Finite Model Reasoning in Description Logics

Diego CalvaneseDipartimento di Informatica e Sistemistica

Universit~ di Roma "La Sapienza"Via Salaria 113, 1-00198 Roma, Italy

calvanese©dis, uniromal, it

Abstract

For the basic Description Logics reasoningwith respect to finite models amounts to reas-oning with respect to arbitrary ones, butfiniteness of the domain needs to be con-sidered if expressivity is increased and thefinite model property fails. Procedures forreasoning with respect to arbitrary models invery expressive Description Logics have beendeveloped, but these are not directly applic-able in the finite case. We first show that wecan nevertheless capture a restricted form offiniteness and represent finite modeling struc-tures such as lists and trees, while still reas-oning with respect to arbitrary models. Themain result of this paper is a procedure toreason with respect to finite models in anexpressive Description Logic equipped withinverse roles, cardinality constraints, and inwhich arbitrary inclusions between conceptscan be specified without any restriction. Thisprovides the necessary expressivity to go bey-ond most semantic and object-oriented Data-base models, and capture several useful ex-tensions.

1 INTRODUCTION

From the start of the discipline, researchers working inKnowledge Representation (KR), have aimed at aug-menting the expressivity of the formalisms used forrepresenting structured knowledge, without comprom-ising the computational properties of the proceduresadopted for reasoning about it. This research has par-alleled the one in Databases (DBs), where expressivedata models have been developed, aimed at represent-ing the data manipulated by commercial applications.

There is a strong similarity between all these ap-proaches, in which the domain knowledge is repres-ented in form of a schema by defining classes, whichdenote subsets of the domain, and by specifying varioustypes of relations between classes, which establish theirstructural properties (Calvanese, Lenzerini, ~: Nardi,1994; Borgida, 1995). In this context it is fundamentalto provide methods for reasoning about such specifica-tions in order to detect inconsistencies and hidden de-pendencies which arise from the specified constraints.This is precisely the objective of all KR systems inthe style of KL-ONE, and of Description Logics (DL)introduced for their formalization. It is gaining an in-creased importance also in DBs, both to support thephase of analysis and design of DB systems, and inthe process of query answering to perform for examplesemantic query optimization.

However, while in DBs it is usually assumed that theunderlying domain is finite, reasoning in DLs is per-formed without making such hypothesis. This seemsto be in contrast with the purpose of a KR system torepresent real world structures, which are inherently fi-nite. It can however be justified by observing that mostDLs studied so far have the finite model property, stat-ing that a consistent schema (or KB) always admits model with finite domain. This implies that one doesnot need to take special care about finiteness of the do-main, since it can always be assumed. Unfortunately,the finite model property does not hold for more ex-pressive logics, as those studied for example in (De Gi-acomo & Lenzerini, 1994a; Buongarzoni, Menghini,Salis, Sebastiani, ~ Straccia, 1995). In particular, thishappens also for the logics which include inverse rolesand functionality of roles, and which therefore have suf-ficient expressivity to represent semantic and object-oriented DB models (Calvanese et al., 1994). For thoselogics it becomes important to provide specific reason-ing methods for both the finite and the unrestrictedcase.

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From: AAAI Technical Report WS-96-05. Compilation copyright © 1996, AAAI (www.aaai.org). All rights reserved.

While for unrestricted reasoning, techniques have beendeveloped to deal with very expressive DLs (De Gi-acomo &; Lenzerini, 1995), decidability of reasoningwith respect to finite models is still open for importantcases. In particular, there is no known method capableof handling schemata in which classes can be defined bynecessary and sufficient conditions, in a logic featuringthe constructs typical of DB models. Such methodsare however necessary if one wants to reason for ex-ample on views in object-oriented models. Moreover,excessively limiting expressivity goes against the spiritof capturing in the schema as much as possible of thesemantics of the modeled reality. Only an expressiveformalism equipped with sound and complete reason-ing procedures provides the user with the necessarytools to reason about relevant aspects of the domainhe needs to represent. In this paper we present suchformalisms in which different forms of finiteness canbe captured.

