Number Restrictions on Complex Roles in Description Logics: A Preliminary Report

Number restrictions are concept constructors that are available in almost all implemented description logic systems. However, even though there has lately been considerable effort on integrating expressive role constructors into description logics, the roles that may occur in number restrictions are usually of a very restricted type. Until now, only languages with number restrictions on atomic roles and inversion of atomic roles, or with number restrictions on intersection of atomic roles have been investigated in detail. In the present paper, we increase the expressive power of description languages by allowing for more complex roles in number restrictions. As role constructors, we consider composition of roles (which will be present in all our languages), and intersection, union and inversion of roles in different combinations. We will present two decidability results (for the basic language that extends A~U by number restrictions on roles with composition, and for one extension of this language), and three undecidability results for three other extensions of the basic language. 1 Motivation and introduction Description logics is a field of knowledge representation in which there is a rather close interaction between theory and practice. On the one hand, there are various implemented systems based on description logics, which offer a palette of description formalisms with differing expressive power [Peltason,1991; Brachman et a/.,1991; MacGregor,1991; Mays et al.,1991; Baader et al.,1994; Bresciani et al.,1995]. On the other hand, the computational properties (like de*This author is supported by the Deutsche Forschungsgemeinschaft under Grant No. Sp 230\ 6-6. cidability, complexity) of various description formalisms have thoroughly been investigated [Nebel,1988; Schmidt-Schauss,1989; Patel-Schneider,1989; Donini et a/.,1991a; 1991b]. These investigation were often motivated by the use of certain constructors in systems or the need for these constructors in specific applications [Baader & Hanschke,1993; Franconi,1994], and the results have influenced the design of new systems. The terminological formalisms of knowledge representation systems based on description logics provide constructors that can be used to build complex concepts and roles out of atomic concepts (unary predicates) and roles (binary predicates). Until recently, the main emphasis, both in implemented systems and in theoretical research, was on constructors for building complex concepts. The need for rich role constructors in certain application domains (such as representing rich schema languages for databases [Calvanese et a/.,1994; 1995], or domains that require the appropriate modeling of part-whole relations [Padgham & Lambrix,1994; Artale et a/.,1994; Sattler,1995]) has triggered research on description languages that also provide for expressive role constructors [Baader,1990; De Giacomo & Lenzerini,1995]. These investigations were facilitated by the observation that the formalisms considered in description logics are very similar to certain modal logics [Schild,1991; De Giacomo & Lenzerini,1994]. In particular, well-known modal logics, such as propositional dynamic logics (PDL) and its extensions [Fischer & Ladner,1979; BenAri et a/.,1982; Harel,1984], provide for role constructors like composition, union, transitive closure, and inversion. Number restrictions are concept constructors that are available in almost all implemented description logic systems. They allow to restrict the number of role successors of an individual w.r.t, a given role. For example, if has-child is an atomic role and person is an atomic concept, then we can describe all persons having at most 2 children by the concept person ~ (_< 2 has-child). In contrast to the rather prominent rSle that number restrictions play in deFrom: AAAI Technical Report WS-96-05. Compilation copyright © 1996, AAAI (www.aaai.org). All rights reserved. scription logics, the corresponding constructors in modal logic--so-called "graded modalities" [Fine,1972; van der HoekD intersection of roles can prohibit that a parent marries his/her own child: (_< 0 has-childYlis-married-to); union and composition can be used to describe that all children have the same name as their parent: (= 1 has-name [_] (has-childohas-name)). Number restrictions on complex roles are not only of interest in toy examples like the family domain used above. Our original motivation for considering these constructs comes from a process engineering application, where planning and optimization of large chemical plants is supported by building process models. The engineering knowledge concerning standard building blocks of these models is to be represented in a description logic system. For example, the concept (device V1 (= 1 controlled-by))describes devices that are controlled by a single control unit. If we want to describe a device such that all devices connected to it are controlled by the same control unit, we need composition in the number restriction: (device [-1 (= 1 connected-to o controlled-by)). To assure that the device itself is also controlled by the same unit controlling the devices connected to it, we additionally need union in the number restriction: (device [-1 (---1 controlled-by U connected-toocontrolled-by)). Inversion of roles comes in if we need the role controls as well. There are also more complex properties of devices and other parts of process models that could be expressed with number restrictions on complex roles. However, to be useful in practice, it is not sufficient to have a description language that can just be used to represent the relevant properties of objects. The description logic system must also be able to reason about the descriptions. As a positive result in this direction, we show that the subsumption and the satisfiability problem for the language AggAf(o), which extends AlL with number restrictions on roles built with composition, are decidable. On the other hand, three extensions of this language turn out to be undecidable: Af-E+with number restrictions on roles built with composition and union; MEg with number restrictions on roles built with composition and intersection; and MEg with number restrictions on roles built with composition, union, and inversion. However, if union and intersection are restricted to role chains of the same length, then we obtain a decidable extension of Af_~. In the next section, we introduce syntax and semantics of the concept and role constructors that will be considered. Section 3.1 describes the algorithm that decides satisfiability of AE_gAf(o)-concepts, and Section 3.2 extends this decidability result to number restrictions on union and intersection of role chains of the same length. The subsequent section sketches the undecidability proofs, which all use a reduction of the domino problem. In Section 5, we mention related decidability and undecidability results from modal and description logics. 2 Concept and role constructors We define syntax and semantics of all the constructors considered in the present paper, and introduce the description languages that will be investigated in more detail. Definition 1 Starting with atomic roles from a set NR of role names, complex roles are built using the role constructors composition (RoS), union (R 0 intersection (R ~ S), inversion (R-I), and transitive closure ( R+ ). The set of Mr_E-concepts is built from a set Nc of concept names using the concept constructors disjunction (C U D), conjunction (C N D), negation (-~C),


