A Tableau Algorithm for ALCN (◦, t)
Fabio Grandi
IEIIT.BO-CNR and DEIS,
Alma Mater Studiorum – Università di Bologna,
Viale Risorgimento 2, I-40136 Bologna, Italy
Email: [email protected]
Abstract
Description Logics have been successfully used as knowledge representation
formalisms in a wide range of application domains. Expressive Description Logics
ALCN (M ) can be defined as extensions of the well-known concept language ALC,
allowing for number restrictions on complex role expressions built with constructors M ⊆ {◦,− , t, u}. Baader and Sattler proved that ALCN (◦) is decidable, but
the addition of other operators may easily lead to undecidability, as it happens
to ALCN (◦, u) and ALCN (◦,− , t). Furthermore, we showed that ALCN (◦)
extended with inverse roles (both in number and in value restrictions) becomes
undecidable, whereas it can be safely extended with qualified number restrictions
without losing decidability. In this work, we further investigate the computational properties of the ALCN family, by showing that ALCN (◦, t) is decidable.
To this end, we provide an effective decision procedure for ALCN (◦, t)-concept
satisfiability (and subsumption) in the form of a tableau-based algorithm.
1
Introduction
Description Logics (DLs) have been successfully used as knowledge representation formalisms in a wide range of application domains [1]. Expressiveness of DLs is based
on the definition of complex concepts and roles, which can be built by means of constructors, starting from a set of (atomic) concept names NC and a set of (atomic)
role names NR. Well-known DLs are ALC [8], which allows for Boolean propositional
constructors on concepts and (universal and existential) value restrictions on atomic
roles, and its extension ALCN [4, 7] introducing (non-qualified) number restrictions
on atomic roles. However, in order to better fulfill requirements of real-world application domains, more expressive extensions of the basic concept languages have
been investigated. One direction along which useful extensions have been sought
is the introduction of complex roles under number restrictions. In fact, considering
role composition (◦), inversion (− ), union (t) and intersection (u), expressive extensions of ALCN can be defined as ALCN (M ) with the adoption of role constructors
M ⊆ {◦,− , t, u} [2].
Several (un)decidability results are available for ALCN (M ) DLs. In [2], Baader
and Sattler showed that concept satisfiability is decidable in ALCN (◦) (for which
they provided a Tableau algorithm), whereas it is undecidable in ALCN (◦, u) and
ALCN (◦,− , t). They also observed that ALCN (− , t, u) is decidable since it can be
65
C, D → A |
AI ⊆ ∆I
>|
>I
⊥|
⊥I
¬C |
(¬C)I
C uD |
(C u D)I
C tD |
(C t D)I
∀R.C |
(∀R.C)I
∃R.C |
(∃R.C)I
∃≥n R |
(∃≥n R)I
∃≤n R |
(∃≤n R)I
R, S → P |
P I ⊆ ∆I × ∆I
R◦S |
(R ◦ S)I
RtS |
(R t S)I
atomic concept
∆I
∅
∆I \ C I
C I ∩ DI
C I ∪ DI
{i ∈ ∆I | ∀j. RI (i, j) ⇒ C I (j)}
{i ∈ ∆I | ∃j. RI (i, j) ∧ C I (j)}
{i ∈ ∆I | ]{j ∈ ∆I | RI (i, j)} ≥ n}
{i ∈ ∆I | ]{j ∈ ∆I | RI (i, j)} ≤ n}
atomic role
= {(i, j) ∈ ∆I × ∆I | ∃k. RI (i, k) ∧ S I (k, j)}
= RI ∪ S I
=
=
=
=
=
=
=
=
=
Figure 1: Syntax and semantics of ALCN (◦, t).
translated into C 2 [3], that is the two-variable FOL fragment with counting quantifiers,
which has proved to be decidable [6].
In [5], we considered further extensions of ALCN (◦) and showed that the addition of role inversion (both in number and value restrictions) leads to undecidability,
whereas concept satisfiability is still decidable when we add qualified number restrictions.
