=T (9y)9x= [ x=t] , which is the same as M; s j=T 9y(9x= )[[x=t] . Then
M; s j=T (9x= [ x=t] )[y=n] for some n 2 N . We can push the substitutions of
y into the components, which is unproblematic because variables are assumed
to be bound just once. Thus we obtain M; s j=T ([y=n]9x y=n = y=n [ x=t[y=n]]]).
By induction, we obtain M; s j=T [y=n] and, hence, M; s j=T 9y.
2. Since 9x is assumed to occur within an odd number of :'s, 9x must be
properly contained in 9y. Let M; s j6=F 9y, that is, for some n 2 N ,
M; s j6=F [y=n]. By induction, M; s j6=F ([y=n]9x y=n = y=n [ x=t[y=n]]]) and,
therefore, M; s j6=F (9x= [ x=t] )[y=n], which implies M; s j6=F 9y()9x= [ x=t]
and, nally, M; s j6=F (9y)9x= [ x=t] .
[
41
]
[
]
[
]
[
]
case B: Since we only allow substitutions at the objective level, no substitutions can
occur in this case and the lemma holds vacuously.
Lemma 8.4 Let be a sentence such that every variable at the objective level is bound
exactly once. Let ~x be a sequence of the existentially quantied variables at the objective
level of . For any situation s and set of situations M and admissible terms ~t for ~x,
if M; s j=T @[~x=~t] then M; s j=T :
Proof : By repeated application of Lemma 8.3. Lemma 8.5 Subjective sentences are two-valued.
Let be a subjective sentence, s a situation, and M a set of situations. Then
M; s j=T i M; s j6=T :
Proof : By a simple induction on the structure of (omitted). We now turn to the proofs of some of the properties of belief listed in Section 3.5.2.
Perfect Introspection
1. j=B BB
2.j=:B B:B
The proof follows immediately from the fact that there is a single, globally accessible
set of situations.
Quantifying-In
3.j=Bxn 9xB
4.j=9xB B9x
5.j=Bxa B9x
6.j=8xB B8x provided is existential-free
7.j=B8x 8xB
The proofs are the same as for the logic BLq of [Lak92c, Lak94]. Here we prove
properties 6 and 7:
To prove property 6, let M j=T 8xB. Then, since is assumed to be existential-free,
for all n 2 N and for all s 2 M , s j=T xn. Therefore, for all s 2 M and for all n 2 N ,
s j=T xn, which implies M j=T B8x.
To prove property 7, let M j=T B8x. Then there are admissible t ; : : :; tk for the existentially quantied variables x ; : : :; xk of such that for all s 2 M , s j=T 8x@[t ; : : :; tk ].
This implies that for all s 2 M and for all n 2 N , s j=T (@[t ; : : :; tk ])xn, which can be
written as s j=T (xn)@[(t )xn; : : :; (tk )xn ]. Therefore, M j=T Bxn for all n 2 N because
the (ti)xn are admissible for xn . M j=T 8xB follows now immediately.
1
1
1
1
1
42
8. j=(B( ) ^ B) B
Let ~x, ~y, and ~z be the existentials in , :, and , respectively. Let M j=T B and
M j=T B( ). If M is empty, the statement holds vacuously. Otherwise,
there are admissible ~t, ~u, and ~v such that for all s 2 M , M; s j=T @[~x=~t] and
M; s j=T (:)@[~y=~u] _ @[~z=~v] . By Lemma 8.4, M; s j=T and, by Lemma 8.5,
M; s j=
6 T :. Now assume that M; s j=T (:)@[~y=~u] for some s 2 M . Then, by
Lemma 8.4, M; s j=T :, a contradiction. Thus for all s 2 M , M; s j=T @[~z=~v] and,
therefore, M j=T B.
9. j=(B ^ :Bfalse) Let M j=T B ^ :Bfalse. Then M is non-empty. Then M j=T follows from
Lemma 8.4 and the fact that M; s j=T is independent of the choice of s.
10. j= B provided is existential-free.
Immediate, since the proviso guarantees that = @.
11. j=B L
Let M be any set of situations. We need to show that if M j=T B then M j=T L.
Thus, let us assume that M j=T B and let ~x be the existentially quantied variables at the objective level of . Then for some admissible ~t and for all s 2 M;
M; s j=T @[~x=~t] . By Lemma 8.4, for all s 2 M; M; s j=T and, therefore, for all
worlds w 2 M; M; w j=T , which immediately implies that M j=T L.
12. j=O B
Follows immediately from the semantics of O.
Proof of Theorem 8
For the proof of the theorem we need several denitions and lemmas. The reader is advised
to skip these on a rst reading and to start with the main proof of the theorem, consulting
the lemmas and denitions as needed.
Denition 28
A standard name n occurs plain in a formula i contains an equality expression
(t = n) or (n = t).