Motivation and introduction
Description logics is a field of knowledge representation in which there is a rather close interaction between theory and practice.On the one hand, there are various implemented systems based on description logics, which offer a palette of description formalisms with differing expressive power [Peltason,1991;Brachman et a/.,1991;MacGregor,1991;Mays et al.,1991;Baader et al.,1994;Bresciani et al.,1995].On the other hand, the computational properties (like de-*This author is supported by the Deutsche Forschungsgemeinschaft under Grant No. Sp 230\ 6-6.cidability, complexity) of various description formalisms have thoroughly been investigated [Nebel,1988;Schmidt-Schauss,1989;Patel-Schneider,1989;Donini et a/.,1991a;1991b].These investigation were often motivated by the use of certain constructors in systems or the need for these constructors in specific applications [Baader & Hanschke,1993;Franconi,1994], and the results have influenced the design of new systems.
The terminological formalisms of knowledge representation systems based on description logics provide constructors that can be used to build complex concepts and roles out of atomic concepts (unary predicates) and roles (binary predicates).Until recently, the main emphasis, both in implemented systems and in theoretical research, was on constructors for building complex concepts.The need for rich role constructors in certain application domains (such as representing rich schema languages for databases [Calvanese et a/.,1994;1995], or domains that require the appropriate modeling of part-whole relations [Padgham & Lambrix,1994;Artale et a/.,1994;Sattler,1995]) has triggered research on description languages that also provide for expressive role constructors [Baader,1990;De Giacomo & Lenzerini,1995].These investigations were facilitated by the observation that the formalisms considered in description logics are very similar to certain modal logics [Schild,1991;De Giacomo & Lenzerini,1994].In particular, well-known modal logics, such as propositional dynamic logics (PDL) and its extensions [Fischer & Ladner,1979;Ben-Ari et a/.,1982;Harel,1984], provide for role constructors like composition, union, transitive closure, and inversion.
Number restrictions are concept constructors that are available in almost all implemented description logic systems.They allow to restrict the number of role successors of an individual w.r.t, a given role.For example, if has-child is an atomic role and person is an atomic concept, then we can describe all persons having at most 2 children by the concept person ~ (_< 2 has-child).In contrast to the rather prominent rSle that number restrictions play in de-scription logics, the corresponding constructors in modal logic--so-called "graded modalities" [Fine,1972; van der Hoek&De Rijke,1995]--have been studied only recently, and thus there are not many results available that could be transferred to description logics.In [De Giacomo&Lenzerini,1994], the problem of adding number restrictions to PDL and various of its extensions has been investigated in detail.However, even though the description languages considered in this work have very expressive formalisms for constructing complex roles, the roles that may occur in number restrictions are restricted to atomic roles and their inverse.To the best of our knowledge, the only other well-investigated concept description language with number restrictions on non-atomic roles is AECAfT~, which allows for intersection of roles in number restrictions.
The present paper is a first attempt to overcome this research deficit.It considers description languages that extend MEg or MEg + (the description logic equivalent to PDL) with number restrictions on complex roles.As role constructors in number restrictions, we will allow for composition (which will be present in all our languages), and intersection, union, and inversion of roles in different combinations.Number restrictions on roles with composition are particularly interesting from a practical point of view since they allow to impose restrictions on role successors for a composed role without explicitly stating restrictions on its atomic components.For example, the restriction person[7 (= 17 has-childo has-child) describes persons that have 17 grandchildren without explicitly saying anything about the number of children, and the number of children of each child.From a theoretical point of view, number restrictions on roles with composition introduce a new level of complexity: the tree model property (which most of the modal logics and description logics investigated in the literature have) is no longer satisfied (see Section 3.1).By adding inversion of roles, we can express that a person has at least 5 siblings: person [7 (> 6 has-child -1 ohas-child); intersection of roles can prohibit that a parent marries his/her own child: (_< 0 has-childYlis-married-to); union and composition can be used to describe that all children have the same name as their parent: Number restrictions on complex roles are not only of interest in toy examples like the family domain used above.Our original motivation for considering these constructs comes from a process engineering application, where planning and optimization of large chemical plants is supported by building process models.The engineering knowledge concerning standard building blocks of these models is to be represented in a description logic system.For example, the concept (device V1 (= 1 controlled-by))describes devices that are controlled by a single control unit.If we want to describe a device such that all devices connected to it are controlled by the same control unit, we need composition in the number restriction: (device [-1 (= 1 connected-to o controlled-by)).
To assure that the device itself is also controlled by the same unit controlling the devices connected to it, we additionally need union in the number restriction: (device [-1 (----1 controlled-by U connected-toocontrolled-by)).Inversion of roles comes in if we need the role controls as well.There are also more complex properties of devices and other parts of process models that could be expressed with number restrictions on complex roles.However, to be useful in practice, it is not sufficient to have a description language that can just be used to represent the relevant properties of objects.The description logic system must also be able to reason about the descriptions.
As a positive result in this direction, we show that the subsumption and the satisfiability problem for the language AggAf(o), which extends AlL with number restrictions on roles built with composition, are decidable.On the other hand, three extensions of this language turn out to be undecidable: Af-E+with number restrictions on roles built with composition and union; MEg with number restrictions on roles built with composition and intersection; and MEg with number restrictions on roles built with composition, union, and inversion.However, if union and intersection are restricted to role chains of the same length, then we obtain a decidable extension of Af_~.
In the next section, we introduce syntax and semantics of the concept and role constructors that will be considered.Section 3.1 describes the algorithm that decides satisfiability of AE_gAf(o)-concepts, and Section 3.2 extends this decidability result to number restrictions on union and intersection of role chains of the same length.The subsequent section sketches the undecidability proofs, which all use a reduction of the domino problem.In Section 5, we mention related decidability and undecidability results from modal and description logics.