On one hand, we show that while keeping the max-imum expressivity, one can add to the language a well-founded constructor which allows one to impose well-foundedness and therefore finiteness of certain struc-tures, without necessarily requiring the whole domainto be finite. By means of the well-founded constructorit becomes possible to define classes that representmodeling structures such as lists and trees, and reasonabout them like about any other class in the schema.This represents a significant improvement with respectto traditional data models, where such modeling struc-tures, if present at all, are ad hoc additions that re-quire a special treatment by the reasoning procedures(Cattell, 1994). The known methods for reasoningwith respect to arbitrary models in expressive DLs arebased on a correspondence between DLs and Proposi-tional Dynamic Logics (PDLs) and exploit the powerfulreasoning methods developed for PDLs (Schild, 1991;De Giacomo & Lenzerini, 1994a). We show that thecorrespondence can in fact be extended to handle alsowell-foundedness, and present a reasoning techniquefor the resulting PDL.

On the other hand, we develop a method to reasonwith respect to finite models on schemata built usinga very expressive DL, and show its decidability in de-terministic double exponential time. The DL we con-sider includes besides number restrictions and inverseroles also general negation of concept expressions. Theschemata are of the most general form, in which onecan express arbitrary inclusions between complex ex-pressions without any restriction at all. This makesour language not only capable of capturing the struc-tural properties of class-based representation formal-isms used in DBs, but provides also the necessary ex-

pressivity to subsume the terminological component ofmost existing concept based systems.

The rest of the paper is organized as follows: In Sec-tion 2 we present the representation formalism we ad-opt, in Section 3 we discuss the issues related to reason-ing about finite modeling structures and in finite mod-els and in Sections 4 and 5 we analyze the reasoningmethods used in both contexts. In Section 6 we outlinepossible directions for future work.

2 THE REPRESENTATIONFORMALISM

In this section, we present the family of class-basedformalisms we deal with. These formalisms exploit De-scription Logics for the definition of schemata, and thelogics we define extend the well known logics of the.4/:-family (Donini, Lenzerini, Nardi, ~ Nutt, 1991a)by several expressive constructs.

2.1 SYNTAX AND SEMANTICS OFDESCRIPTION LOGICS

The basic elements of Description Logics (DLs) areconcepts and roles, which denote classes and binaryrelations, respectively. Complex concept and role ex-pressions are built, starting from a set of concept androle names, by applying certain constructors. EachDL is characterized by the set of allowed construct-ors. Table 1 shows the concept and role forming con-structors we consider in this paper. The first columnspecifies the name of each constructor and the secondcolumn specifies a letter that is used to denote eachconstructor when naming a logic. The basic logic weconsider is .A£, which contains only the constructorsto which no letter is associated. All other logics wedeal with include the constructors of ,4£. Each logicis named with a string that starts with .A£ and in-cludes the letters that are associated to the additionalconstructors in the logic.

The syntax of each constructor in concept and roleexpressions is shown in the third column of the table.Concept names are denoted by C and complex conceptexpressions by E. Role names and complex role ex-pressions are denoted by P and R, respectively, mdenotes a positive and n a nonnegative integer.

Concepts are interpreted as subsets of a domain androles as binary relations over that domain. More pre-cisely, an interpretation Z := (Az, .z) consists of a setA2: (the domain of Z) and a function .z (the interpret-ation function of 77) that maps every concept name to a subset Cz of Az, every role name P to a sub-

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I Constructor Name Syntax Semantics

concept name Ctop Tatomic negation -~Cconjunction Ex n E2universal quantification VR.Eunqualified existential quantification 3Rexistential quantification $ 3R.Edisjunction H E1 tJ E2general negation C -~Enumber restrictions Af 3->mR

qualified number Qrestrictionswell-founded Wrole value map }2

role nameinverse Zunion 7~concatenation 7~ R1 o R2reflexive transitive closure 7~ R*identity T~ id( E)difference T~ RI \ R~