In this paper, we attack the main problem left open in this framework: the decidability of ALCN (◦, t). In particular, we will show in Sec. 2 that ALCN (◦, t)-concept
satisfiability is decidable and provide an effective decision procedure in the form of a
tableau-based algorithm. ALCN (◦, t) is a quite interesting language, as it is powerful
enough to express intersection of role chains, if they have a finite number of successors. For example, the concept description ∃≥2 R u ∃≥2 S ◦ T u ∃≤2 (R t S ◦ T ) defines
domain objects which have two R-successors which are also (S ◦ T )-successors. For
this reason, ALCN (◦, t) does not have the tree model property (which is usually a
guarantee of a good computational behaviour [9]). However, unlike other expressive
Description Logics without the tree model property which were proved undecidable
(e.g. ALCN (◦, u)), ALCN (◦, t) does not have the ability to express infinity axioms
and, thus, remains decidable. In fact, an easy consequence of the results that will be
proved in this paper is that ALCN (◦, t) has the finite model property.
The syntax rules at the left hand side of Fig. 1 inductively define valid concept
and role expressions for ALCN (◦, t). As far as semantics is concerned, concepts are
interpreted as sets of individuals and roles as sets of pairs of individuals, as usual.
Formally, an interpretation is a pair I = (∆I , ·I ), where ∆I is a non-empty set of
individuals (the domain of I) and I is a function (the interpretation function) which
maps each concept to a subset of ∆I and each role to a subset of ∆I × ∆I , such that
the equation at the right hand side of Fig. 1 are satisfied. The concept description C
is satisfiable iff there exist an interpretation I such that C I 6= ∅; in this case, we say
that I is a model for C. The concept description D subsumes the concept description
C (written C v D) iff I I ⊆ DI for all interpretations I; concept descriptions C and
D are equivalent iff C v D and D v C. In ALC and its extensions, subsumption can
be reduced to concept satisfiability and vice versa: C v D iff C u ¬D is unsatisfiable
and C is satisfiable iff not C v A u ¬A, where A is an arbitrary concept name.
66
2
Decidability of ALCN (◦, t)
We will show in this Section how an effective decision procedure for ALCN (◦, t)concept satisfiability can be provided as a tableau-based algorithm. To this end,
we consider ALCN (◦, t)-concept descriptions in Negation Normal Form (NNF [8]),
where the negation sign is allowed to appear before atomic concepts only. In fact,
ALCN (◦, t)-concept descriptions can be transformed into NNF in linear time via application of the same rules which can be used for ALCN (pushing negations inwards):
¬∃≤n R → ∃≥n+1 R
¬∃R.C → ∀R.¬C
¬∃≥n R → ∃≤n−1 R (⊥ if n = 0)
¬∀R.C → ∃R.¬C
in addition to the absorption rule for double negations and De Morgan’s laws for u
and t. We define the concept descriptions obtained in this way as in NNF.
Furthermore, in complex role expressions appearing under number restrictions,
which involve role compositions and disjunctions, we can make use of the following
substitution rules (pushing disjunctions outwards):
R ◦ (S1 t S2 ) → R ◦ S1 t R ◦ S2
(S1 t S2 ) ◦ R → S1 ◦ R t S2 ◦ R
Such substitutions are semantically safe. In fact, for the former we have:
(R ◦ (S1 t S2 ))I
¡
¢
= {(i, j) ∈ ∆I × ∆I | ∃k. RI (i, k) ∧ S1I (k, j) ∨ S2I (k, j) }
¡
¢
= {(i, j) ∈ ∆I × ∆I | ∃k. RI (i, k) ∧ S1I (k, j) ∨ RI (i, k) ∧ S2I (k, j) }
¡
¢ ¡
¢
= {(i, j) ∈ ∆I × ∆I | ∃k. RI (i, k) ∧ S1I (k, j) ∨ ∃k. RI (i, k) ∧ S2I (k, j) }
= {(i, j) ∈ ∆I × ∆I | ∃k. RI (i, k) ∧ S1I (k, j)} ∪
{(i, j) ∈ ∆I × ∆I | ∃k. RI (i, k) ∧ S2I (k, j)}
= (R ◦ S1 t R ◦ S2 )I
Similarly, also the latter can be proved. Obviously, the substitutions are also correct
if R, S1 , S2 are complex role expressions. Hence, via repeated applications of the
rules above, we can transform (in the worst case in exponential time) any general
role expression R(◦, t) appearing under number restrictions into a disjunction of role
chains:
R(◦, t) →
c
t R1k ◦ R2k ◦ · · · ◦ R`kk
k=1
and consider ALCN (◦, t) role expressions rewritten as disjunctions of role chains to
be in Role Normal Form (RNF). Hence, in the following, we assume to deal with
ALCN (◦, t)-concept descriptions in NNF with role expressions in RNF.