Concept and role constructors
We define syntax and semantics of all the constructors considered in the present paper, and introduce the description languages that will be investigated in more detail.As usual in description logics, the extensions of concepts and roles involving the constructors introduced above are defined inductively on the structure of complex concepts and roles.
Definition 2 An interpretation I = ( A z, .z)consists of a set A z, called the domain of l, and an extension ]unction .zthat maps every concept to a subset of A z, and every (complex) role to a subset of z ×A z such that the followings equalities are satisfied: For this reason, we may restrict our attention to the satisfiability problem, both in the decidability and in the undecidability proofs.

3
Decidability results In the first part of this section, we show that satisfiability of AECAf(o)-concepts is decidable.This result is extended in the second part to a description language where, additionally, union and intersection of role chains of the same length are allowed in number restrictions.

.AfLAf(o) is decidable
We present a tableau-like algorithm for deciding satisfiability of MECAf(o)-concepts.The algorithm and the proof of its correctness are very similar to existing algorithms and proofs for languages with number restrictions on atomic roles [Hollunder et al.,1990;Hollunder&Baader,1991].It should be noted, however, that the presence of number restrictions on role chains of the form R1 oR2 o...oRn with n > 1 has as consequence that the finite models generated by the algorithm need no longer be tree models.A tree model of a concept C is an interpretation such that (1) every element of the model can be reached from an initial (root) element, which is an instance of C, via role chains, (2) the root does not have a role predecessor, and (3) every other element has exactly one role predecessor.The following .Atl2Af(o)-concept is satisfiable, but it obviously does not have a tree model: Nevertheless, the models that will be generated by our algorithm are very similar to tree models in that properties (1) and ( 2) are still satisfied, and every chain from the root to an element has the same length (even though there may exist more than one such chain).This fact will become important in the proof of termination.
As usual, we assume without loss of generality that all concepts are in negation normal form (NNF), i.e., negation occurs only immediately in front of concept names.The basic data structure our algorithm works on are constraints: Definition 3 Let 7 = {x, y, z,...} be a countably infinite set of individual variables.A constraint is of the form xRy, x:D, or x#y, 1 1We consider such inequalities as being symmetric, i.e., if x ~ y belongs to a constraint system, then y # x (implicitly) belongs to it as well.
where R is a role name, x, y are individual variables, and D is an Af_CAf(o)-concept in NNF.constraint system is a set of constraints.For a constraint system S, let Ts C T denote the individual variables occuring in S.
An interpretation Z is a model of a constraint system S iff there is a mapping ~ : TS -+ AZ such that Z, 7r satisfy each constraint in S, i.e., (~(x)  The completion algorithm works on a tree where each node is labelled with a constraint system.It starts with the tree consisting of a root labelled with S = {x0 : C0}, where Co is the AECAf(o)-concept in NNF to be tested for satisfiability.
A rule can only be applied to a leaf labelled with a clash-free constraint system.Applying a rule S ~ Si, for 1 < i < n, to such a leaf leads to the creation of n new successors of this node, each labelled with one of the constraint systems Si.The algorithm terminates if none of the rules can be applied to any of the leaves.In this situation, it answers with "Co is satisfiable" iff one of the leaves is labelled with a clash-free constraint system.
Correctness of this algorithm is an immediate consequence of the following facts: Lemma 4 Let Co be an AfLAf(o)-concept in NNF, and let S be a constraint system obtained by applying the completion rules to {x0:C0}.Then 1.For each completion rule T~ that can be applied to S, and for each interpretation Z we have Z is a model of S iff Z is a model of one of the systems S~ obtained by applying T~.
2. If S is a complete and clash-free constraint system, then S has a model.

If S contains a clash, then S does not have a model.
4. The completion algorithm terminates when applied to (x0 : Co}. Indeed, termination shows that after finitely many steps we obtain a tree such that all its leaf nodes are labelled with complete constraint systems.If Co is satisfiable, then {x0 : Co } is also satisfiable, and thus one of the complete constraint systems is satisfiable by (1).By (3), this system must be clash-free.Conversely, one of the complete constraint systems is clash-free, then it is satisfiable by (2), and because of (1) implies that {x0 : Co} is satisfiable.Consequently, the algorithm is a decision procedure for satisfiability of AECAf(o)-concepts: Theorem 5 Subsumption and satisfiability of AECAf ( o )-concepts is decidable.
Proof of Part 1 of Lemma 4: We consider only the rules concerned with number restrictions, since the proof for Rules 1-4 is just as for AC_C.

Number restriction:
Assume that the rule is applied to the constraint x :(:> n Rio...oRm), and that its application yields Since S is a subset of S', any model of S' is also a model of S.