Cz C_ Az

Az

Az \ z

{o I Co’: (o, o’) ̄ ~ -+ o’ ¯E~}{o 13o’: (o,o’) ̄ ~}

{o 13o’: (o, o’) ̄ ~ ̂ o’̄ Es}E~ u ElAs \ Es

{o I ~{o’1 (o, o’) ̄ ~} >m}3--mR.E {o I ~{o’ t (o, o’) ̄ s ̂ o’̄ U} > .~}3 0: (o. o~+,) ~ ~}

(R1 C_ R~) {o I {o’ I (o,o’) ̄ z} c{o’1 (o,o’) ¯ nzp pZ C_ As x As

R- {(o, o’)I (o’, o) ~}R~ u R~ R~ u R~

{(o, o) I o ̄ Ez}

Table 1: Syntax and semantics of the concept and role forming constructors.

set pZ of Az x Az, and concept and role expressionsaccording to the last column of Table 11.

Most of the constructors we have presented are wellknown in DLs, and have a correspondence in modelingconstructs used both in Frame Based Systems and inDB models. This is not the case for the constructorwf, called well-founded, which has rarely been con-sidered in KR, but which proves particularly usefulfor expressing modeling structures that are inherentlyfinite.

2.2 .A£-SCHEMATA

Using concept expressions of a DL, intensional know-ledge can be specified through schemata. Given anA£-DL £., an £-schema is a triple 8 := (C,P,T),where C is a finite set of concept names, P is a finiteset of role names, and 7- is a finite set of assertions.Each such assertion has one of the forms

C 3 E (primitive concept specification)

C - E (concept definition)

where C E C and E is a concept expression of £ in-volving only names of C U P. In the following, when

IUS denotes the cardinality of a set S.

not specified otherwise, we assume that 8 := (C, 7), T),and that C and P are the sets of concept and role namesthat appear in 7-.

Primitive concept specifications are used to specify ne-cessary conditions for an object to be an instance of aconcept, while concept definitions specify both neces-sary and sufficient conditions. More formally, an inter-pretation Z satisfies an assertion C ~ E if Cz C Ez,

and it satisfies an assertion C - E if Cz = Ez. Aninterpretation that satisfies all assertions in a schema ,9is called a model of 8. A finite model is a model withfinite domain. A concept expression E is (finitely) con-sistent in 8 if $ admits a (finite) model Z such thatEz ~k O, and E1 is (finitely) subsumed by E2 in 8 ifEz C_ Ez for every (finite) model Z of ,9.

Both in KR and in DB models several different as-sumptions on the form of a schema are made, eitherimplicitly or explicitly:

1. For each concept name there is at most one asser-tion in which that name appears in the left-handside.

2. The schema contains no cycles2.

2See (Nebel, 1991) for a formal definition of cycle in schema.

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3. The schema contains only primitive concept spe-cifications (primitive schema).

Assumption 1 corresponds to the natural requirementthat all the information concerning a class is containedin one place, which contributes to a better structuringof the represented knowledge (Nebel, 1990; Baader,1990; Lecluse &: Richard, 1989; Bergamaschi & Sar-tori, 1992). It may however be too restrictive, espe-cially in those cases where one wants to state additionalconstraints that are not specific to a certain class:

Assumption 2 of acyclicity (which is meaningful onlyin the presence of assumption 1) is made in most ex-isting concept-based KR systems, although imposingthis condition strongly limits the expressive power ofthe system (Nebel, 1990). Many real world conceptscan in fact be expressed naturally only in a recursiveway and thus through cycles. The reason for not ad-mitting cycles is twofold. On one hand cycles greatlyincrease the computational complexity of reasoning ona schema (Baader, 1990; Calvanese, 1996a), and on theother hand, there is still no agreement in the field onwhich semantics to adopt in their presence. Besidesdescriptive semantics, which is the semantic specifica-tion described above, so called fixpoint semantics havebeen considered (Nebel, 1991). Descriptive semantics,which is the one we use, has been advocated to be theone which gives the most intuitive results (Nebel, 1991;Buchheit, Donini, Nutt, & Schaerf, 1995), and it is alsothe only one that can be adopted for the most gen-eral type of schemata, called free, in which none of thethree conditions above is required to hold3. (Baader,1990) argues that from a constructive point of viewgreatest fixpoint semantics should be preferred. Leastfixpoint semantics, on the other hand, seems the mostappropriate for the definition of inductive structures.We will see in Section 3 that we can obtain a similarresult by using the well-founded constructor togetherwith descriptive semantics. We remark that recentlythere have been also proposals for an integration ofthe different types of semantics by including fixpointconstructors in the logic (Schild, 1994; De Giacomo& Lenzerini, 1994b) and interpreting cycles with de-scriptive semantics.