Denition 29 For any formulas and , let N; be the set of all standard names occurring in or .
Lemma 8.6 Let : N ! N be a surjective function over the standard names. is
extended to apply to terms and formulas in the usual way. Let N_ N such that jN (
_
restricted to N_ ) is bijective. (Such a N_ always exists because is assumed to be surjective.)
We write _ instead of jN_ .
For any situation s =< T; F; d > dene a situation s =< T ; F ; d > s.t.
(~n) 2 T (F ) i P( (~n)) 2 T (F ) for every primitive formula P(~n):
P
43
Let d be such that for all non-rigid closed terms t, d (t) = _ 1 (d( (t))). (It is easy to
show that such d exists.) Finally, given any set of situations M , let M = fs j s 2 M g.
Then for all objective such that every =-expression in is variable free and every
standard name that occurs plain in is a member of N_ ,
M ; s j=T i M; s j=T ()
M ; s j=F i M; s j=F ()
Proof : The proof is by induction on the structure of .
First, it is easy to see that for any closed term u, (d (u)) = d( (u)). For, if u is non-rigid,
this follows immediately from the construction of d . If u is a standard name, then the
fact that denotation functions are the identity function when applied to standard names
implies (d (u)) = (u) = d( (u)).
case atomic: Let = P(~t), where P is not the equality predicate.
M ; s j=T P(~t) i P(d (~t)) 2 T i P( (d (~t))) 2 T (by denition of T )
i P(d( (~t))) 2 T i M; s j=T P( (~t)). The case M ; s j=F P(~t) is treated the
same way.
Let us now turn to the additional base case involving equality. Let t and t0 be closed
terms. Since = is two-valued (an =-expression has true-support i it does not have
false-support), it suces to prove the case of true-support.
Let s j=T t = t0. By the denition of =, d (t) = d (t0) and, therefore, (d (t)) =
(d (t0)). Since (d (t)) = d( (t)) and (d (t0)) = d( (t0)), d( (t)) = d( (t0)), which
implies s j=T (t = t0).
Conversely, let s j=T (t = t0):
case i: both t and t0 are non-rigid.
Then, since d( (t)) = d( (t0)), d (t) = _ (d( (t))) = _ (d( (t0))) = d (t0),
which implies s j=T (t = t0).
case ii: both t and t0 are standard names.
Then, since both t and t0 occur plain, t and t0 are members of N_ . From d( (t)) =
d( (t0)) and the denition of denotation functions we obtain (t) = (t0). Then
t and t0 are the same because _ = jN is bijective, and s j=T (t = t0) follows.
case iii: t is a standard name and t0 is non-rigid.
Then d (t0) = _ (d( (t0))) = _ ( (t)) = t (since t 2 N_ ),
which implies s j=T (t = t0).
case iv: t is non-rigid and t0 is a standard name.
This case is symmetric to case (iii).
The cases for : and _ follow easily by induction.
case 9x: Let M ; s j=T 9x. Thus for some n 2 N , M ; s j=T xn and, by induction,
M; s j=T (xn ). (Note that the induction hypothesis applies because =-expressions in
are assumed to be variable free.) M; s j=T (xn ) can be rewritten as M; s j=T ()x n ,
from which M; s j=T (9x) follows.
1
1
_
1
1
( )
44
Conversely, assume M; s j=T (9x). Then for some n 2 N , M; s j=T ()xn . Since is surjective, there is an n0 2 N such that n = (n0). Therefore, M; s j=T (xn0 ) and,
by induction, M ; s j=T xn0 , which immediately implies M ; s j=T 9x.
A similar argument proves that M ; s j=F 9x i M; s j=F (9x).
case B: Let M ; s j=T B.
Then for some admissible ~t and for all s0 2 M , M ; s0 j=T s @[~t] .
By induction, M; s0 j=T (s @[~t] ) for all s 2 M . Since ds ( (t)) = (ds (t)) for all
closed terms t, we obtain that for all s0 2 M , M; s0 j=T ()s @[ (~t)]], from which
M; s j=T (B) follows.
Conversely, let M; s j=T (B). Then there are admissible ~t such that for all s0 2 M ,
M; s0 j=T ()s @[~t] . Since is surjective, there are admissible ~t0 such that (~t0) = ~t.
Therefore, for all s0 2 M , M; s0 j=T ()s @[ (~t0)]]. Again using the fact that ds ( (t)) =
(ds (t)) for all closed terms t, we obtain that for all s0 2 M , M; s0 j=T (s @[~t0] ).
Thus, by induction, for all s0 2 M , M ; s0 j=T s @[~t0] , from which M ; s j=T B
follows. Lemma 8.7 Let be an ordinary, objective, and existential-free sentence such that all =expressions in are closed (no variables). Let be an ordinary, objective, and quantierfree formula with free variables ~x = x1 ; : : :; xk . Let J = fn1 ; : : :; nk g be a set of standard
names occurring nowhere in or .