Number restriction:
Assume that the rule can be applied to x:(< n Rio...oRm) ¯ S, and let Z together with the valuation r : ~'s --+ Az be a model of S. On the one hand, since the rule is applicable, x has more than n R1 o... oRmsuccessors in S. On the other hand, Z, r satisfy x :(< m Rio...oRm) ¯ S, and thus there are two different Rio...oRm-successors yl, y2 of x in S such that r(yl) = ~r(y2).Obviously, this implies (Yl ~ y2) ¢ S, which shows that Sy 1,y2 = S[y2/yl] is one of the constraint systems obtained by applying Rule 6 to x :(< n Rio...oRm).In addition, Proof of Part 2 of Lemma 4: Let S be a complete and clash-free constraint system that is obtained by applying the completion rules to {Xo : Co}.We define a canonical model Z of S as follows: A z := Ts and for allAEArc: x E A z iff x : A E S, for allRENR: (x,y) ER z iff xRyES.
In addition, let 7r : TS -~ AZ be the identity on TS.We show that Z, r satisfy every constraint in S.
By definition of E, a role constraint of the form xRy is satisfied by Z, lr iff xRy E S.More generally, y is an Rio...oRm-successor of x in S iff y is an R1 o...oRmsuccessor of x in Z.We show by induction on the structure of the concept C that every concept constraint x : C E S is satisfied by Z, zr.Again, we restrict our attention to number restrictions since the induction base and the treatment of the other constructors is just as for Af_C.
Consider x:(> n Rlo...oRm) E S. Since is complete, Rule 5 cannot be applied to x:(_> n Rio...oRm), and thus x has at least n Rio...oRmsuccessors in S, which are also R1 o... o Rmsuccessors of x in Z.This shows that Z, zr satisfy x:(> n Rio...oR,,).
Constraints of the form x :(< n R1 o...oRm) E are satisfied because S is clash-free and complete.In fact, assume that x has more than n Rio...oRmsuccessors in Z. Then x also has more than n Rio ...oR,n-successors in S. If S contained inequality constraints yi ~ yj for all these successors, then we would have a clash.Otherwise, Rule 6 could be applied.For the termination proof, the following observations, which are an easy consequence of the definition of the completion rules, are important: Let mo be the maximal role depth of Co.Because of the first fact, every individual x in a constraint system S (reached from {xo : Co) by applying completion rules) has a unique role level level(x), which is its distance from the root node Xo, i.e., the unique length of the role chains that connect Xo with x.Because of the second fact, the level of each individual is an integer between 0 and m0.
In the following, we define a mapping ~ of constraint systems S to 5(too + 1)-tuples of nonnegative integers such that S --+ S' implies ~(S) ~-~(S~), where ~notes the lexicographic ordering on 5mo-tuples.Since the lexicographic ordering is well-founded, this implies termination of our algorithm.In fact, if the algorithm did not terminate, then there would exist an infinite sequence So --+ $1 --+ ..., and this would yield an infinite descending ~--chain of tuples.
Thus, let S be a constraint system that can be reached from {x0 : Co} by applying completion rules.We define ,,(s) where ~t := (k~,l, kt,2, kt,3,kt,4,kt,5) and the components k~,i are obtained as follows: In the following, we show for each of the rules of Figure 1 that S --~ S ~ implies ~(S) ~-n(S').
1. Conjunction" Assume that the rule is applied to the constraint x:C1 ~ C2, and let S ~ be the system obtained from S by its application.Let := level(x).First, we compare ~t and ~, the tuples respectively associated with level ~ in S and SC Obviously, the first components of ~t and ~ agree since the number of individuals and their levels are not changed.The second component of ~ is smaller than the second component of ~t: [C1 [7 C2[n,u is removed from the sum, and replaced by a number that is not larger than IC11n,u + IC21n,u (depending on whether the top constructor of C1 and 6'2 is disjunction or conjunction, or some other constructor).Since tuples are compared with the lexicographic ordering, a decrease in this component makes sure that it is irrelevant what happens in later components.For the same reason, we need not consider tuples nm for m > ~.Thus, assume that m < ~.In such a tuple, the first three components are not changed by application of the rule, whereas the remaining two components remain unchanged or decrease.Such a decrease can happen if level(y) = and S contains co nstraints yRx, y :(VR.Ci) (or :(3R.Ci)).
2. Disjunction" This rule can be treated like the conjunction rule.

Value restriction"
Assume that the rule is applied to the constraints x :(VR.C),xRy, and let S t be the system obtained from S by its application.Let ~ := level(x).Obviously, this implies that level(y) = level(x) + 1 On level £, the first three components of ~t remain unchanged; the fourth remains the same, or decreases (if S contains constraints zSy and z :(3S.C) for an individual z with level(z) = ~); and the fifth decreases by at least one since the constraints x :(VR.C), xRy are no longer counted.It may decrease by more than one if S contains constraints zSy and z :(VS.C) for an individual with level(z) = Because of this decrease at level g, the tuples at larger levels (in particular, the one for level level(x) 1, where the re might be an increase), need not be considered.The tuples of levels smaller than e are not changed by application of the rule.In particular, the third component of such a tuple does not change since no role constraints or inequality constraints are added or removed.