The Tableau algorithm we are going to introduce manipulates, as basic data structures, ABox assertions involving domain individuals.
Definition 1 Let NI be a set of individual names. An ABox A is a finite set of
assertions of the form C(a) –concept assertion– or R(a, b) –role assertion– where C is
67
a concept description, R a role name, and a, b are individual names. An interpretation
I, which additionally assigns elements aI ∈ ∆I to individual names a, is a model of
an ABox A iff aI ∈ C I (resp. (aI , bI ) ∈ RI ) for all assertions C(a) (resp. R(a, b)) in
A. The ABox A is consistent iff it has a model. The individual a is an instance of
the description C w.r.t. A iff aI ∈ C I holds for all models I of A. We also consider
in a ABox inequality assertions of the form a 6= b, with the obvious semantics that an
interpretation I satisfies a 6= b, iff aI 6= bI . Inequality assertions are assumed to be
symmetric, that is saying that a 6= b ∈ A is the same as saying b 6= a ∈ A.
Definition 2 The individual y is a (R1 ◦R2 ◦· · ·◦R` )-successor of x in A iff ∃y2 y3 . . . y`
variables in A such that {Ri (yi , yi+1 ) | 2 ≤ i ≤ `−1}∪{R1 (x, y2 ), R` (y` , y)} ⊆ A. The
individual y is a R(◦, t)-successor of x in A, where R(◦, t) = tck=1 R1k ◦ R2k ◦ · · · ◦ R`kk ,
iff y is a (R1k ◦ R2k ◦ · · · ◦ R`kk )-successor of x for some k (1 ≤ k ≤ c).
Notice that, owing to this definition, role successors in A are also successors in every
model I of A: if I satisfies A, and y is an (R1 ◦ R2 ◦ · · · ◦ Rm )-successor (resp. R(◦, t)successor) of x in A, then y I is an (R1 ◦R2 ◦· · ·◦Rm )-successor (resp. R(◦, t)-successor)
of xI in I.
Definition 3 An ABox A contains a clash iff, for an individual name x ∈ NI, one of
the two situations below occurs:
• {A(x), ¬A(x)} ⊆ A, for a concept name A ∈ NC;
• (∃≤n R(◦, t))(x) ∈ A and x has p R(◦, t)-successors y1 , . . . , yp with p > n such
that {yi 6= yj | 1 ≤ i < j ≤ p} ⊆ A, for role names {R | R is in R(◦, t)} ⊆ NR
and an integer n ≥ 0.
To test the satisfiability of an ALCN (◦, t) concept C in NNF with role expressions in RNF, the proposed algorithm works as follows. Starting from the initial ABox
{C0 (x0 )}, it applies the completion rules in Fig. 2, which modify the ABox. It stops
when no rule is applicable (when a clash is generated, the algorithm does not immediately stops but it always generate a complete ABox). An ABox A is called complete
iff none of the completion rules is any longer applicable. The algorithm answers “C is
satisfiable” iff a complete and clash-free ABox has been generated. The ALCN (◦, t)algorithm is non-deterministic, due to the t-, ≥- and ≤-rules (for instance, the t-rule
non-deterministically chooses which disjunct to add for a disjunctive concept).
In particular, the Tableau algorithm in Fig. 2 is basically an extension of the
algorithm for ALCN (◦) proposed in [2]. The main difference is in the treatment of
the ≥-rule. In particular, when processing an assertion (∃≥n R(◦, t))(x), the new
R(◦, t)-successors which are generated for x are shared out among the different role
chains which make up R(◦, t) in RNF. In particular, if R(◦, t) = tck=1 R1k ◦R2k ◦· · ·◦R`kk
and x already has in A:
p1
(R11 ◦ R21 ◦ · · · ◦ R`11 )-successors: y1 , y2 , . . . , yp1 ,
p2 (R12 ◦ R22 ◦ · · · ◦ R`22 )-successors: yp1 +1 , yp1 +2 , . . . , yp1 +p2 ,
..