Let I = fn j n 2 N and n occurs plain in or g [ J . Then
j=B 8~xB i j=B B [~x=~n] for all ~n 2 I k :
Proof : The only-if direction follows directly from the meaning of 8.
To prove the if direction, let us assume that j=B B [~x=~n] for all ~n 2 I k and j6=B 8~xB .
Then for some set of situations M and ~n0 = n01; : : :; n0k , M j=T B and M j6=T B [~x=~n0]. Let
J^ = fn^ 1; : : :; n^lg be the largest subset of fn01; : : :; n0k g such that J^\ I = fg. By assumption,
J^ is non-empty. Thus there is a set J~ = fn~ 1; : : :; n~ lg s.t. J~ J and J~\fn01; : : :; n0k g = fg.
Note that, by the denition of I , if n^i occurs in or , then n^ i does not occur plain in or .
Let us dene a surjection : N ! N such that
1: (n) = n 8n 2 N; (the standard names in and )
2: (~ni ) = n^i 8n~ i 2 J~
3: (nj ) = nj 8nj 2 J J~
To apply Lemma 8.6, let N_ N contain N; and fn j n 2 J and (n) 62 N; g such
that _ = jN_ is bijective. (Since N; and fn j n 2 J and (n) 62 N; g are disjoint, such
a _ always exists.)
M j=T B implies 8s 2 M; s j=T () because is existential-free and = (). If
we dene s for every s 2 M as in Lemma 8.6, the lemma implies that 8s ; s j=T . If
we let M = fs j s 2 M g, M j=T B follows.
Also, by assumption M j6=T B [~x=~n0]. First, we show that there is an ~n00 2 I k s.t.
(~n00) = ~n0.
45
case i: n0i 2 I .
Then choose n00i = n0i. For, if n0i 2 N; and n0i occurs plain in or , (n0i) = n0i by
construction of (condition 1.). Otherwise, if n0i 2 J , then n0i 62 J~, then (n0i ) = n0i
as well by construction of (condition 3.).
case ii: n0i 62 I .
Then n0i = n^ j for some n^j 2 J^. Then let n00i = n~j (2 J ).
Thus M j6=T (B [~x=~n00]) for some ~n00 2 I k , which implies that there is an s 2 M
such that s j6=T [~x=~n00] and, consequently, M j6=T B [~x=~n00] contradicting the assumption
that j=B B [~x=~n] for all ~n 2 I k. Lemma 8.8 Let and be ordinary objective sentences without equality such that is
existential-free and contains no universally quantied variables.
Let ~x = hx1; : : : ; xk i be a sequence of the existentially quantied variables in .
Then there is a computable, nite set of substitutions ~t for ~x such that for any substitution
of closed terms ~t0 for ~x,
j=B B @[~x=~t0] i ~t0 is a ground instance of some member of .
Proof : VWithout loss of generality, let = 8~z V i be in prenex conjunctive normal form.
Let 0 = j be the matrix of in conjunctive normal form.
By a result in [Lak94], j=B B i there is a sequence of closed terms ~t such that for
every j there are ~u and i such that i[~z=~u] j [~x=~t].
With that it is not hard to show that there is a nite set of most general substitutions
of ~x such that any ~t that satises the above subsumptions is an instance of a member of .
The details of how to compute can be found in Patel-Schneider's t-entailment algorithm
( [Pat87], page 37{38). Theorem 8 Let be an ordinary objective sentence without equality. Let be an ordinary
basic sentence without equality such that any free variable x in a subformula B in is a
universally quantied variable in . In other words, only universally quantied variables
may participate in quantifying-in. Then the validity of O B is decidable.
Proof :
Let ~x be a sequence of the existentially quantied variables at the objective level of .
Let ~y be a sequence of the universally quantied variables at the objective level of .
As in the proof of Theorem 7 on page 29, j=O B i j=B0 Bk k< 0 , where 0
is with all existentials replaced by sk-terms. Note that, in contrast to Theorem 7,
k k< 0 may contain equality expressions other than (#1 = #1) due to free variables in
subformulas B . In particular, given the restrictions on , it is easy to see that k k< 0
contains only equality expressions of the form (t = t0), where t and t0 are either standard
names or universally quantied variables. Similar to the proof of the previous theorem, it
suces to establish the decidability of the validity problem for B0 B 0 for an arbitrary
objective and existential-free 0 and any objective 0, whose equality expressions contain
only standard names or universally quantied variables. A simple induction proof then
establishes that k k< 0 can be eectively computed.
[
[
]
]
[
[
]
46
]
For any such 0, let 00 denote its matrix. Let n; : : : ; nl be distinct standard names
occurring nowhere in N;0 and let I = fn j n occurs plain in 0g [ fn; : : :; nl g.