Existential restriction:
Assume that the rule is applied to the constraint x :(3R.C), and let S' S U {xRy, y : C} be the system obtained from S by its application.Let e := level(x).Obviously, this implies that level(y) = level(x) + 1 The first two components of ~e obviously remain unchanged.The third component may decrease (if y is the first successor for an at-least restriction) or it stays the same.Since the fourth component decreases, the possible increase of the fifth component is irrelevant.For the same reason, the increase of the first component of ~e+i is irrelevant.Tuples of levels smaller than t~ are not increased by application of the rule.All components of such a tuple remain unchanged, with the possible exception of the third component, which may decrease.

Number restriction:
Assume that the rule is applied to the constraint x :(> n Ri o...oRm) E S, let S' be the system obtained by rule application, and let ~ = level(x).As for Rule 4, the first two components of ~e remain the same.In addition, there is a decrease in the third component of t~e, since the new individual z can now be added to the maximal sets of explicitly distinct Ri o...o Rm-successors of x.Note that these sets were previously smaller than n (because even the set of all Rio...oRm-successors of x was smaller than n).For this reason, the possible increase in the fifth component of ~e and in the first components of tuples of levels larger than t are irrelevant.Tuples of levels smaller than ~ are either unchanged by application of the rule, or their third component decreases.

Number restriction:
Assume that the rule is applied to the constraint x :(< n Ri o...oRm) E S, let S ~ = Sy 1,~2 be the system obtained by rule application, and let e = level(x).On level ~ + m, the first component of the tuple ~e+,~ decreases.Thus, possible increases in the other components of this tuple are irrelevant.Tuples associated with smaller levels remain unchanged or decrease.In fact, since Yi in S ~ has all its old constraints and the constraints of y2 in S, some value restrictions or existential restrictions for individuals of the level immediately above level + m may become satisfied (in the sense that the corresponding rule no longer applies).Since no constraints are removed, previously satisfied value restrictions or existential restrictions remain satisfied.The third component of tuples of smaller level cannot increase since the individuals Yi, Y2 that have been identified were not related by inequality constraints. ..
For languages where number restrictions may contain--in addition to composition--union or intersection of roles, an important property used in the above termination proof is no longer satisfied: It is not possible to associate each individual generated by a tableau-like procedure with a unique role level, which is its distance to the "root" individual x0 (i.e., the instance Xo of Co to be generated by the tableau algorithm).Indeed, in the concept the number restriction enforces that an R-successor of an instance of Co is also an R o R-successor of this instance.For this reason, an R-successor of the root individual must be both on level 1 and on level 2, and thus the relatively simple termination argument that was used above is not available for these larger languages.However, as we shall show in Section 3.2, this termination argument can still be used if union and intersection are restricted to role chains of the same length.Without this restriction, satisfiability may become undecidable: in Section 4, we show that satisfiability is in fact undecidable for A~A [(o, 9).For A52A/'(o, decidability of satisfiability is still an open problem.

An extension of the decidability result
The algorithm given in Section 3.1 will be extended such that it can also treat union and intersection of role chains that have the same length.The proof of soundness, completeness and termination of this extended algorithm is very similar to the one for the basic algorithm, and will thus only be sketched.
In the remainder of this section, a complex role is The satisfiability algorithm is extended by adding two new rules 5a and 5b to handle number restrictions (_> n T~) for complex roles with union or intersection, and by substituting rule 6 by a new rule 6' that is able to handle the new types of complex roles.To formulate the new rules, we must extend the notion of Obviously, this definition is such that role successors in S are also role successors in every model of S: if Z, ~r satisfy S, and y is an T~-successor of x in S for a complex role T~, then lr(y) is an 7~-successor of 7r(x) in/:.The new rules are described in Figure 2. The rules 5a, 5b are added to the completion rules, whereas rule 6' substitutes rule 6 in Figure 1.
To show that the new algorithm obtained this way decides satisfiability of concepts for the extended language, we must proof that the four parts of Lemma 4 still hold.
1. Local correctness of the rules 5a, 5b and 6' can be shown as in the proof of Part 1 of Lemma 4 above.
2. The canonical model induced by a complete and clash-free constraint system is defined as in the proof of Part 2 of Lemma 4. The proof that this canonical model really satisfies the constraint system is also similar to the one given there.Note that our notion of an ~-successor of a complex role 7~ in a constraint system was defined such .
. that it coincides with the notion of an 7~-successor in the canonical model I induced by the constraint system.
The proof that a constraint system containing a clash is unsatisfiable is the same as the one given above.Note that this depends on the fact that role successors in a constraint system are also role successors in every model of the constraint system.
Because the role chains in complex roles are of the same length, it is easy to see that Lemma 6 still holds.Thus, we can define the same measure a(S) as above for all constraint systems obtained by applying the extended completion rules to {xo : C0}.It is easy to see that the proof that S --+ S' implies to(S) ~-~(S') can be extended to the new rules.
It should be noted that the proof given above was already formulated in a more general way than necessary for the language considered there.In fact, we have only used that all role chains connecting two individuals have the same length (which is still satisfied for the extended language), and not that these role chains also have the same name (which is only satisfied for AEgAf(o)).
The following theorem is an immediate consequence of these observations: Theorem 7 Subsumption and satisfiability is decidable for the language that extends AECAf (o) by number restrictions on union and intersection of role chains of the same length.