Assumption 3 is sometimes made in concept-based KRsystems (Buchheit et al., 1995) and it is usually impli-cit in DB models, in which the classes are specifiedonly through necessary conditions (Calvanese et al.,

3The expressivity of free schemata defined in this wayis equivalent to the one of schemata constituted by ]reeinclusion assertions, which have the form E1 __. E~, whereboth E1 and E2 are arbitrary concept expressions with norestriction at all.

1994). Although it strongly limits expressivity, reas-oning on primitive schemata is nevertheless intractableeven in the simplest setting, where we use only theconstructors of .A~ (which are common to almost allrepresentation formalisms) and the schema is acyclic(Buchheit et al., 1995). In fact assumption 3 reducesworst case complexity of reasoning on a schema onlyif relatively weak languages are adopted (Calvanese,1996a).

FROM FINITE MODELINGSTRUCTURES TO FINITEMODELS

Both in Kl% and in DBs representation formalisms havebeen developed which offer powerful structuring cap-abilities and procedures to reason about a specifica-tion. However, while in DBs the common assump-tion is that the modeled domain is finite, this has sel-dom been an issue in KR in general and in DLs inparticular, where the developed reasoning methods donot take care of finiteness. For most of the DLs thathave been considered till now this does not representa limitation, neither for reasoning on concept expres-sions (Donini et al., 1991a; Donini, Lenzerini, Nardi,& Nutt, 1991b) nor on schemata (Nebel, 1991; Buch-heit et al., 1995), since for these formalisms the fi-nite model property holds. It states that if a schema(or concept expression) admits a model, then it alsoadmits one with a finite domain, and therefore reas-oning with respect to unrestricted ~aodels amounts toreasoning with respect to finite ones. This does nothold anymore for the more expressive logics studiedrecently, which include inverse roles and functionalityof roles, since these constructors in combination withcycles in the schema cause the finite model property tofail to hold.

Example 1 Consider the following schema:

Guard ~ 3shields.Guard17 3-

singleton concepts are used to specify concepts thatare consistent only in an infnite domain.

For the most expressive DLs (and in particular forthose that lack the finite model property), methodsfor reasoning with respect to arbitrary models havebeen developed, which are based on a correspond-ence between DLs and Propositional Dynamic Logics(PDLs). PDLs are formalisms specifically designed forreasoning about program schemes (Kozen & Tiuryn,1990), and their correspondence with DLs was de-scribed frst in (Schild, 1991) and extended to moreexpressive logics in (De Giacomo & Lenzerini, 1994a).The correspondence allows one to reduce concept con-sistency to satisfiability of a formula of some PDL,and to use the advanced methods developed for PDLsto solve this latter task. These methods exploit thefundamental tree model property, shared by all PDLs,which states that every satisfiable formula admits par-ticular models that have the form of an (infinite) treeof bounded degree. If the domain has to be finite,however, the existence of tree-like models is not guar-anteed, and the known reasoning methods are not ap-plicable.

It is nevertheless possible to express restricted forms offiniteness, even in those logics that lack the finite modelproperty, by imposing conditions that force certainstructures in the domain to be well-founded, withoutassuming the whole interpretation domain to be finite.This allows us to deal with finite modeling structures,while exploiting the powerful techniques for reasoningwith respect to unrestricted models. Well-foundednesscan be imposed by using wf(R), which expresses thatthere is no infinite sequence of objects, connected oneto the next by means of role R. In terms of PDLs,wf(R) corresponds to a negated repeat formula-~A(R)(Streett, 1982), which is in turn equivalent to the leastfixpoint expression4 pX.VR.X (Kozen, 1983). Thuswe can use such constructor to express finite structuresthat admit a least-fixpoint definition. We illustrate thison the example of binary trees.

Example 2 We can define a binary tree inductivelyas the least set bin-tree such that:

* a node having no successors is a bin-tree,

¯ a node whose successors are all bin-trees, is a bin-tree,

4A least fixpoint expression pX.f(X), where X is aconcept variable and f is a monotone operator, is inter-preted as the least set Xz of objects in Az satisfyingXz -- (f(X)) z. See (Schild, 1994; De Giacomo & Len-zerini, 1994b) for more details.

and where additionally a node is constrained to haveat most one predecessor and at most two successors(plus possibly additional properties such as a label).

The inductive part of the definition can be captured bythe least fixpoint expression

pX.((Node n 3-

relation can be captured by taking the transitive clos-ure of a basic_part_of role, while asymmetricity andfiniteness (i.e. well-foundedness) are imposed by an as-sertion of the form T ___ wf(basic_part_of). Noticealso that by means of role value maps on role nameswe can capture different specializations of the part-ofrelation.

The constructors that cause the loss of the finite modelproperty are all necessary in order to capture the char-acteristics of the most important representation form-

alisms used both in KR and in DBs (Calvanese et al.,1994). In fact, neither the Entity-Relationship modelnor expressive object-oriented models have the finitemodel property, and since the assumption of a finitedomain is essential in this context, specialized meth-ods capable of reasoning in these formalisms with re-spect to finite models are needed. This seems to bemore difficult than reasoning with respect to arbitraryones. As already noticed, if the domain has to be fi-nite the tree model property does not hold, and thecorrespondence with PDLs cannot be exploited. Infact, decidability of finite concept consistency and fi-nite subsumption for more expressive formalisms is stillan open problem, and there are no known methods tohandle concept definitions or even (qualified) existen-tial quantification in conjunction with the constructorsthat cause the finite model property not to hold. Themain result of this paper, presented in Section 5, is amethod to reason with respect to finite models on freeA£dQZ-schemata. The possibility in such schematato make use of full negation and of concept definitionsprovides indeed the necessary expressivity to go bey-ond the basic features of DB models and capture andreason upon several useful constructs that have beenproposed as extensions to existing models. Relevantexamples are:

¯ Classes defined through necessary and sufficient con-ditions, which correspond to so called views inobject-oriented DBs (Abiteboul & Bonner, 1991).Using free assertions5 it is also possible to specifyonly sufficient conditions for an object to be an in-stance of a class, and more general forms of integrityconstraints.

¯ Classes defined as the union of other classes.

¯ ISA relations between relationships in the Entity-Relationship model (Di Battista &; Lenzerini, 1993).

¯ Negative assertions about classes, such as disjoint-ness, or non-participation in a relationship (Di Bat-tista & Lenzerini, 1993).

5See footnote 3.

Athlete ~ Vpartic.CompR 3S4partic RVwins.RunningCompN Hwins.Final

Comp ~ Vpartic-.AthleteFinal ~ Comp R 3~lwins-

RunningComp ~ CompFinal R RunningComp ~ 3~6partic-

Figure 1: A schema with finitely inconsistent concepts

Example 3 The schema S shown in Figure 1 mod-els the participation of athletes in sports competitions.The last (free) inclusion assertion represents a con-straint that is not associated to a specific conceptname. It turns out, however, that $ contains a mod-eling error that causes Athlete to be finitely incon-sistent. In fact, the conditions imposed on Athleteshould only be required for (fast) runners that win final. Therefore it is not correct to assume that onlysuch athletes can participate in a competition. Noticethat the inconsistency above would not be detected bya reasoning procedure that ignores finiteness, since allconcepts in S are consistent. ¯

Before discussing in Section 5 the method to reasonwith respect to finite models, we briefly illustrate inSection 4 how to reason with respect to arbitrary mod-els in a very expressive DL which includes the well-foundedness constructor, and therefore allows us torepresent finite modeling structures.

4 REASONING ABOUT FINITEMODELING STRUCTURES

We now sketch a method to reason with respect toarbitrary models on free schemata expressed in aDL that includes all constructors shown in Table 1,and in particular the well-founded constructor that al-lows us to define finite modeling structures. Someof the other constructors, however, and in particularrole-value maps, intersections of complex roles, andnumber restrictions on complex roles, make reasoninghighly undecidable if they are used without restrictions(Schmidt-Schauf3, 1989; Harel, 1983). Therefore, syntactically restrict the use of these problematic con-structors in a way similar to what done in (De Giac-omo & Lenzerini, 1995) for the logic CATS. The logicAfT, in which we distinguish between basic roles, de-noted by the letter B, and arbitrary complex roles, isdefined as follows:

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The semantics for .A/:T class and role expressions isdefined as in Table 1, considering that basic roles arejust particular types of complex roles.

Decidability in deterministic exponential time of reas-oning in an object-oriented model with the same ex-pressivity as free .A/:T-schemata was already shownin (Calvanese, De Giacomo, & Lenzerini, 1995). briefly sketch the idea underlying the proof, which ex-ploits the correspondence with PDLs: Concepts of DLscorrespond to formulae of PDLs, roles correspond toprograms, and (ahnost) every constructor of DLs hasits counterpart in PDLs.

We can reduce the problem of reasoning on a free ACT-schema to the problem of verifying the satisfiability ofa formula of R.FCPDL, which is basic PDL augmentedwith the converse operator on programs (correspondingto inverse roles), local functionality on both direct andinverse atomic programs (which corresponds to func-tional restrictions, to which existential quantificationsare reduced by "reifying" basic roles) and the repeatconstructor (which corresponds to well-foundedness).The details of the reduction can be found in (Calvanese,1996b).

Proposition 4 Given a free .A~.T-schema S and aconcept expression E, one can construct in polynomialtime in ISI + [E] an rtFCPDL-formula fS,E such thatfS,E is satisfiable if and only if E is consistent in S.

In order to prove the desired decidability result,however, we cannot exploit known decision proceduresfor PDLs, since the decidability of R.FCPDL is open. Toshow decidability of R.FCPDL we reduce satisfiability ofa formula to nonemptiness of the language accepted bya finite automaton on infinite objects derived from theformula (Thomas, 1990). This is a standard techniquethat provides tight upper bounds for various modaland temporal logics (Emerson &: Jutla, 1989; Vardi

Wolper, 1994) and logics of programs (Vardi, 1985;Vardi &: Wolper, 1986; Vardi &; Stockmeyer, 1985), andwe extend it to R.FCPDL. It is based on the tree modelproperty which holds also for R.FCPDL. In particular,the automaton we obtain from a R.FCPDL formula f isa hybrid automaton HI := (TI, W$), that combines anautomaton on infinite trees T! of exponential size in Ifl,with an automaton on infinite words Wf of polynomialsize in Ifl and which handles the repeat sub-formulae.Exploiting the results described in (Emerson ~: Jutla,1989), the nonemptiness problem for such a hybrid treeautomaton can be solved in deterministic time that is

polynomial in the number of states of T$ and exponen-tial in the number of states of W/.

This allows us to establish the desired upper boundfor satisfiability in R.FCPDL, and therefore for unres-tricted model reasoning in Af-.T. By the well knownEXPTIME lower bound for reasoning on free ~’Z:--schemata6 the complexity bound is tight.

Theorem 5 Unrestricted concept consistency andconcept subsumption in free .ACT-schemata areEXP TIME- complete.

5 REASONING WITH RESPECTTO FINITE MODELS

In this section we present a technique for reasoningwith respect to finite models in /:-schemata. Themethod extends the one proposed in (Calvanese et al.,1994) to reason on primitive AZ://AfT:-schemata, and isbased on the same idea of constructing from a schemaa system of linear inequations, and relating the exist-ence of finite models of the schema to the existenceof particular solutions of the system. The method wepresent decides concept consistency (and therefore alsosubsumption) in free .A£CQZ-schemata. Due to thepresence of arbitrary free inclusion assertions, and thehigher expressivity of the underlying language (in par-ticular the presence of existential quantification) theconstruction of the system of inequations is much moreinvolved than for primitive .A£UAfZ-schemata.

We describe the method for reasoning on free .A/:CAfi/:-schemata and sketch then how it must be extendedto deal with qualified number restrictions. We usethe term literal, ranged over by L, to denote eithera concept name or a concept name preceded by thesymbol "-~". It is easy to see that each free schemacan be transformed in linear time into an equivalentnormalized schema, where each assertion has the formL _ E, and E has one of the forms

L1 151 H L2 I VR.L1 13R.LI I 3>-"n I 3

and an element Q E 7)x2 cx2c a compound role. 5ranges over compound concepts and Q over compoundroles. A compound assertion has one of the forms

b ___ 3>-ran, b 24 3-

which the first element is a compound concept, and the

second element is a function ~ : T¢ --+ 22c that asso-ciates to each role and inverse role a set of compoundconcepts. We introduce two functions that allow us torefer to the components of bicompound concepts:

- The function cc : Bc,’p --+ 2c returns the first com-

ponent D of a bicompound concept (/9, ~o).

- The function ccs : Bc,.pxT¢ --~ 22c returns for a

bicompound concept /? := (D, 9) and a role

the set of compound concepts assigned to R by thesecond component ~o of B.

An element of "PxBe,7, xBc,~, is called a bicompoundrole. B and U range over bicompound concepts andbicompound roles respectively.

The intuition behind the definition of bicoffnpound con-cepts and roles is the following: If 2c(B) = D, thenall instances of B are instances of D. Let o be suchan instance. Then for each compound concept D’ in

ccs(/?, R), there is an instance o’ of D’ connected too via role R, while for the compound concepts notin ccs(/~, R) there is no such instance. In analogy

compound roles, a bicompound role (P, B1, B2) is in-terpreted as the restriction of role P to the pairs whosefirst component is an instance of/~1 and whose secondcomponent is an instance of/~2.

A bicompound assertion has one of the forms

~5 24 3->mR, ~ 24 3-m~=R, where m=ox := max{m I3L ̄ cc(~) : (L 24 3->mR) ¯

- if for some L ¯ cc(2) there is an assertion (L 3

(C,’P,B,U,T) be the biexpansion of S. Then qs

contains one unknown V~r()~) for each bicompoundconcept or role )~ in B UU, and consists of the follow-

ing linear homogeneous inequations:

¯ For each B E /~, for each P E P, for each D Eccs(/3, P) the inequation

Var(B) ~ Var((F, B,/32)).A

(P,B,B2)EL! [ cc(B2)=D

¯ For each /3 E /~, for each P E ~o, for each b Eccs(/3, P-) the inequation

Var( ) < Var((F,A

(P, BI,B)~ld [ cc(B1)=D

¯ For each bicompound assertion (B 2< 3>-mR) E 7-the inequation

m. Vat(B) _< S(B, R), where

S(B,P) := ~ Var((P,B, B2)),

(P,B,B2)EIA

S(/3, P-) := Z Var((F, B1,

(P,B1 ,B)ELI

¯ For each bicompound assertion (/3 _ 3--nR) E the inequation

n. Var( ) > R).It is now possible to relate a solution of ~s to a fi-nite model of S, in which each bicompound conceptand bicompound role has a number of instances thatis given by the value that the solution assigns to thecorresponding unknown. Since in a model of S eachbicompound role that refers (in its second or third com-ponent) to a bicompound concept that has an emptyextension, also has an empty extension, this condi-tion reflects in an additional requirement that the solu-tion must satisfy. A solution of ~s is acceptable, if,whenever it assigns value 0 to an unknown correspond-ing to a bicompound concept /3, then it assigns alsovalue 0 to all unknowns corresponding to bicompoundroles that have/3 as their second or third component.We can now state the main theorem concerning finiteconcept consistency.

Theorem 6 A concept C E C is finitely consistent inS if and only if

~Eh’l c¢(~)

admits an acceptable nonnegative integer solution.

As shown in (Calvanese, 1996b), by applying linearprogramming techniques one can decide the existenceof acceptable nonnegative integer solutions in polyno-mial time in the size of the system. Since ~/s can beconstructed in time which is at most double exponen-tial in ]S], and since in free A~:CAfZ-schemata we caneasily reduce both consistency of an arbitrary conceptexpression and concept subsumption to consistency ofa single concept name, we obtain the following worstcase upper bound.

Theorem 7 Finite concept consistency and conceptsubsumption in free .4f_.CAfZ-schemata can be decidedin worst case deterministic double exponential time.

The method as described above cannot deal directlywith qualified number restrictions, since the differ-entiation introduced by bicompound concepts is toocoarse. In fact, we need to make a separation in bicom-pound classes based not only on the existence but alsoon the number of links of a certain type. It turns outthat it is in fact sufficient to consider only intervals ofnumbers of links, where the ranges of these intervalsare given by the numbers that effectively appear in theschema. In this way it is still possible to keep the sizeof the resulting "extended" biexpansion double expo-nential in the size of the schema. The resulting "exten-ded" bicompound assertions can then again be codedby means of a system of linear inequations, and we canprove the counterpart of Theorem 6 for the system de-rived from a free ~4£:CQZ-schema. We omit furtherdetails and state only the final result.

Theorem 8 Finite concept consistency and conceptsubsumption in free A£CQZ-schemata can be decidedin worst case deterministic double exponential time.

6 CONCLUSIONS

In this paper we have discussed the issues concerningfinite model reasoning at the intensional level in ex-pressive DLs that lack the finite model property. Twodistinct ways of dealing with finiteness have been pro-posed:

1. We have introduced in the language a specificconstructor to express well-foundedness, and haveshown how known methods for reasoning with re-spect to arbitrary models can be extended to dealwith it. This provides an EXPTIME decisionprocedure for concept consistency and subsump-tion.

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2. We have developed a technique to reason with re-spect to finite models on schemata of the mostgeneral form expressed in a DL capable of cap-turing most known known data models. The pro-posed algorithm works in worst case deterministicdouble exponential time.

We are currently extending our work in several direc-tions. A major point is the extension of the reasoningtechniques to deal also with extensional knowledge, i.e.with assertions stating properties of specific objects.In the unrestricted case, the reasoning technique for.A£.T discussed in Section 4 can be directly integratedwith the work in (De Giacomo, 1996), still obtainingan EXPTIME upper-bound. In the finite case, theproposed technique can be applied under specific as-sumptions which are rather restrictive but common inDBs. However, a general method for finite model reas-oning in expressive DLs on a schema together withassertions on individuals is still open.

We aim also at increasing the expressivity of theschema definition language while still preserving decid-ability. Concerning point 1 above, we have mentionedin Section 3 that the well-founded constructor is equi-valent to a particular type of least-fixpoint expression.A natural extension would be to allow arbitrary nestedfixpoints as in the p-calculus (Kozen, 1983) togetherwith inverse roles and number restrictions. Concern-ing point 2, it can be shown that the reasoning methoddeveloped for .A£CQZ can be extended to deal withsome of the constructors of .A£T, and in particular

basic roles (i.e. union, intersection, and difference ofrole names and inverse roles) and role value maps onbasic roles. Qualified number restrictions turn out tobe indispensable for this. If the addition of reflexivetransitive closure preserves decidability also in the fi-nite case is still open.

Acknowledgements

I would like to thank Maurizio Lenzerini and GiuseppeDe Giacomo for many inspiring discussions and FranzBaader for his hospitality at the RWTH Aachen, wherepart of this work was carried out.

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