1. j=B0 B 0 i there are admissible terms ~t for ~x such that j=B0 B 0@[~x=~t]
(Lemma 6.6).
2. Let ~t be any admissible sequence of terms for ~x.
j=B0 B 0@[~x=~t]
i j=B0 B8~y 00[~x=~t]
by Lemma 6.2
0
00
~
i j=B 8~yB [~x=t]
since j=B8~y 00[~x=~t] 8~yB 00[~x=~t]
l
0
00
i for all ~n 2 I ; j=B B( [~x=~t])[~y=~n] by Lemma 8.7.
1
1
3. Next we show that one can always eliminate occurrences of equality expressions from
00[~y=~n]. Let ~n 2 I l.
From the denition of k:k< 0 and the restrictions on it follows that any equality
expression in 00[~y=~n] is of the form (n = m) for standard names n and m. Since
these have a xed interpretation, we can replace them by true and false accordingly
and then simplify the formula as in the proof of the previous theorem. Thus 00[~y=~n]
reduces to true, false, or a sentence without occurrences of =. In the latter case,
call 00[~y=~n] non-trivial.
4. Let ~n 2 I l be such that 00[~x=~n] is non-trivial. By Lemma 8.8, there is a nite set (~n)
of most general substitutions for ~x such that for any closed terms ~t = ht ; : : : ; tki,
j=B0 B( 00[~x=~t])[~y=~n] i ~t is a ground instance of a member of (~n):
^ ~n), which allows us later (part 5c) to decide
We now transform each (~n) into a set (
whether admissible terms ~t exist that work for all ~n 2 I l.
Without loss of generality, let no two members of (~n) share any variables and let
all variables be dierent from ~y. Let ht ; : : : ; tk i 2 (~n), where ti is a substitution
for the existentially quantied variable xi of 0 for 1 i k.
[
]
1
1
A term t^i is called compatible with ti i
1. for any variable y 2 fy ; : : :; ylg that occurs in ti, xi is bound in 0 within the
scope of y.
2. ^ti[~y=~n] = ti.
1
Let ~a = ha ; : : : ; ali be distinct constants occurring nowhere in 0 or 0. Let
^ ~n) = fht^ [~y=~a]; : : :; ^tk [~y=~a]i j ht ; : : :; tk i 2 (~n)
(
and t^i is compatible with ti for 1 i kg:
(Note that the members of ^ (~n) are obtained from those of (~n) by replacing some
of the standard names in ~n by the corresponding constants in ~a. This is done to
^ ~n).)
allow the unication of members of dierent (
Since (~n) is nite and for any ti there are only nitely many ways to obtain a
^ ~n) is also nite.
compatible t^i, (
1
1
1
47
5. We are now ready to prove the decidability of the question: does there exist a sequence
of admissible ~t such that for all ~n 2 I l, j=B0 B( 00[~x=~t])[~y=~n]?
(a) First check whether there are ~n 2 I l such that 00[~y=~n] reduces to false. In
such a case, j6=B0 B( 00[~x=~t])[~y=~n] for all admissible ~t. Therefore, by part 1
and 2, j6=B0 B 0 and we are done.
(b) For any ~n 2 I l such that 00[~y=~n] reduces to true, j=B0 B( 00[~x=~t])[~y=~n] for
all admissible ~t.
(c) Thus it suces to show the decidability of the question: does there exist a
sequence of admissible ~t such that for all ~n 2 I l with a non-trivial 00[~x=~n],
j=B0 B( 00[~x=~t])[~y=~n].
We now show that such a ~t exists i one can pick one element from each of the
^ (~n) and unify them. Since the ^ (~n) are nite and can be eectively computed,
decidability follows.
^ ~n) and assume the ~t~n unify. Then there
i. If direction: for all ~n, let ~t~n 2 (
exists a most general instance ~t of all the ~t~n and let ~t0 be a ground instance
of ~t obtained by replacing the variables by arbitrary standard names. By
construction of the ^ (~n), ~t0[~a=~y] is admissible for ~x in 0, and for all ~n 2 I l
with a non-trivial 00[~x=~n], j=B0 B( 00[~x=(~t0[~a=~y])])[~y=~n].
ii. Only-if direction: let ~t be admissible such that for all ~n 2 I l with a nontrivial 00[~y=~n], j=B0 B( 00[~x=~t])[~y=~n].
Thus for every ~n 2 I l with a non-trivial 00[~x=~n], ~t[~y=~n] is an instance of
some ~t~n 2 (~n). In other words, the ~t~n unify. Then for each ~t~n there is an
^ ~n) such that ~t[~y=~a] is a unier of these ~t~0n.
~t~0n 2 (
This completes the proof of Theorem 8. 48
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