4
Undecidability results We will use a reduction of the domino problem-a well-known undecidable problem [Knuth,1968;Berger,1966] often used in undecidability proofs in logic--to show that concept satisfiability is undecidable for three extensions of the decidable language ALUAf(o) considered in the previous section.
Īn order to reduce the domino problem to satisfiability of concepts, we must show how a given tiling system 7) can be translated into a concept E~ (of the language under consideration) such that Ez) is satisfiable iff 7) allows for a compatible tiling.This task can be split into three subtasks, which we will first explain on an intuitive level, before showing how they can be achieved for the three concept languages under consideration.
Task 1: It must be possible to represent a single "square" of IN x IN, which consists of points (n, m), (n, m + 1), (n + 1, m), and (n + 1, The idea is to introduce roles X, Y, where X goes one step into the horizontal (i.e.x-) direction, and Y goes one step into the vertical (i.e.y-) direction.
The concept language must be expressive enough to describe that an individual (a point (n, m)) exactly one X-successor (the point (n + 1,m)), exactly one Y-successor (the point (n,m ÷ 1)), and that the X o Y-successor coincides with the YoX-suceessor (the point (n + 1,m + 1)).Task 2: It must be possible to express that a tiling is locally correct, i.e., that the X-and Y-successors of a point have an admissible domino type.The idea is to associate each domino type Di with an atomic concept Di, and to express the horizontal and vertical matching conditions via value restrictions on the roles X, Y.
Task 3: It must be possible to impose the above local conditions on all points in IN x IN.This can be achieved by constructing a "universal" role U and a "start" individual such that every point is a U-successor of this start individual.The local conditions can then be imposed on all points via value restrictions on U for the start individual.
Task 2 is rather easy, and can be realized using the Af_C-concept Cz) given in Figure 3.The first conjunct expresses that every point has exactly one domino type, and the value restrictions in the second conjunct express the horizontal and vertical matching conditions.
Task 1 can be achieved in any extension of Af_CAf(o) with either union or intersection of roles in number restrictions: see the concepts Cu and Cn in Figure 3.
Task 3 is easy for languages that extend Af~+ and more difficult for languages without transitive closure.The general idea is that the start individual s is an instance of the concept Ez) to be constructed.From this individual, one can reach via U the origin (0, 0) of IN × IN, and all points that are connected with the origin via arbitrary X-and Y-chains.
(1) In extensions of AEC + we can use an atomic role R to reach the origin, and the complex role R U (Ro(X Y)+) to reach every point.Thus, the tiling system can be translated into the Af~+Af(o, U)-concept We can even restrict the complex role in the value restriction to a simple transitive closure of an atomic role.To achieve this, we make sure that the X-and the Y-successors of a point are also R-successors of this point.This allows us to use R + in place of R U (Ro (X u Y)+) as "universal" role: see the concept ') in Figure 3.
(2) In A£gAf(o, -1 ), we explicitly int roduce a r ol name U for the "universal" role, and use number restrictions involving composition, union, and inversion of roles to make sure that the start individual is directly connected via U with every point: see the concept E(v 2) in Figure 3.The number restrictions inside the value restriction make sure that every point p that is reached via U from the start individual satisfies the following: Its X-successor and its Y-successor each have exactly one U-predecessor, which coincides with the (unique) U-predecessor of p, i.e., the start individual.Thus, the X-successor and the Y-successor of p are also U-successors of the start individual.