.
pc
(R1c ◦ R2c ◦ · · · ◦ R`cc )-successors: yp1 +···+pc−1 +1 , yp1 +···+pc−1 +2 , . . . , yp1 +···+pc−1 +pc ,
68
u-rule:
if
1. (C1 u C2 )(x) ∈ A and
2. {C1 (x), C2 (x)} 6⊆ A
then
t-rule:
if
A0 := A ∪ {C1 (x), C2 (x)}
1. (C1 t C2 )(x) ∈ A and
2. {C1 (x), C2 (x)} ∩ A = ∅
then
∃-rule:
if
1. (∃R.C)(x) ∈ A and
2.
then
∀-rule:
if
then
if
there is no y with {R(x, y), C(y)} ⊆ A
A0 := A ∪ {R(x, y), C(y)} where y is a fresh variable
1. (∀R.C)(x) ∈ A and
2.
≥ -rule:
A0 := A ∪ {D(x)} for some D ∈ {C1 , C2 }
there is a y with R(x, y) ∈ A and C(y) 6∈ A
A0 := A ∪ {C(y)}
c
1. (∃≥n ( t R1k ◦ R2k ◦ · · · ◦ R`kk ))(x) ∈ A and
k=1
2. for each k (1 ≤ k ≤ c), x has exactly pk (R1k ◦ · · · ◦ R`kk )-successors
yp1 +···+pk−1 +1 , . . . , yp1 +···+pk−1 +pk , with p1 + · · · + pc = p < n
such that {yi 6= yj | 1 ≤ i < j ≤ p} ⊆ A
then
choose dk ≥ 0 (1 ≤ k ≤ c) such that d1 + d2 + · · · + dc = n − p,
k
k
k
k
k
A0 := A ∪ {R1k (x, zi2
), R2k (zi2
, zi3
), R3k (zi3
, zi4
), . . .
k
. . . , R`kk (zi`
, zd1 +···+dk−1 +i ) | 1 ≤ k ≤ c, 1 ≤ i ≤ dk }
k
∪{zi 6= zj | 1 ≤ i < j ≤ n − p}
∪{yi 6= zj | 1 ≤ i ≤ p, 1 ≤ j ≤ n − p}
where
k
zij
(for 1 ≤ k ≤ c, 1 ≤ i ≤ dk , 2 ≤ j ≤ `k ) and
P
zi (for 1 ≤ i ≤ n − p) are ck=1 dk `k fresh variables
≤ -rule:
if
1. no other rule is applicable
2. (∃≤n R(◦, t))(x) ∈ A and
3. there are more than n variables y1 , . . . , yp in A
which are R(◦, t)-successors of x such that
{yi 6= yj } ∩ A = ∅ for some i, j (1 ≤ i < j ≤ p),
then
for some pair yi , yj (1 ≤ i < j ≤ p) such that {yi 6= yj } ∩ A = ∅
A0 := [yi /yj ]A (i.e. occurrences of yi are replaced by yj in A0 )
Figure 2: The Completion Rules for ALCN (◦, t)
69
which are all distinct, and with p1 + p2 + · · · + pc = p < n, then the algorithm nondeterministically chooses to split the new n−p
successors
of x among the c different role
¡
¢
chains in R(◦, t), which can be done in n−p+c−1
different
ways. This is equivalent
c
to choose one integer solution of the equation d1 + d2 + · · · + dc = n − p and assign dk
successors of x to the k-th role chain (with dk ≥ 0, 1 ≤ k ≤ c).
The other difference is in the application of the ≤-rule, which is delayed until no
other rule is applicable. This is done to yield a sufficient condition to show termination
of the algorithm. As a matter of fact, the execution of the algorithm can be subdivided
into two phases: in Phase 1 only the first five rules are applied, whereas in Phase 2 the
≤-rule only is applied. In Phase 1, the algorithm generates a (finite) tree structure
rooted at x0 , whereas, in Phase 2, some nodes of the structure are “glued” together
by the application of the ≤-rule.
Lemma 1 Let C0 be an ALCN (◦, t)-concept in NNF with role expressions in RNF,
and let A be an ABox obtained by applying the completion rules to {C0 (x0 )}. Then:
1. For each completion rule R that can be applied to A and for each interpretation
the following equivalence holds: I is a model of A iff I is a model of the ABox
A0 obtained by applying R.
2. If A is a complete and clash-free ABox, then A has a model.
3. If A is complete but contains a clash, then A does not have a model.
4. The completion algorithm terminates when applied to {C0 (x0 )}.
As a matter of fact, termination (4) yields that after finitely many steps we obtain
a complete ABox. If C0 is satisfiable, then {C0 (x0 )} is also satisfiable and, thus, at
least one of the complete ABoxes that the algorithm can generate is satisfiable by (1).
Hence, such an ABox must be clash-free by (3). Conversely, if the application of the
algorithm produces a complete and clash-free ABox A, then it is satisfiable by (2) and,
owing to (1), this implies that {C0 (x0 )} is satisfiable. Consequently, the algorithm is
a decision procedure for satisfiability of ALCN (◦, t)-concepts.
Proof of Part 1 of Lemma 1 We consider only the ≥-rule and ≤-rule, as the proof
for the first four rules is the same as for ALC.
≥-rule. Assume that the rule is applied to the constraint (∃≥n R(◦, t))(x) with
R(◦, t) = tck=1 R1k ◦ R2k ◦ · · · ◦ R`kk and that its application yields:
k
k
k
k
k
A0 = A ∪ {R1k (x, zi2
), R2k (zi2
, zi3
), R3k (zi3
, zi4
), . . .
k
. . . , R`kk (zi`
, zd1 +···+dk−1 +i ) | 1 ≤ k ≤ c, 1 ≤ i ≤ dk }
k
∪{zi 6= zj | 1 ≤ i < j ≤ n − p} ∪ {yi 6= zj | 1 ≤ i ≤ p, 1 ≤ j ≤ n − p}
Since A is a subset of A0 , any model of A0 is also a model of A. Conversely, assume that I is a model of A. On the one hand, since I satisfies (∃≥n R(◦, t))(x),
xI has at least n R(◦, t)-successors in I. On the other hand, since the ≥-rule
is applicable to (∃≥n R(◦, t))(x), x has exactly p R(◦, t)-successors y1 , . . . , yp ,
with p < n, in A. In particular, y1 , . . . , yp1 are (R11 ◦ · · · ◦ R`11 )-successors,
70
yp1 +1 , . . . , yp1 +p2 are (R12 ◦ · · · ◦ R`22 )-successors, . . . , yp1 +···+pc−1 +1 , . . . , yp are
(R1c ◦· · ·◦R`cc )-successors. Thus, there exists n−p R(◦, t)-successors b1 , . . . , bn−p
of xI in I such that bi 6= yjI for all i, j (1 ≤ i ≤ n − p, 1 ≤ j ≤ p). At least
for one of the possible choices for d1 , d2 , . . . , dc , we have that b1 , . . . , bd1 are
(R11 ◦ · · · ◦ R`11 )-successors, bd1 +1 , . . . , bd1 +d2 are (R12 ◦ · · · ◦ R`22 )-successors, . . . ,
bd1 +···+dc−1 +1 , . . . , bn−p are (R1c ◦· · ·◦R`cc )-successors. For this choice, and for all k
(1 ≤ k ≤ c) and i (1 ≤ i ≤ dk ), let {bki2 , . . . , bki`k } ⊆ ∆I be such (xI , bki2 ) ∈ (R1k )I ,
(bki2 , bki3 ) ∈ (R2k )I , . . . , (bki`k , bi ) ∈ (R`kk )I . We define the interpretation of the new
k )I = bk , . . . , (z k )I = b
variables added by the ≥-rule as (zi2
i`k (1 ≤ k ≤ c, 1 ≤
i2
i`k
I
i ≤ dk ), and zi = bi (1 ≤ i ≤ n − p). Obviously, I satisfies A0 .
≤-rule. Assume that the rule is applied to the constraint (∃≤n R(◦, t))(x) ∈ A and
let I be a model of A. On the one hand, since the rule is applicable, x has more
than n R(◦, t)-successors in A. On the other hand, I satisfies ∃≤n R(◦, t)(x)
and, thus, there are two different R(◦, t)-successors yi , yj of x in A such that
yiI = yjI . Obviously, this implies that yi 6= yj 6∈ A and, thus, A0 = A[yi /yj ]
is the ABox obtained by applying the ≤-rule to (∃≤n R(◦, t))(x). In addition,
since yiI = yjI , I satisfies A0 . Conversely, assume that A0 = A[yi /yj ] is obtained
from A by applying the ≤-rule, and let I be a model of A0 . If we consider an
interpretation I so that yjI = yiI for the additional variable yj that is present in
A then obviously I satisfies A.
¤
Proof of Part 2 of Lemma 1 Let A be a complete and clash-free ABox that
is obtained by applying the completion rules to {C0 (x0 )}. We define the canonical
interpretation IA of A as follows:
1. The domain ∆IA of IA consists of all the individual names x ∈ NI in A.
2. For all concept names C ∈ NC we define C IA := {x | C(x) ∈ A}.
3. For all role names R ∈ NR we define RIA := {(x, y) | R(x, y) ∈ A}.
4. For all individual names xIA := x
Hence, it can be easily shown that IA satisfies every constraint in A (the proof is
almost equal to the similar proof for ALCN (◦) in [2]).
By definition, IA satisfies all the role assertions of the form R(x, y), iff R(x, y) ∈ A.
More generally, y is an (R1 ◦ · · · ◦ Rm )-successor of x in A iff y is an (R1 ◦ · · · ◦ Rm )successor of x in IA and, thus, y is an R(◦, t)-successor of x in A iff y is an R(◦, t)successor of x in IA . Furthermore, y 6= z implies y IA 6= z IA by construction of IA .
By induction on the structure of concept descriptions, it can be easily shown that IA
satisfies the concept assertions as well, provided that A is complete and clash-free.
Again, we restrict our attention to number restrictions, since the induction base and
the treatment of other constructors is the same as for ALC.
• Consider (∃≥n R(◦, t))(x) ∈ A. Since A is complete, the ≥-rule cannot be
applied to (∃≥n R(◦, t))(x) and, thus, x has at least n R(◦, t)-successors in A,
which are also R(◦, t)-successors of x in IA . Hence, x ∈ (∃≥n R(◦, t))IA
71
• Constraints with the form (∃≤n R(◦, t))(x) ∈ A are satisfied since A is clashfree and complete. In fact, assume that x has more than n R(◦, t)-successors
in IA . Then x has more than n R(◦, t)-successors also in A. If A contained
inequality constraints yi 6= yj for all these successors, then we would have a
clash. Otherwise, the ≤-rule could be applied.
¤
Proof of Part 3 of Lemma 1 Assume that A contains a clash. If {A(x), (¬A)(x)}
⊆ A, then clearly no interpretation can satisfy both constraints. Thus assume that
(∃≤n R(◦, t))(x) ∈ A and x has p > n R(◦, t)-successors y1 , . . . , yp with {yi 6= yj |
1 ≤ i < j ≤ p} ⊆ A. Obviously, this implies that, in any model I of A, xI has p > n
R(◦, t)-successors in I, which shows that I cannot satisfy (∃≤n R(◦, t))(x).
¤
Proof of Part 4 of Lemma 1 We must show that the Tableau algorithm that tests
satisfiability of ALCN (◦, t)-concepts always terminates. In the following, we consider
only ABoxes A that are obtained by applying the completion rules to {C0 (x0 )}.
The execution of the algorithm is composed of two separate phases: Phase 1, in
which the first five completion rules are applied, and Phase 2, in which the ≤-rule only
is applied. The separation between the two phases is sharp: Phase 2 cannot begin
before Phase 1 has terminated, since the ≤-rule cannot be applied while other rules are
applicable. It is also easy to see that no further application of the first five completion
rules can be triggered once Phase 2 has started1 : no new individuals and constraints
are added to A by the application of the ≤-rule, only variables are merged and all the
constraints that were satisfied at the end of Phase 1 remain satisfied. In particular,
the ≤-rule can only identify two variables yi and yj for which no inequality yi 6= yj is
explicitly present in A and, thus, the variables zj which were introduced as R(◦, t)successors by the application of the ≥-rule during Phase 1 (for which inequalities were
explicitly added to A) cannot be identified during Phase 2, and the (∃≥n R(◦, t))(x)
constraints which were satisfied during Phase 1 remain satisfied.
Finiteness of Phases 1 and 2 is then proved owing to the Definition and Lemma
which follow.
Definition 4 Let A be an ABox. The graph GA induced by A is a directed graph
GA = (N, E, L) with labels on nodes and edges defined as:
N = {x ∈ NI | x occurs in A}
L(x) = {C | C(x) ∈ A}
E = {(x, y) | R(x, y) ∈ A}
L(x, y) = {R | R(x, y) ∈ A}
It is easy to see (e.g. by induction on the number of rule applications) that, for any
ABox A generated during Phase 1 from the initial ABox {C0 (x0 )}, the induced graph
GA is a tree rooted at x0 , for any node x ∈ N , L(x) contains subconcepts of C0 and for
any edge (x, y) ∈ E, L(x, y) contains a single role R occurring in C0 . For a concept C,
1
(ADDED AFTER DL’03) Actually, this claim is wrong. We thank very much Franz Baader
and Ulrike Sattler for having pointed it out. As it is, the algorithm does not always produce a complete
ABox. By removing precondition 1. in the ≤-rule, the algorithm produces a complete ABox but is
not guaranteed to terminate. Decidability of ALCN (◦, t) remains an open problem.
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we define size(C) as the total number of symbols (operators, concept and role names)
in it and depth(C) (maximal role depth) as follows:
depth(A) = depth(¬A) := 0
for A ∈ NC
depth(C1 u C2 ) = depth(C1 t C2 ) := max{depth(C1 ), depth(C2 )}
depth(∃R.C) = depth(∀R.C) := 1 + depth(C)
depth(∃≥n (tck=1 R1k ◦ R2k ◦ · · · ◦ R`kk )) := max{`k | 1 ≤ k ≤ c}
Hence, as an easy consequence of the definition of the first five completion rules, every
variable x 6= x0 that occurs in A is an (R1 ◦ · · · ◦ Rm )-successor of x0 for some role
chain of length m, with m ≥ 1. Furthermore, m ≤ depth(C0 ), since for each constraint
C(x) ∈ A, 0 ≤ depth(C) ≤ depth(C0 ) − m.
Lemma 2 For the graph GA induced by A during Phase 1, the following facts hold:
1. For every node x, the size of L(x) is bounded by the number of subconcepts in
C0 and, thus, by size(C0 )
2. The length of a directed path in GA is bounded by depth(C0 )
3. The out-degree of GA is bounded by size(C0 ) · N
where N = max{n | ∃≥n R(◦, t) occurs in C0 }.
Proof of Lemma 2 For the first two facts, the proof is straightforward. For the third
fact, branches from a node x are generated by the application of the ∃-rule (which
generates one branch) and of the ≥-rule (which generates at most N branches). Such
rules are triggered, respectively, by ∃R.C and ∃≥n R(◦, t) concepts which occur in
L(x). Thanks to Fact 1, the number of such concepts is bounded by size(C0 ). Hence,
the out-degree is globally bounded by size(C0 ) · N .
¤
As a consequence of Lemma 2, Phase 1 has a finite length. Each rule application
either adds a new element to L(x) or adds additional nodes to GA . Since constraints
are never deleted and individuals are never deleted or identified during Phase 1, an
infinite sequence of rule applications would either lead to an infinite number of nodes
in the tree or to an infinite label of one of the nodes, which contradicts the bounds
derived in Lemma 2.
During the execution of Phase 2, each application of the ≤-rule decreases by one
the number of nodes in the directed graph GA (which may lose the tree shape and
even acyclicity). Since constraints are never added and new individuals are never
created, an infinite sequence of rule applications would imply an infinite number of
nodes in GA after the end of Phase 1, which contradicts the bounds of Lemma 2. As
a consequence, also Phase 2 has a finite length.
This completes the proof of Part 4 of Lemma 1.
¤
Corollary 1 Concept satisfiability (and subsumption) for ALCN (◦, t) is decidable,
and the Tableau algorithm based on the completion rules in Fig. 2 is an effective
decision procedure.
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3
Conclusions
In this paper we studied expressive Description Logics ALCN (◦, t), allowing for number restrictions on complex roles built with the composition and disjunction operators. We proved decidability of ALCN (◦, t)-concept satisfiability (and subsumption)
by means of a tableau-based algorithm. In our future work, we will also consider other
issues concerning reasoning in ALCN (◦, t), which include the characterization of its
complexity, the design of optimized algorithms, and the treatment of other tasks (e.g.
consistency with respect to a knowledge base).
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