(3) For AECAf(o, [7), a similar construction is possible we introduce role names R and T. The intuition is that T plays the r61e of the inverse of R (except for one individual), and the "universal" role corresponds to the composition RoToR: The start individual s (which Related work and open problems Propositional dynamic logic (PDL), which corresponds to our language AC-.C + has been shown to be decidable in [Fischer&: Ladner,1979], and decidability of its extension by deterministic programs, DPDL, is shown in [Ben-Ari et al.,1982].In principle, the use of deterministic programs corresponds to introducing a restricted form of number-restrictions, namely (< 1 R) for atomic roles R. Adding inversion (of atomic roles) DPDL has a drastic consequence: the finite model property is lost, i.e., there are satisfiable formulae (concepts) that do not have a finite model.Nevertheless, satisfiability is still decidable [Vardi,1985] (EXPTIME complete, like all the other decision problems for PDL and its extensions mentioned until now).It should be noted, however, that in these languages inversion does not occur in the number-restrictions, since only atomic programs are asserted to be deterministic.In [De Giacomo,1995], general number restrictions and Boolean operators for roles are added to PDL with inversion, and (ExPTIME) decidability is shown by a rather ingeneous reduction to the decision problem for PDL.In this work, atomic roles and their inverse my occur in number restrictions, but not more complex roles.In addition, the complement of roles is built relative to a fixed role any, which need not be interpreted as the universal role.Thus, it does not yield the classical negation of roles.If one adds number restrictions on atomic roles and their intersections to Af_d, satisfiability for the obtained language is still decidable with a PSPACS-algorithm [Donini et a/.,1991a].Certain modal logics and concept description languages (e.g., AC__C) can be translated into first-order logic such that only two different variable names occur in the formulae obtained by this translation (see, e.g., [Borgida,1996]).Thus, decidability of subsumption and other inference problems for these languages follows from the known decidability result for Z:~, i.e., first-order logic with two variables and without function symbols [Mortimer,1975].Recently, this decidability result has been extended to C2, i.e., predicate logic with 2 variables and counting quantifiers [Griidel et al.,].As an immediate consequence, satisfiability and subsumption for A52Af(I l, I-1, .%-1), the extension of ACC by number restrictions with inversion and Boolean operators on roles, is still decidable.It should be noted, however, that expressing composition of roles in predicate logic requires more than two variables.
In this paper, we have shown that the language AECA/'(o), which adds number restrictions on roles with composition to AOT, is still decidable.It is not clear, however, whether there exists a PSPACE algorithm for the problem.The one presented above is EXPTIME, and since different role paths need to be joined together, the trace method developed in [Schmidt-Schaut] & Smolka,1991] cannot directly be applied.Almost all extensions of AO2Af(o) by union, intersection, and inversion of roles were shown to be undecidable (unless restricted to role chains of the same length).Only decidability for Af__CAf(o, t_}) is still open.For Aft7 + however, the extension by composition and union could already be shown to be undecidable.
As related undecidability results, one can mention undecidability of the extension of DPDL by intersection of roles (which does not occur in the number restrictions, however) [Harel,1984].In [Hanschke,1992], an extension of Ag_C by so-called existential and universal agreements on role chains is shown to be undecidable.It is easy to see that existential (universal) agreements can be simulated by number restrictions involving composition and intersection (union).

Figure 1
Figure1introduces the completion rules that are used to test AECAf(o)-concepts for satisfiability.The completion algorithm works on a tree where each node is labelled with a constraint system.It starts with the tree consisting of a root labelled with S = {x0 : C0}, where Co is the AECAf(o)-concept in NNF to be tested for satisfiability.A rule can only be applied to a leaf labelled with a clash-free constraint system.Applying a rule S ~ Si, for 1 < i < n, to such a leaf leads to the creation of n new successors of this node, each labelled with one of the constraint systems Si.The algorithm terminates if none of the rules can be applied to any of the leaves.In this situation, it answers with "Co is satisfiable" iff one of the leaves is labelled with a clash-free constraint system.
¯a role chain ~ = Rio...oRn, or ¯an intersection T~ = Ri o... o Rn n $1 o... o Sn of two role chains of the same length, or ¯a union T~ = Rio...oR,, U Sio...oSn of two role chains of the same length.
Figure 2: The additional completion rules.
Definition 8 A tiling system 7) = (D, H, V) is ven by a non-empty set D = {D1,... ,D~} of domino types, and by horizontal and vertical matching pairs H C_ D x D, V C_ D × D. The domino problem asks for a compatible tiling of the first quadrant IN x IN of the plane, i.e., a mapping t : IN x IN --+ D such that for all m, n E IN: