F M Donini M Lenzerini D Nardi A Schaerfweb.cs.iastate.edu/~honavar/reasoning-survey.pdf · yle Jon...

�� F� M� Donini� M� Lenzerini� D� Nardi� A� Schaerf

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Wright� Jon R�� Elia S� Weixelbaum� Gregg T� Vesonder� Karen E� Brown�Stephen R� Palmer� Jay I� Berman� and Harry H� Moore� �� A Knowledge�Based Con�gurator that Supports Sales� Engineering� and Manufacturing atAT�T Network Systems� AI Magazine ��

Yen� John� Robert Neches� and Robert MacGregor� �� CLASP IntegratingTerm Subsumption Systems and Production Systems� IEEE Transactions onKnowledge and Data Engineering ��

��

Reasoning in Description Logics � ��

Levesque ��

Reiter� R� �� On Asking What a Database Knows� In Symposium on Com�putational Logics� ed� J� W� Lloyd� �� Springer�Verlag� ESPRIT BasicResearch Action Series�

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Mylopoulos� J�� P� A� Bernstein� and E� Wong� �� A Language Facility forDesigning Database�Intensive Applications� ACM Transactions on DatabaseSystems ��

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Patel�Schneider� P� F�� and Bill Swartout� �� Working Version �Draft� De�scription Logic Speci�cation from the KRSS E�ort� June� UnpublishedManuscript�

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Quillian� M� Ross� �� Word Concepts A Theory and Simulation of Some BasicCapabilities� Behavioral Science �� Republished in Brachman and

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Donini� Francesco M�� Daniele Nardi� and Riccardo Rosati� ��b� Non��rst�order features in concept languages� In Proc� of the th Conf� of the ItalianAssociation for Arti�cial Intelligence AI�IA�� Lecture Notes InArti�cial Intelligence� No� �� Springer�Verlag�

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MacGregor� Robert� Deborah McGuinness� Eric Mays� and Tom Russ �ed�� Working Notes of the AAAI Fall Symposium on Issues on Description Logics�Users meet Developers� Boston� USA�

McAllester� David� �� Unpublished Manuscript�

Minsky� Marvin� �� A Framework for Representing Knowledge� In MindDesign� ed� J� Haugeland� The MIT Press� A longer version appeared inThe Psychology of Computer Vision �� Republished in Brachman andLevesque ��

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J� Doyle� E� Sandewall� and P� Torasso� �� Bonn� Morgan Kaufmann�Los Altos�

De Giacomo� Giuseppe� and Maurizio Lenzerini� ��a� Boosting the Corre�spondence between Description Logics and Propositional Dynamic Logics� InProc� of the �th Nat� Conf� on Arti�cial Intelligence AAAI�� AAAI Press�The MIT Press�

De Giacomo� Giuseppe� and Maurizio Lenzerini� ��b� Concept Language withNumber Restrictions and Fixpoints� and its Relationship with ��Calculus� InProc� of the ��th European Conf� on Arti�cial Intelligence ECAI��

Devambu� Premkumar� and Diane Litman� �� Plan�Based TerminologicalReasoning� In Proc� of the nd Int� Conf� on the Principles of KnowledgeRepresentation and Reasoning KR�� ed� James Allen� Richard Fikes� andErik Sandewall� �� Morgan Kaufmann� Los Altos�

Donini� Francesco M�� Bernhard Hollunder� Maurizio Lenzerini� Alberto Mar�chetti Spaccamela� Daniele Nardi� and Werner Nutt� ��a� The Complexityof Existential Quanti�cation in Concept Languages� Arti�cial IntelligenceJournal ��

Donini� Francesco M�� Maurizio Lenzerini� Daniele Nardi� and Werner Nutt��a� The Complexity of Concept Languages� In Proc� of the nd Int�Conf� on the Principles of Knowledge Representation and Reasoning KR�� ed� James Allen� Richard Fikes� and Erik Sandewall� �� MorganKaufmann� Los Altos�

Donini� Francesco M�� Maurizio Lenzerini� Daniele Nardi� and Werner Nutt��b� Tractable Concept Languages� In Proc� of the �th Int� Joint Conf�on Arti�cial Intelligence IJCAI�� Sydney�

Donini� Francesco M�� Maurizio Lenzerini� Daniele Nardi� and Werner Nutt�� The Complexity of Concept Languages� Technical Report RR��Deutsches Forschungszentrum f�ur K�unstliche Intelligenz �DFKI��

Donini� Francesco M�� Maurizio Lenzerini� Daniele Nardi� Werner Nutt� and An�drea Schaerf� ��b� Adding Epistemic Operators to Concept Languages� InProc� of the �rd Int� Conf� on the Principles of Knowledge Representationand Reasoning KR�� Morgan Kaufmann� Los Altos�

Donini� Francesco M�� Maurizio Lenzerini� Daniele Nardi� Werner Nutt� and An�drea Schaerf� ��a� Queries� Rules and De�nitions as Epistemic Sentencesin Concept Languages� In Proc� of the ECAI Workshop on Knowledge Rep�resentation and Reasoning� �� Lecture Notes In Arti�cial Intelligence�No� �� Springer�Verlag�

Donini� Francesco M�� Maurizio Lenzerini� Daniele Nardi� Werner Nutt� and An�drea Schaerf� ��a� Adding Epistemic Operators to Concept Languages�Technical report� Dipartimento di Informatica e Sistemistica� Universit�a diRoma �La Sapienza�� In preparation�

Donini� Francesco M�� Maurizio Lenzerini� Daniele Nardi� and Andrea Schaerf��b� Deduction in Concept Languages From Subsumption to InstanceChecking� Journal of Logic and Computation ��

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Italy� Technical Report� Universit�a di Roma �La Sapienza�� Dipartimento diInformatica e Sistemistica� RAP ��

Borgida� Alexander� and Peter F� Patel�Schneider� �� A Semantics and Com�plete Algorithm for Subsumption in the CLASSIC Description Logic� Journalof Arti�cial Intelligence Research ��

Brachman� Ronald J� �� On the Epistemological Status of Semantic Networks�In Associative Networks� ed� Nicholas V� Findler� �� Academic Press�

Brachman� Ronald J� �� Reducing� CLASSIC to Practice Knowledge Rep�resentation Meets Reality� In Proc� of the �rd Int� Conf� on the Principles ofKnowledge Representation and Reasoning KR�� Morgan Kauf�mann� Los Altos�

Brachman� Ronald J�� and Hector J� Levesque� �� The Tractability of Sub�sumption in Frame�Based Description Languages� In Proc� of the th Nat�Conf� on Arti�cial Intelligence AAAI��

Brachman� Ronald J�� and Hector J� Levesque� �� Readings in KnowledgeRepresentation� Morgan Kaufmann� Los Altos�

Brachman� Ronald J�� Victoria Pigman Gilbert� and Hector J� Levesque� �� AnEssential Hybrid Reasoning System Knowledge and Symbol Level Accountsin KRYPTON� In Proc� of the �th Int� Joint Conf� on Arti�cial IntelligenceIJCAI�� Los Angeles�

Brachman� Ronald J�� and James G� Schmolze� �� An Overview of the KL�ONE Knowledge Representation System� Cognitive Science ��

Buchheit� Martin� Francesco M� Donini� Werner Nutt� and Andrea Schaerf� ��a�Terminological Systems Revisited Terminology � Schema � Views� In Proc�of the �th Nat� Conf� on Arti�cial Intelligence AAAI�� Seattle�USA�

Buchheit� Martin� Francesco M� Donini� Werner Nutt� and Andrea Schaerf� ��A Re�ned Architecture for Terminological Systems Terminology � Schema� Views� Technical Report RR�� Deutsches Forschungszentrum f�urK�unstliche Intelligenz �DFKI��

Buchheit� Martin� Francesco M� Donini� and Andrea Schaerf� �� DecidableReasoning in Terminological Knowledge Representation Systems� Journal ofArti�cial Intelligence Research ��

Buchheit� Martin� Manfred A� Jeusfeld� Werner Nutt� and Martin Staudt� ��b�Subsumption between Queries to Object�Oriented Databases� InformationSystems �� Special issue on Extending Database Technology�EDBT��

Calvanese� Diego� Maurizio Lenzerini� and Daniele Nardi� �� A Uni�ed Frame�work for Class Based Representation Formalisms� In Proc� of the th Int�Conf� on the Principles of Knowledge Representation and Reasoning KR�� ed� J� Doyle� E� Sandewall� and P� Torasso� �� Bonn� MorganKaufmann� Los Altos�

Cohen� William W�� and Haym Hirsh� �� Learning the CLASSIC DescriptionLogics Theorethcal and Experimental Results� In Proc� of the th Int� Conf�on the Principles of Knowledge Representation and Reasoning KR�� ed�

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Baader� Franz� Hans�J�urgen B�urkert� Jochen Heinsohn� Bernhard Hollunder�J�urgen M�uller� Bernard Nebel� Werner Nutt� and Hans�J�urge Pro�tlich� ��Terminological Knowledge Representation A Proposal for a TerminologicalLogic� Technical Report TM�� Kaiserslautern� Germany DeutschesForschungszentrum f�ur K�unstliche Intelligenz �DFKI��

Baader� Franz� and Bernhard Hollunder� �� A Terminological KnowledgeRepresentation System with Complete Inference Algorithm� In Proc� of theWorkshop on Processing Declarative Knowledge� PDK�� LectureNotes In Arti�cial Intelligence� No� �� Springer�Verlag�

Baader� Franz� and Bernhard Hollunder� �� Embedding Defaults into Ter�minological Knowledge Representation Formalisms� In Proc� of the �rd Int�Conf� on the Principles of Knowledge Representation and Reasoning KR�� Morgan Kaufmann� Los Altos�

Baader� Franz� and Bernhard Hollunder� �� How to Prefer More Speci�cDefaults in Terminological Default Logic� In Proc� of the ��th Int� Joint Conf�on Arti�cial Intelligence IJCAI�� Chambery� France� MorganKaufmann� Los Altos�

Baader� Franz� Bernhard Hollunder� Bernhard Nebel� Hans�J�urgen Pro�tlich� andEnrico Franconi� �� An Empirical Analisys of Optimization Techniques forTerminological Representation Systems� In Proc� of the �rd Int� Conf� on thePrinciples of Knowledge Representation and Reasoning KR�� Morgan Kaufmann� Los Altos�

Baader� Franz� Maurizio Lenzerini� Werner Nutt� and Peter F� Patel�Schneider�ed�� Working Notes of the �� Description Logics Workshop� Bonn�Germany� Deutsches Forschungszentrum f�ur K�unstliche Intelligenz �DFKI��D��

Beck� H� W�� S� K� Gala� and S� B� Navathe� �� Classi�cation as a QueryProcessing Technique in the CANDIDE Semantic Data Model� In Proc� ofthe �th IEEE Int� Conf� on Data Engineering ICDE��

Bell� John L�� and Moshe Machover� �� A Course in Mathematical Logic�North�Holland Publ� Co�� Amsterdam�

Bergamaschi� Sonia� and Bernhard Nebel� �� Acquisition and Validation ofComplex Object Database Schemata Supporting Multiple Inheritance� Ap�plied Intelligence ��

Bergamaschi� Sonia� and Claudio Sartori� �� On Taxonomic Reasoning inConceptual Design� ACM Transactions on Database Systems ��

Borgida� Alexander� �� From Type Systems to Knowledge RepresentationNatural Semantics Speci�cations for Description Logics� Journal of Intelligentand Cooperative Information Systems ��

Borgida� Alexander� Ronald J� Brachman� Deborah L� McGuinness� and LoriAlperin Resnick� �� CLASSIC A Structural Data Model for Objects� InProc� of the ACM SIGMOD Int� Conf� on Management of Data� ��

Borgida� Alexander� Maurizio Lenzerini� Daniele Nardi� and Bernhard Nebel�ed�� Working Notes of the �� Description Logics Workshop� Rome�

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tively used in the construction of knowledge�based applications in severaldomains �see for example Wright et al� �� like conguration and soft�ware engineering� Recently we have seen attempts to use them in the eldsof machine learning �see for example Cohen and Hirsh �� and planning�see for example Devambu and Litman �� Weida and Litman �� Ar�tale and Franconi �� where the basic formalism is extended to deal withtime and action��

Related �elds� Finally there are close relationships between Descrip�tion Logics and formalisms developed in other areas of computer science�In particular there is a strict relationship with logics of programs origi�nally studied for program verication and analysis �see Schild �� Schild�� De Giacomo and Lenzerini ��a�� As we said in Sections and �this correspondence has provided new useful tools for studying descrip�tion logics equipped with a very expressive concept language as well asinsights on the denitional mechanisms of the TBox� Moreover there arestrong links with the eld of Databases and in particular with Semanticand Object�oriented Data Models �see for example Bergamaschi and Nebel�� Bergamaschi and Sartori �� Borgida �� Calvanese et al� ��Research in this direction can be signicant not only for the exchange oftechniques and results but may have a strong impact on the developmentof tools for database specication and design�

Acknowledgements

This work has been supported by the Esprit Basic Research Action ��Compulog �� and by the Italian National Research Council as part of theProgetto Finalizzato Sistemi Informatici e Calcolo Parallelo Sottoprogetto� LdR Ibridi�

References

Abrial� J� R� �� Data Semantics� In Data Base Management� ed� J� W� Klimbieand K� L� Ko�eman� �� North�Holland Publ� Co�� Amsterdam�

Artale� Alessandro� and Enrico Franconi� �� A computational Account for aDescription Logic of time and action� In Proc� of the th Int� Conf� on thePrinciples of Knowledge Representation and Reasoning KR�� ed� J� Doyle�E� Sandewall� and P� Torasso� �� Bonn� Morgan Kaufmann� Los Altos�

Attardi� Giuseppe� and Maria Simi� �� Consistency and Completeness ofomega� a Logic for Knowledge Representation� In Proc� of the �th Int�Joint Conf� on Arti�cial Intelligence IJCAI�� Vancouver� BritishColumbia�

Baader� Franz� �� Terminological Cycles in KL�ONE�based Knowledge Rep�resentation Languages� Technical Report RR�� Kaiserslautern� Ger�many Deutsches Forschungszentrum f�ur K�unstliche Intelligenz �DFKI�� Anabridged version appeared in Proc� of the �th Nat� Conf� on Arti�cial Intel�

ligence AAAI�� pp� ��

��

� � F� M� Donini� M� Lenzerini� D� Nardi� A� Schaerf

by two parts� axioms about concepts in the TBox representing intensionalknowledge and assertions about individuals in the ABox representing ex�tensional knowledge� We arranged the presentation by considering di�erentTBox�ABox settings and looking at corresponding reasoning problems�

However research on Description Logics has made several other contri�butions as the reader can nd in a series of workshops specically dedicatedto this subject �Nebel et al� �� MacGregor et al� �� Baader et al� ��Borgida et al� �� In the following we brie�y mention some of the issueswhich were outside the scope of this survey�

Implementation issues� The implementation of DL�systems involvesmany other aspects besides the computational analysis of reasoning prob�lems� Engineering of knowledge representation systems is illustrated forexample in �Brachman �� More specically with regard to the im�plementation of reasoning procedures the tableaux�based reasoning tech�niques which we have considered in this survey should be equipped withspecialized strategies for rule selection and application� Another approachto the characterization of reasoning methods is to look at existing imple�mentations and then try to characterize the computations done by the sys�tem possibly using a non�standard semantics �Borgida and Patel�Schneider��

Non��rst�order features� The analysis developed in this paper waslimited to deal with the rst�order formalization of knowledge representa�tion systems based on classes which means that we have addressed neithernonmonotonic nor behavioral features�

There is a large body of research on nonmonotonic reasoning withintaxonomies starting from the use of defaults in Semantic Networks andFrames� Some of this research specically addresses this issue in the frame�work of Description Logics �see for example Baader and Hollunder ��Quantz and Royer �� Padgham and Nebel �� Baader and Hollunder�� Straccia �� Padgham and Zhang �� Donini et al� �� b��

Knowledge representation systems typically provide several forms ofbehavioral features� Some of them such as methods or daemons closelyfollow the object�oriented programming paradigm others take the formof procedural rules and are borrowed from production rule systems� Theuse of production rules and forward reasoning can be nicely formalized inDescription Logics by using the epistemic operator dened in Section � tocharacterize what is known by the system �Donini et al� ��b Donini etal� ��a��

Applications� We have not discussed the elds of application wherethe representation and reasoning techniques discussed so far can be prof�itably applied� Description Logics as well as their predecessors SemanticNetworks and Frames have been initially developed for applications in nat�ural language understanding� However Description Logics can be e�ec�

�


least and greatest xpoint semantics� With regard to the descriptive se�mantics Subsumption is both coNP�hard and in PSPACE but it is notknown if it is complete for PSPACE�

�� Free TBoxes

Free TBoxes are sets of concept inclusions �C �� D� and concept equations�C

�� D� �see Section �� Reasoning in free TBoxes requires di�erent tech�

niques than those studied for primitive and simple TBoxes� This is alsoconrmed by complexity results� for the language FL� while reasoning insimple TBoxes adopting descriptive semantics is in PSPACE reasoning infree TBoxes is EXPTIME�hard �McAllester �� However the distinc�tion disappears for languages containing union and negation which cansimulate a whole free TBox in one primitive concept specication inter�preted under descriptive semantics �Buchheit et al� ��

For the sake of brevity we do not delve into the details of reasoningtechniques� We simply note that there are two main approaches� The rstapproach is based on an extension of the Tableaux calculus �see Buchheitet al� �� while the second one is based on the correspondence betweendescription logics and logics of programs� Such a correspondence rstdiscussed in �Schild �� has been proved extremely useful for devisingSubsumption algorithms in free TBoxes expressed in very powerful lan�guages� Recent work on this subject can be found in �De Giacomo andLenzerini ��a Schild ��

� Reasoning in a complete DL�system

The ultimate goal of the research about reasoning in Description Logics isto reason on a complete DL�system � � hT �Ai where T is a TBox and Ais an ABox both expressed in a concept language L� Although this is themost general problem related to reasoning in Description Logics researchon this subject has developed only recently� Similarly to the case of freeTBoxes there are two main approaches to the development of suitabletechniques for reasoning in complete DL�systems� One of the approachesis based on extending the tableaux�based calculus presented in Section ��to deal with both intensional and extensional knowledge and the otherone is based on exploiting the correspondence between Description Logicsand logics of programs� The interested reader is referred to �Buchheit etal� �� and �De Giacomo and Lenzerini ��a� for an account of theseapproaches�

� Conclusions

We have surveyed the most important issues related to reasoning problemsarising in the construction of knowledge representation systems based onDescription Logics� We have characterized such DL�systems as composed

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ing equation singles out a unique subset of �I hence it de�nes the conceptA� For Denition � � the least xpoint semantics leads to identify A with�� The same holds for Human in Denition �� It is evident that the leastxpoint semantics does not help in the above examples� On the other handthe least xpoint semantics is particularly suitable for providing inductivedenitions of concepts� Consider the case of a single�source directed acyclicgraph �DAG��

�� The EMPTY�DAG is a DAG �base step��

�� A NODE that has connections and all connections are DAG is aDAG �inductive step��

�� Nothing else is a DAG�

We can easily write a denition statement re�ecting the rst two conditions�

Dag�� EmptyDagt �Node u �ARCu �ARC�Dag�

and to enforce the third condition we need to take the smallest solutionof the corresponding equation�

Greatest Fixpoint Semantics� A�� F �A� species that A is to

be interpreted as the greatest solution �if it exists� of the correspondingequation� Again if the operator associated with the denition statementis monotonic then for any interpretation I satisfying the denition state�ment the corresponding equation singles out a unique subset of �I� ForDenition � � the greatest xpoint semantics leads to interpret A as theclass of all the individuals having a child in A� More generally the greatestxpoint semantics is suitable for dening classes of individuals whose struc�ture is circularly repeating� As another example the class FoB �FOreverBlond� of individuals who are blond and generation after generation havesome descendant who is FoB is dened by

FoB�� Blondu �CHILD�FoB

With the above denition statement we want to denote the class of allindividuals satisfying the equation corresponding to the above denitionstatement that is we want its greatest solution� On the other hand if weinterpret the two denition statements

Human�� Mammalu �PARENTu �PARENT�Human

Horse�� Mammalu �PARENTu �PARENT�Horse

with the greatest xpoint semantics we obtain a rather unintuitive re�sult� for any interpretation I satisfying the above denition statementsHumanI � HorseI�

The computational properties of the three semantics are studied in�Baader �� Nebel �� It turns out that for the language FLN�Subsumption in cyclic TBoxes is PSPACE�complete with respect to both

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�� Cyclic simple TBoxes

Consider the following cyclic simple TBox T

B�� A u �P �B�

and the initial interpretation I� such that� �I�

� fa� b� c� dg AI�

�fa� b� c� dg and P I

�

� f�a� b�� c� d�� d� d�g� It is easy to see that thereare two ways to extend I� to a model I for T �

�� I� is extended to I such that BI � fa� bg

�� I� is extended to I such that BI � fa� b� c� dg�

This example shows that given an initial interpretation for a cyclic TBoxT there is no unique way to extend it to a model for T i�e� to determinethe denotation of dened concepts�

In order to discuss the problems related to cyclic denitions considera TBox T constituted by a single cyclic denition A

�� C where C is

an expression containing A� Semantically A�� C is a sort of equation

specifying that for any interpretation I the subsets of �I which can betied to the concept A must be a solution of AI � CI�

Notice that we can associate to a denition statement an operator fromsubsets of �I to subsets of �I such that the solutions of the equation cor�respond to the xpoints of the operator� Consider the following denition�

A�� CHILD�A� �

we can associate to it the operator� �S�fa � �I j �b��a� b� � CHILDI � b �Sg for any interpretation I� In what follows we write F �� to denote suchan operator�

In �Baader �� Nebel �� De Giacomo and Lenzerini ��a� threeways of interpreting cyclic denitions are discussed according to one of thefollowing semantics�

Descriptive Semantics� A�� F �A� is a constraint stating that A has

to be some solution of the corresponding equation� For example Deni�tion � � states that the individuals in the class A have a child in the classA and the individuals that have a child in the class A are themselves in theclass A where A is no further specied� In general the denition statementA

�� C in the descriptive semantics is equivalent to fA �� C� C �� Ag� It

follows that the descriptive semantics is not appropriate for dening con�cepts� On the other hand it allows one to express necessary and su�cientconditions for concepts like in�

Human�� Mammalu �PARENTu �PARENT�Human��

Least Fixpoint Semantics� A�� F �A� species that A is to be inter�

preted as the smallest solution �if it exists� of the corresponding equation�If the operator associated with the denition statement is monotonic thenfor any interpretation I satisfying the denition statement the correspond�

��


that A directly uses an atomic concept B in T if the expression fT �A�contains B� Let uses �in T � denote the transitive closure of the relationdirectly uses �in T �� A simple TBox T is said to be cyclic if some atomicconcept uses itself in T acyclic otherwise� Intuitively cycles correspond torecursive denition of concepts�

We can now return to the above question and analyze the two cases ofacyclic and general simple TBoxes �or simply TBoxes� respectively�

�� Acyclic simple TBoxes

Acyclic simple TBoxes �or acyclic TBoxes for short� are ubiquitous in DL�systems� Indeed almost all existing systems do not allow the specica�tion of cyclic TBoxes� Nevertheless formal properties of acyclic TBoxeshave been investigated only recently� Nebel ��b� shows that for acyclicTBoxes the answer to the above question is positive� Given an acyclic TBoxT any initial interpretation can be extended to at most one model of T �This means that there is no ambiguity in interpreting a dened concepton the basis of the interpretation for primitive concepts� Moreover it iseasy to see that Subsumption in simple TBoxes can be reduced to Sub�sumption in an empty terminology� Indeed if we want to check whether Cis subsumed by D in an acyclic TBox T we can iteratively substitute ev�ery occurrence of any dened concept A in C and D by the correspondingexpression fT �A� obtaining the two expressions C�� D�� Since T is acyclicthis process terminates in a nite number of iterations and it holds thathT � �i j� C v D if and only if h�� i j� C� v D��

This result shows that reasoning with acyclic TBoxes can be reducedto reasoning with concept expressions� However Nebel ��b� also showsthat the above method may lead to a cost of the Subsumption algorithmwhich is exponential in the size of the TBox� Surprisingly the same papershows that intractability does not stem from the method but from theinherent complexity of the problem� determining Subsumption betweenC and D in an acyclic TBox T is co�NP�complete even if the languageused to express C�D and T is less expressive than FL�� The techniqueused to establish this result is interesting on its own and is based onreducing equivalence between nondeterministic nite state automata to twoSubsumption tests in a TBox encoding the two automata�

The reason why this result is surprising is that it was believed thatSubsumption in a TBox is not harder than Subsumption of concept de�scriptions� Nebel ��b� argues that indeed the exponential worst case isunlikely to occur and this would explain why Subsumption algorithms inacyclic simple TBoxes are well behaved in practice�

��


the Database setting where models for the TBox correspond to Databasestates and therefore are assumed to be nite�

�� Simple TBoxes

Recall that a concept denition is a statement of the form�

A�� C

�called the denition of A� and is intended to provide a denitional accountof the concept whose name is A� A simple TBox T is a set of conceptdenitions that obeys the following syntactic restriction� for every conceptname A there is at most one denition of A in T �

Informally dening a concept A means singling out a set of necessaryand su�cient conditions for an object to belong to the extension of A� Alldenitions however must ultimately rely on some primitive notions thatare left undened� Therefore in simple TBoxes we distinguish between so�called primitive and dened concepts where the former are those conceptswhose meaning is assumed somehow understood �and are therefore leftundened in the TBox� whereas the latter are those concepts that have acharacterization in the TBox in terms of the set of primitive concepts androles� The form of the TBox determines the sets of primitive and denedconcepts� dened �resp� primitive� concepts are those that do �not� have adenition in the TBox�

Intuitively for any simple TBox T we would like to be able to assign aunique interpretation to the dened concepts in T once we have xed theinterpretation of the primitive ones� In order to capture such an intuitionwe introduce the notion of initial interpretation� Let us call initial an in�terpretation I � ��I� �I� whose interpretation function maps every atomicconcept that is primitive in T to a subset of �I and every role to a subsetof �I �I� Given an initial interpretation I we obtain an interpreta�tion for T by extending the interpretation function �I to all concepts androles in T �including complex and dened concepts as well complex roles��The denotation of complex concepts and roles must obey the conditionsreported in Tables � and � respectively� Moreover it must assign the samedenotation to A and C for every denition A

�� C in T � Now the question

is�

Given an initial interpretation� is there a unique way to determine the interpre�tation of de�ned concepts�

We will see that the answer to this question depends on the form of theTBox and in particular on whether the TBox contains cycles or not�

In order to precisely dene the notion of cycle observe that to a simpleTBox T we can associate a total function fT from atomic concepts toconcept expressions dened as follows� fT �A� � C if A

�� C � T and

fT �A� � A if A is primitive in T � Now following Nebel �� we say

��


where each statement is used to specify a necessary condition on the in�stances of the concept whose name is A namely that they are instances ofthe concept C as well�

The semantics of a primitive concept specication is given in Section�� Intuitively each specication is used to restrict the set of admissibleinterpretations to those that obey the inclusion condition expressed in thespecication�

Primitive TBoxes provide a characterization of Frames Semantic Net�works and several Database Models� Indeed the structure of the inten�sional component of a knowledge base expressed in each of these formalismscan be correctly represented in terms of a set of primitive concept specica�tions� Two recent papers �Buchheit et al� ��a Calvanese et al� �� ad�dress the problems of developing suitable reasoning techniques and studyingthe computational complexity of reasoning in primitive TBoxes�

In �Buchheit et al� ��a� an architecture for DL�systems is presentedconstituted by three components� One of these components called schemais just a set of primitive concept specications in a language obtained fromFL� by adding the two constructors �� P �� P � where P is anatomic role� The paper shows that Subsumption between atomic conceptsin a schema can be decided in polynomial time� However if the schemalanguage is enriched with inverse roles� or with the negation of atomicconcepts Subsumption becomes NP�hard� In the paper it is also observedthat if the DL�system architecture is carefully designed primitive conceptspecications on one hand are proved very useful in domain modeling andon the other hand do not increase the complexity of reasoning� This claimis supported by three case studies where a primitive TBox in the style de�scribed above is added to three working systems namely ConceptBasekris and classic�

In �Calvanese et al� �� primitive TBoxes using more powerful lan�guages are investigated� The main language considered in that paper calledALUNI is obtained from ALUN by adding the constructor for inverseroles� Inverse roles are shown essential in order to express the modelingconstructors used in Semantic and Conceptual Database Models� It is alsoshown that primitive TBoxes expressed in this language exhibit an inter�esting property� a concept may be satisable in a TBox T only in innitemodels i�e� models with innite domains� In other words �unrestricted�Concept Satisability is di�erent from Finite Concept Satis�ability i�e� theproblem of checking if a concept has a nonempty extension in at least onenite model of the TBox� Observe that nite Satisability is important in

�Given a role R� its inverse is usually denoted as R�� and its semantics is �R��I �f�d�� d�� j �d�� d�� RIg�

��


a set must be computed �possibly in a recursive way due to nestedoccurrences of K�

�� when the K operator is in front of a role name KP this is equiva�lent to a role of the form f�a�� b�� an� bn�g� Such a role is notexpressible even in languages with O�

Despite the above di�erences using a technique similar to the one usedfor reasoning in ALCO in �Donini et al� �� a� we prove that InstanceChecking in ALC using ALCK as query language is a PSPACE�completeproblem�

The above result proves that answering ALCK�queries is not �worst�case� computationally harder than answering ALC�queries� This allowsus to conclude that the use of epistemic operators in ALC does not sub�stantially increase the complexity in query answering� This is extremelyinteresting especially in light of the expressive power gained� Next weshow also a case in which the epistemic operator decreases the complexity�

In the examples given above we show that the use of K may allowus to express queries ruling out the case analysis� In particular this isdone by replacing concepts of the form �R�C with concepts of the form�KR�KC� In �Donini et al� �� a� it is shown that such a replacementcan lessen the complexity of reasoning� More precisely considering anABox in AL it is possible to answer ALEK�queries in polynomial timew�r�t� j�j �data complexity� provided that the only qualied existentialquantications are of the form �KR�KE� Conversely recall that answeringgeneral ALE queries in coNP�hard as shown in Section ��

� Reasoning with the TBox

In this section we deal with the problem of reasoning with a set of TBox�statements� As we said in Section � TBox�statements are used to specifythe intensional component of a knowledge base i�e� properties of classesand relationships between classes� The basic reasoning services to be pro�vided on TBoxes are Concept Satisability and Subsumption� Here weaddress these two problems in the hypothesis that the knowledge base �has the form hT � �i i�e� in the absence of ABox�assertions� In particularwe consider three types of TBox�statements and correspondingly we dis�tinguish between three kinds of TBoxes� primitive TBoxes simple TBoxesand free TBoxes�

�� Primitive TBoxes

Recalling denitions of Section �� a primitive TBox is a set of statements�called primitive concept specications� of the form�

A �� C

��


teaches only intermediary courses as well as models in which he teachesalso courses that are not intermediary�

Query � instead asks whether every course that is known to be taughtby John is also known to be an intermediary course� Since the only coursestaught by John are CS�� and CS�� and they are indeed intermediarycourses the answer to Query � is YES� Note that this is a form closed�worldreasoning since the universal quantication takes into account only knownindividuals�

Example � We now consider two queries involving nested quantiers�Query � has been already discussed in Section �� we report it here forconvenience� and Query � is its modied version with the epistemic oper�ator�

Query � j� �TEACHES��ENROLLED�Gradu �ENROLLED��Grad��john��Answer YES�

Query � j� �KTEACHES�K��ENROLLED�Gradu �ENROLLED��Grad��john��Answer� NO�

As we already said the answer of Query � is YES due to a case anal�ysis� On the other hand Query � asks whether John is known to teach acourse that is known to be an instance of the concept �ENROLLED�Grad u�ENROLLED��Grad� The courses known to be taught by John are CS��and CS�� and therefore none of them is within the conditions requiredby the query�

Query � shows how in some cases the rst�order semantics might notagree with the intuitive reading of a query� In fact most people tend to readQuery � as requiring the reasoning pattern that is actually associated withthe semantics of Query �� For this reason in our opinion it is importantto have the operator K which allows us to express both types of queries�

Several other queries and other possible uses of the epistemic operatorare discussed in �Donini et al� ��b Donini et al� ��a��

�� Complexity Results

In this section we address the computational complexity of answering epis�temic queries� To this aim we realize an interesting relation between rea�soning with epistemic concepts and reasoning with concepts involving col�lections of individuals�

The intuition underlying such relation is that a concept of the formKCcan be considered equivalent to the concept fa�� ang where a�� anare exactly the individuals for which � j� C�ai� holds� However reasoningwith K is more complex than reasoning with O for two reasons�

�� when using O the set fa�� ang is given while when using K such

��


Figures � and � by replacing I with I�W and �I with � �for example�C uD�I�W � CI�W �DI�W ��

�� for the epistemic constructor the following equations are satised

�KC�I�W ��J�W

�CJ �W�

�KP �I�W ��J�W

�PJ �W��

Some issues typical of rst�order modal systems arise� Such issues con�cern the interpretation structures and are dealt with by the following as�sumptions�

every interpretation is dened over a xed domain called � �Com�mon Domain Assumption� that is �I � � for any I�

for every interpretation the mapping from the individuals into thedomain elements called � is xed �Rigid Term Assumption� that isaI�W � ��a��

Notice that since the domain is xed independently of the interpre�tation it is meaningful to refer to the conjunction of the extensions of aconcept in di�erent interpretations� It follows that KC is interpreted inW as the set of objects that are instances of C in every interpretationbelonging to W� In this sense KC represents those objects known to beinstances of C in W�

An ABox � in a concept language L logically implies an assertion � inLK written � j� � if � is true in every pair �I�M�� where M�� theset of models of � and I is an element of M��

Given an L�ABox � an LK�concept C and an individual a the answerto the query C�a� posed to � is YES if � j� C�a� NO if � j� �C�a� andUNKNOWN otherwise�

Our goal here is to show that the use of epistemic operators in queriesallows for a more sophisticated interaction with the DL�system� For thispurpose we consider the ABox � of Example �� and we provide variouskinds of queries that can be posed to it using the language ALEK� Inparticular in order to understand the role of the epistemic operator Kwe consider both ALE queries and modied versions of them including K�The comparison between their respective semantics highlights the role ofK in the query language�

Example �� We consider two queries that involve universal quantiers�

Query � j� �TEACHES�IntCourse�john�� Answer� UNKNOWN�

Query � j� �KTEACHES�KIntCourse�john�� Answer� YES�

Query � asks whether every course taught by John is an intermediaryone� The answer is UNKNOWN because there are models for � in which John

��


� Queries with an Epistemic Operator

One of the strengths of concept languages is that they are given a set�theoretic rst�order semantics� However rst�order semantics for otheraspects is also their weakness since it leaves out several aspects of prac�tical systems that are generally described in terms of procedural or non�monotonic mechanisms� Therefore it seems appropriate to enrich such asemantics both to explore novel language features and to account for as�pects that cannot be described in a standard rst�order framework� Theneed of some enrichment of this sort is discussed in the literature �see forexample Doyle and Patil �� Woods �� and can be easily recognizedby looking at recent DL�systems such as classic and clasp �Yen et al��

Although here we are not concerned with a full treatment of these issueswe present a non�rst�order extension of concept languages that providesspecial forms of reasoning in the ABox� In �Donini et al� ��b Donini etal� ��a Donini et al� �� a� it is proposed to enrich concept languageswith an epistemic operator dened in the style of �Levesque �� Reiter�� Lifschitz �� In particular we focus our attention on the use ofthe epistemic operator to dene a more powerful query language�

We rst provide some examples that show how the epistemic querylanguage allows one to address both aspects of the external world as rep�resented in the knowledge base and aspects of what the knowledge baseknows about the external world� We then discuss the complexity of rea�soning�

�� Semantics

In this section we present epistemic concept languages which are conceptlanguages extended with an epistemic operator� In details an epistemicconcept language includes the concept constructor KC and the role con�structor KP where C and P are a concept and a role respectively� As anotation we call LK the concept language obtained by adding the aboveconstructors to a concept language L� Intuitively KC denotes the set ofindividuals which are known to be instances of the concept C i�e� thoseindividuals that are in the extension of C in every model for the knowledgebase �and similarly for KP ��

The semantics of the epistemic operator is an adaptation to the frame�work of concept languages of the one proposed in �Lifschitz ��

An epistemic interpretation is a pair �I�W� where I is an interpretationand W is a set of interpretations such that

�� concept and role names are interpreted independently of W� that isAI�W � AI and P I�W � P I�

�� the interpretation of the non�epistemic constructors is obtained from

�


�� Complexity of Instance Checking and Consistency

An extensive analysis of the complexity of ABox�reasoning in several lan�guages is carried out by Donini et al� ��b�� The analysis singles out sev�eral interesting properties� Firstly Consistency is shown to be in the samecomplexity class as Concept Satisability for all the considered languages�Secondly di�erent results are obtained for Instance Checking dependingupon the language used to express the ABox� In particular�

�� Languages with polynomial Subsumption such as AL and ALN preserve their tractability although in general some additional workwith respect to Subsumption is required�

�� The new source of complexity singled out for ALE in the previoussection does not arise in the other languages in which Subsumptionis NP�complete such as ALR� In fact although ALR can simulatequalied existential quantication in ABox assertions �Section ��it cannot express qualied existential quantication in the query con�cept neither explicitly nor implicitly� It follows that Instance Check�ing in ALR is NP�complete exactly like Subsumption in ALR�

�� With regard to other languages Instance Checking is again in thesame complexity class as Subsumption� This happens either becauseno navigation through the knowledge base is allowed �e�g� ALU� orbecause the language is rich enough �e�g� ALC� so that Subsumptionis already computationally demanding�

An analysis of the complexity of Instance Checking for languages includ�ing the constructor O has been started by Lenzerini and Schaerf �� andhas been carried out extensively by Schaerf ��b��

The analysis shows that in languages with O all reasoning serviceshave the same complexity� This is due to the fact that individuals can nowappear in concept expressions hence the source of complexity related totheir treatment has already an impact when reasoning on concept expres�sions�

The corresponding complexity results also show that reasoning withthe construct O is generally hard� When O is added to languages such asALALE complexity of reasoning increases� Even in those cases in whichreasoning in a language including O is in the same complexity class ofreasoning in the language without O �e�g� ALCO ALC� the algorithmstaking into account O are generally more complex �in terms of both timeand space� than the algorithms that do not deal with O�

Finally we remark that results presented here for O are based ona standard rst�order semantics� Semantics employed in actual systemscan be a variant of pure rst�order semantics �see e�g� Borgida and Patel�Schneider �� However there is no general agreement on whether suchnon�standard semantics are appropriate and understandable by users�

��


respect to the size of F such result proves that Instance Checking in ALEis coNP�hard in the size of the knowledge base�

Schaerf �� also shows that Instance Checking in ALE is in �p� with

respect to data complexity� We conjecture that the problem is actuallycomplete for �p

� �i�e� it is �p��hard too�� Instead if combined complexity is

considered Donini et al� ��b� show that Instance Checking in ALE isPSPACE�complete�

These results single out several interesting properties�

�� Reasoning in ALE su�ers from an additional source of complexitywhich makes Instance Checking not polynomially reducible to Sub�sumption and �unlike Subsumption in ALE� not solvable by meansof one single level of nondeterministic choice�

�� The problem of Instance Checking in ALE is in �p� if data complexity

is considered and is PSPACE�complete if combined complexity isconsidered� This shows that in order to have a signicant complexitymeasure of reasoning problems in DL�systems we must pay attentionto which is the crucial data of the application�

�� With respect to combined complexity ALE is in the same class�PSPACE� as more complex languages �e�g� ALCNR see Buch�heit et al� �� Therefore whenever the expressivity of ALE isrequired other constructors can be added without any increase ofthe �worst�case� computational complexity�

�� The source of complexity identied in this section is not related to thepossibility of nesting an arbitrary number of existential and universalquantiers�like the one pointed out in Section �� for Subsumptionin ALE � In fact it leads to intractability even using only two nestedexistential quantiers and no universal ones� This is an importantobservation since long chains of nested quantiers seem not to appearfrequently in practice�

With regard to Point � one may object that there are cases in whichthe reasoning process underlying this source of complexity is not in theintuition of the user as pointed out by Query �� Therefore such processshould be ruled out by the reasoner �and by the semantics�� On the con�trary there are cases in which this kind of reasoning seems to agree withthe intuition and therefore it must be taken into account� In Section ��we address the issue of the appropriate semantics for queries involving ex�istential quantications� In particular we show how is possible to achievea more sophisticated control on the semantics by means of an epistemicoperator in the query language�

��


now the set of models for � in which Grad�susan� holds� In each of thesemodels the course CS�� is taken by both a graduate �Susan� and an un�dergraduate �Peter�� Similarly consider the set of the remaining modelsfor � i�e� the ones in which �Grad�susan� holds� It is easy to see that inevery model for this set the course CS�� this time is taken by both agraduate �Mary� and an undergraduate �Susan��

In conclusion in every model for � either CS�� or CS�� is in theextension of �ENROLLED�Gradu �ENROLLED��Grad� It follows that in everymodel for � the assertion � is true proving that the correct answer is YES�

We now show that the kind of reasoning required in the example makesinstance checking in ALE coNP�hard with respect to data complexity�

The proof is based on a reduction from a suitable variation of the propo�sitional satisability problem �SAT� to Instance Checking in ALE � We de�ne ��CNF formula on an alphabet P as a CNF formula F such thateach clause of F has exactly four literals� two positive and two negativeones where the propositional letters are elements of P �ftrue� falseg� Fur�thermore we call ��SAT the problem of checking whether a ��CNFformula is satisable� The problem ��SAT is proved NP�complete bySchaerf ��

Given a ��CNF formula F � C� �C� � � � � �Cn where Ci � Li��

Li�� L

i�� L

i�� we associate with it an ABox �F and a concept Q

as follows� �F has one individual l for each letter L in F one individualci for each clause Ci one individual f for the whole formula F plus twoindividuals true and false for the corresponding propositional constants�The roles of �F are Cl� P�� P�� N�� N� and the only concept name is A�

�F � f A�true��A�false��

Cl�f� c�� Cl�f� c�� Cl�f� cn��

P��c�� l�� P��c�� l

�� N��c�� l

�� N��c�� l

��

� � �

P��cn� ln�� P��cn� l

n�� N��cn� l

n�� N��cn� l

n�� g

Q � �Cl��P��A u �P��A u �N��A u �N��A��

Intuitively the membership to the extension of A or �A corresponds tothe truth values true and false respectively and checking if �F j� Q�f�corresponds to checking if in every truth assignment for F there exists aclause whose positive literals are interpreted as false and whose negativeliterals are interpreted as true i�e� a clause that is not satised�

Schaerf �� proved that a ��CNF formula F is unsatisable ifand only if �F j� Q�f�� Since for any ��CNF formula F Q is xedindependently of F and �F can be computed in polynomial time with

��


through the assertions on roles in the ABox� For example the query� �ndthe individuals who teach a course in which a graduate student is enrolled can be expressed by the ALE�concept �TEACHES��ENROLLED�Grad�

We rst consider data complexity of Instance Checking in ALE� Tothis regard it is easy to show that such complexity is at least as high asthe complement of Satisability and therefore is NP�hard� We now provethat it is coNP�hard too� Since Subsumption in ALE is NP�complete aconsequence of this result is that �assuming NP �� coNP� Instance Checkingfor ALE is strictly harder than Subsumption�

This unexpected result shows that Instance Checking in ALE su�ersfrom a new source of complexity which does not show up when reasoningwith concept expressions �see Section �� This source of complexity isrelated to the use of qualied existential quantication in the concept rep�resenting the query which makes the behavior of the individuals heavilydependent on other individuals in the ABox� Let us illustrate this pointby means of an example�

Example �� Let � be the following ABox�

� � f Professor�john�� TEACHES�john� cs�� TEACHES�john� cs��IntCourse�cs�� IntCourse�cs��ENROLLED�cs�� mary�� ENROLLED�cs�� susan��ENROLLED�cs�� susan�� ENROLLED�cs�� peter��Grad�peter�� Grad�mary�g

The ABox � is also shown in graphical form in Figure �� It can be easilyveried that � is satisable and that it has indeed several di�erent models�In fact it does not have complete knowledge about the represented world�For example � doesn!t know whether Susan is a graduate or not�

Consider the following query to the ABox ��

Query � j� �TEACHES��ENROLLED�Gradu �ENROLLED��Grad��john��

Query � asks whether John teaches a course in which both a grad�uate and an undergraduate are enrolled� At a supercial reading ofthe query it might seem that the answer should be NO� The answer NO

is supported by the fact that none of the courses taught by John isknown to be in the requested conditions i�e� � logically implies neither�ENROLLED�Grad u �ENROLLED��Grad�cs�� nor�ENROLLED�Grad u �ENROLLED��Grad�cs�� Nevertheless the correct an�swer is YES and in order to get it one must reason by case analysis� Aswe have already remarked the knowledge base does not provide the infor�mation as to whether Susan is a graduate or an undergraduate� howeverin every model she must be either one or the other� This fact ensures thatin every model for � either Grad�susan� or �Grad�susan� holds� Consider

��


JJJJ�

��

��

��R

��

JJJJ�

��

IntCourse

�Gradmary susan peter

TEACHES

ENROLLED

Grad

cs�� cs�� IntCourse

Professor

john

FIGURE � A pictorial representation of the ABox

data complexity of Instance Checking the complexity of checking if� j� D�a� with respect to j�j�

combined complexity of Instance Checking the complexity of checkingif � j� D�a� with respect to j�j� jDj

Combined complexity is the one usually considered in the literature inconcept languages� It is a good measure of the actual cost of InstanceChecking in the cases in which the ABox and the query are comparable insize�

On the other hand data complexity is important when the size of thequery is negligible with respect to the size of the ABox� Although such ameasure is widely accepted in the database community in the frameworkof concept languages the assumption that the size of the query is negligibleis not always reasonable�

It is worth noticing that combined complexity is always higher or equalto data complexity� In fact the complexity with respect to j�j and jDjobviously includes as a particular case that in which jDj is small withrespect to j�j�

�� Instance Checking in ALE

We start our analysis on ABox�reasoning by studying Instance Checking inthe particular language ALE which shows interesting properties� ALE isrich enough to express both signicant classes of assertions in the ABox andmeaningful queries� In particular by virtue of qualied existential quan�tication it is possible to express queries requiring some form of navigation

��


OR�source OR�sourceabsent present

AND�source AL�ALN ALU �ALUNabsent PTIME coNP�complete

AND�source ALE�ALR�ALER all other AL�languages but ALENpresent NP�complete PSPACE�complete

TABLE � Complexity of Subsumption in AL�languages

is Instance Checking� We consider the Instance Checking problem in thecase of an empty TBox that amounts to checking whether the ABox impliesthat an individual is an instance of a given concept�

In particular we address the question of whether Instance Checkingcan be solved by means of Subsumption algorithms� The basic techniqueunderlying most of the approaches to check whether � j� C�a� consists inretrieving all the assertions in the knowledge base � relevant to a given indi�vidual a collecting them into a single concept called most speci�c concept

of a �MSC�a�� and then checking whether C subsumes MSC�a�� Thistechnique called Abstraction"Subsumption has been broadly exploitedin actual systems �see Kindermann �� Quantz and Kindermann ��Nebel ��a��

The Abstraction"Subsumption idea is applicable to a number of lan�guages� However there are cases where in order to check whether � j�D�a� it is necessary to consider assertions about other objects in the knowl�edge base di�erent from a and the above method is no longer applicable�We discuss these cases in Sections �� and ��

Subsequently we show in Section �� how to enrich concept languageswith an epistemic operator so as to distinguish the knowledge about theworld and knowledge about the state of the knowledge base� In particu�lar we demonstrate that the epistemic operator introduces a sophisticatedquery mechanism that can be used to decrease the complexity of InstanceChecking�

�� Complexity Measures

The complexity of a problem is generally measured with respect to the sizeof its whole input� For instance in Section � the complexity of Subsumptionhas been measured w�r�t� the sum of the size of the candidate subsumerand that of the candidate subsumee� However in Instance Checking twoinputs of di�erent kind are given namely the ABox and the query� For thisreason di�erent kinds of complexity measures can be considered similarlyto what has been suggested by Vardi �� in database query answering�

De�nition �� Given an ABox � a concept D an individual a we call�

��


quantier of the QBF in the following way�

Di ��

��P �A� u ��P ��A� if Qi � �

�P �� if Qi � �

The concept Ci� is obtained from the clause Gi using the concept A when a

boolean variable occurs positively in Gi �A when it occurs negatively andusing l nestings of universal role quantication to encode the variable Xl�In detail let k be the maximum index of all boolean variables appearingin Gi� Then for l � ��k� �� one denes

Cil ��

��

�R��A tCil�� if Xl appears positively in Gi

�R��A tCil�� if Xl appears negatively in Gi

�R�Cil�� if Xl does not appear in Gi

and the last concept of the sequence is dened as

Cik ��

��R�A if Xk appears positively in Gi

�R��A if �Xk appears negatively in Gi�

It can be shown that each trace in a constraint system corresponds to atruth assignment to the boolean variables and that all traces of a constraintsystem form a set of truth assignments consistent with the prex� There�fore we can conclude that Satisability in ALC is PSPACE�hard� Combin�ing this result with the polynomial�space calculus given for ALCNR oneobtains that Satisability �and Subsumption� in ALCNR are PSPACE�complete and that the exponential�time behavior of the calculus cannotbe improved unless PSPACE�PTIME�

Using the simulation of qualied existential quantication via subrolesone can use the same reduction to prove that Satisability and Subsumptionin ALUR are PSPACE�hard �and thus PSPACE�complete�� With a similarreduction Donini et al� ��a� proved that also Satisability in ALNRis PSPACE�hard� Observe that all these languages contain both sources ofcomplexity�

This intuition carries over also when one or both sources of complexityare absent� In this way the complexity analysis of reasoning in the AL�languages was completed by Donini et al� ��a�� The only languagenot completely classied is ALEN where both sources of complexity arepresent �it can be proved both NP� and coNP�hard� but their combinationin a PSPACE�hardness proof has not been found yet�

The results are summarized in Table �� The complexity classes men�tioned refer to Subsumption� For the complexity of Satisability NP andcoNP should be interchanged�

� Reasoning with the ABox

In this section we analyze in detail the problem of reasoning within anABox� The basic reasoning task for retrieving information from an ABox

��


causes an exponential number of possible refutations to be searched through�each refutation being a clash in a trace�� This source of complexity is notevident in propositional calculus but a similarproblem appears in predicatecalculus where the interplay of existential and universal quantiers maylead to innite models� A proof of coNP�hardness of Satisability �so NP�hardness of Subsumption� has been found using this source of complexitydescribed by Donini et al� ��a��

It is interesting to observe that analogously to Examples �� alsothis source of complexity can appear without the explicit presence ofqualied existential and universal quantication� In fact the pattern�P �C� u �P �C� u �P �D can also be simulated by role intersection as

��P uQ�� u �Q��C� u ��P uQ�� u �Q��C� u �P �D

This simulation was used to prove coNP�hardness of Satisability of ALRby Donini et al� ��a��

In a language containing both sources of complexity one might expect tocode any problem involving the exploration of polynomial�depth AND�ORgraphs� The computational analog of such graphs is the class APTIME�problems solved in polynomial time by an alternating Turing machine�which is equivalent to PSPACE �e�g� see Johnson �� pg�� A well�known PSPACE�complete problem is Quantied Boolean Formulae �QBF��Given a sentence of the form

�Q�X��Q�X�� QnXn�#F �X�� Xn�$

where each Qi is a quantier �either � or �� and F �X�� Xn� is a booleanformula with boolean variables X�� Xn decide whether the sentenceis true� The problem remains PSPACE�complete if F is in �CNF i�e�conjunctive normal form with at most three literals per clause� We callprex the string of quantiers and matrix of the QBF the formula F �

This problem can be encoded in an AND�OR graph using AND�nodesto encode ��quantiers and OR�nodes for ��quantiers� We use this anal�ogy to illustrate the reduction which was rst published by Schmidt�Schau% and Smolka �� in a slightly di�erent form�

Without loss of generality we assume that the matrix of the QBF isin �CNF and that in each clause do not appear both a literal and itscomplement� We translate the QBF �Q�X�� QnXn�#G�� Gm$ intoan ALC�concept

C � D uC�� u � � �uC

n� �

in which all concept are formed using the atomic concept A and the atomicrole P �

The concept D encodes the prex and is of the form D� u �P ��D� u�P �� Dn�� u �P �Dn� � � �� where for i � ��n each Di corresponds to a

��


As to the time needed by this calculus both the nondeterministic ver�sion and the deterministic one work in exponential time� to prove that sucha calculus is really optimal we need a reduction from a PSPACE�completeproblem� We consider this in the next subsection�

�� Sources of Complexity

Note that the deterministic version of the calculus for ALCNR can be seenas exploring an AND�OR tree where an AND�branching corresponds to the�independent� check of all successors of a variable while an OR�branchingcorresponds to the choices of application of a nondeterministic rule�

Realizing that one can see that the exponential�time behavior of thecalculus is due to two independent origins� The AND�branching respon�sible for the exponential size of a single constraint system and the OR�branching responsible for the exponential number of di�erent constraintsystems� We call these two di�erent combinatorial explosions sources of

complexity�The OR�branching is due to the presence of disjunctive constructors�

The obvious disjunctive constructor is t hence ALU is a good sublanguageto study this source of complexity� We have seen also that disjunction canbe introduced by combining role restrictions and universal quantication inExample �� and by combining number restrictions and role intersectionin Example �� This source of complexity is the same that makes proposi�tional satisability NP�hard� in fact Satisability in ALU can be triviallyproved NP�hard by rewriting propositional letters as atomic concepts � asu and � as t� Many proofs of coNP�hardness of Subsumption were foundexploiting this source of complexity �Levesque and Brachman �� Nebel�� by reducing an NP�hard problem to non�Subsumption�

The AND�branching is more subtle� Its exponentiality is due to theinterplay of qualied existential and universal quantiers hence ALE isnow a minimal sublanguage of ALCNR with these features� One can seethe e�ects of this source of complexity by expanding the constraint systemfx � Cg when C is the following concept �taken from Schmidt�Schau% andSmolka ��

�P ��C��u�P��C��u�P��P��C��u

�P��C��u�P�� Pn�Cn�u

�Pn�Cn�u�Pn�D� � � ��

A clash may be found independently in each trace i�e� in each branch of thetree of variables given by the successor relation� This source of complexity

��


The basic property of traces in ALC is that a constraint system con�tains a contradiction if and only if it contains a trace that contains a con�tradiction� We will call this property independency of traces� Traces areindependent if the graph associated with a constraint system is a tree �withx as root�� Whenever in a language L traces are independent constraintsystems can be checked for Satisability just by looking at their traces oneat a time� If each trace has polynomial size with respect to the concept Cto be checked we have a PSPACE algorithm for Satisability in L�

The constraint systems built by the above propagation rules are notexactly trees since role intersection forces the introduction of many arcsbetween two variable nodes �see the generating rules �� Howeverif we consider the successor relation between variables it indeed forms atree because no variable is a direct successor of more than one variable�Moreover the depth of such a tree is bounded by the number of nestedquantiers in the concept �counting also the �at�least restrictions as exis�tential quantiers�� Hence generating depth�rst such a tree�shaped con�straint system needs just polynomial space since the depth is bounded bythe length of the concept� Such a depth�rst construction can be achievedby using the following strategy for the application of rules�

�� apply a generating rule only if no nongenerating rule is applicable�

�� when a generating rule is applicable to more than one concept con�straint choose the concept constraint with the most recently gener�ated variable�

In order to actually use polynomial space one should also discard con�straints y � D yPz y �

�� z whenever there is no rule applicable to any

successor and any predecessor of y� Since this would also modify the con�ditions of application of rules �namely �� we do notdelve into the details of such a modication�

Note that till now we devised a calculus working in polynomial spacewhich is nondeterministic� a deterministic version would require to explorein turn all possible applications of the nondeterministic rules looking forchoices that do not lead to a clash� However also such an explorationcan be done depth�rst using backtracking� The number of backtrackingpoints that need to be memorized is no more than the number of constraintsof the form y � D� tD� and y � �� nR� and this number can be provedto be polynomially bounded by the length of the initial concept� HenceSatisability of ALCNR�concepts can be decided in polynomial space�

Noting that Subsumption in ALCNR can be rephrased as Satisabilitywe can conclude the main theorem of this subsection�

Theorem �� Satis�ability and Subsumption in ALCNR can be decided

with polynomial space�

�


for the presence of clashes inside� The three important properties that mustbe proved for such a calculus are soundness completeness and termination�We sketch each one in turn�

Soundness� Intuitively we prove that the calculus does not add un�necessary contradictions� That is deterministic rules always preserve theSatisability of a constraint system and nondeterministic rules have alwaysa choice of application that preserves Satisability� The proof �see Doniniet al� �� resembles soundness proofs for tableaux methods �e�g� Fitting�� Lemma �� The only non�standard constructors are number re�strictions� Without number restrictions the generating rules could operateby introducing new constants all di�erent from one another� However thiswould not preserve Satisability e�g� in the following example�

x � �P �C u �P �D u �� P �

after having applied two times the�u�rule and two times the��rule theresulting constraint system contains the constraints fxPy� xPz� x � �� P �g�we omit the others� and is unsatisable if y� z are di�erent constants�Using variables and I�assignments instead the resulting constraint systemis still satised by an I�assignment which maps y and z into the sameelement�

Termination� A constraint system is complete if no propagation ruleapplies to it� A complete system derived from a system S is also called acompletion of S� Completions are reached when there is no innite chain ofapplications of rules� Intuitively this can be proved by using the followingargument� all rules but �� and �� are never applied twice on the sameconstraint� these two rules in turn are never applied to a variable x moretimes than the number of the direct successors of x which is bounded bythe length of a concept� nally each rule application to a constraint y � Cadds constraints z � D such that D is a strict subexpression of C�

Completeness� If S is a completion of fx � Cg and S contains no clashthen it is always possible to construct an interpretation for C on the basisof S such that CI is nonempty� The proof is a straightforward inductionon the length of the concepts involved in each constraint�

We now analyze the complexity of such a calculus and how we canrene it to get an optimal upper bound�

�� Traces and Completions

The notion of constraint system was introduced by Schmidt�Schau% andSmolka �� who analyze the language ALC� The key observation madeby Schmidt�Schau% and Smolkawas that in ALC a constraint system can bepartitioned into traces� Recalling that constraint systems can be regardedas graphs traces correspond to paths in the graph that starts from thevariable x of the initial constraint system fx � Cg�

��


�� S �� fxP�y� � � � � xPky� y � Cg � S

if �� x � �R�C is in S�� R � P� u � � �uPk�� y is a new variable�� there is no z such that z is an R�successor of

x in S and z � C is in S � S �� fxP�yi� � � � � xPkyi j i � ��ng�fyi �

�� yj j i� j � ��n� i �� jg�S

if �� x � � nR� is in S�� R � P� u � � �uPk�� y�� yn are new variables�� there do not exist n pairwise separated R�

successors of x in S�� S �� S#y�z$

if �� x � �� nR� is in S�� x has more than n R�successors in S�� y� z are two R�successors of x which are not separated

We call the rules �t and �� nondeterministic rules since they canbe applied in di�erent ways to the same constraint system� All the otherrules are called deterministic rules� Moreover we call the rules �� and�� generating rules since they introduce new variables in the constraintsystem� All other rules are called nongenerating ones�

One can verify that rules are always applied to a system S becauseof the presence in S of a given constraint x � C �condition �� When noconfusion arises we say that a rule is applied to the constraint x � C or tothe variable x �instead of saying that it is applied to the constraint systemS��

While building a constraint system we can look for evident contra�dictions to see if the constraint system is not satisable� We call thesecontradictions clashes� A clash is a constraint system having one of thefollowing forms�

fx � �g

fx � A� x � �Ag where A is a concept name�

fx � �� nR�g � fxP�yi� � � � � xPkyi j i � ��n� �g� fyi �

�� yj j i� j � ��n� �� i �� jg

where R � P� u � � �u Pk�

A clash is evidently an unsatisable constraint system hence any constraintsystem containing a clash is obviously unsatisable� The last case of clashdeals with the situation in which a variable has an at�most restriction anda set of R�successors that cannot be identied because they are pairwiseseparated�

The purpose of the calculus is to generate a constraint system and look

��


A pair �I� �� satis�es the concept constraint x � C if ��x� � CI therole constraint xPy if ��x�� y�� P I the inequality constraint x �

�� y if

��x� �� y�� A constraint system S is satis�able if there is a pair �I� ��that satises every constraint in S�

Obviously a concept C is satisable if and only if the constraint sys�tem fx � Cg is satisable� In order to check a constraint system S forSatisability our technique adds constraints to S until either an evidentcontradiction is generated or an interpretation satisfying it can be obtainedfrom the resulting system� Constraints are added on the basis of a suitableset of so�called propagation rules�

Before providing the rules we need some additional denitions whichare evident considering variables in a constraint system as nodes in a graphand constraints xPy as labeled arcs in this graph� Let S be a constraintsystem and R � P� u � � � u Pk �k �� be a role� We say that y is anR�successor of x in S if xP�y� � � � � xPky are in S� We say that y is a direct

successor of x in S if for some role R y is an R�successor of x� We calldirect predecessor the inverse relation of direct successor� If S is clear fromthe context we simply say that y is an R�successor or a direct successor ofy �or direct predecessor�� Moreover we call successor the transitive closureof the relation �direct successor and we call predecessor its inverse�

We denote by S#y�z$ the constraint system obtained from S by replacingeach occurrence of the variable y by the variable z�

We say that x and y are separated in S if the constraint x �� y is in S�

The propagation rules of the Calculus for ALCNR are�

�� S �u fx � C�� x � C�g � S

if �� x � C� uC� is in S�� x � C� and x � C� are not both in S

�� S �t fx � Dg � S

if �� x � C� tC� is in S�� neither x � C� nor x � C� is in S�� D � C� or D � C�

�� S �� fy � Cg � S

if �� x � �R�C is in S�� t is an R�successor of x�� y � C is not in S

��


S �� fxRy� y � Cg � S

if x � �R�C is in S there is no z such that both xRz andz � C are in S and y is a new variable

Following Schmidt�Schau% and Smolka �� we call the generic inter�pretation being built a constraint system as it is a set of constraints thatincrementally specify the properties that the generic interpretation beingbuilt must satisfy�

There is an interesting property of rst�order interpretations of conceptlanguages� Since concepts and roles are interpreted as unary and binarypredicates an interpretation can be viewed as a labeled directed graph�Each node is a variable �a generic element of the interpretation� and eacharc is a relationship among variables that must hold in the interpretation�Labels on arcs are primitive roles and labels on nodes are sets of primitiveconcepts and negations of primitive concepts� This property is not su��cient to allow any result on graph problems to be transferred to conceptlanguages� a graph problem has always a given graph as input whereashere we must nd a graph�if any�that satises the concept� Neverthe�less the property is useful to intuitively explain some structural propertiesof constraint systems within a given concept language as exemplied insubsection ��

�� Case Study A Calculus for ALCNR

As an example of how to devise a tableaux�based deduction method forreasoning in concept expressions we provide a calculus for the expressivelanguage ALCNR�

We introduce an alphabet of variable symbols V whose elements aredenoted by the letters x� y� z� A constraint is a syntactic entity of one ofthe forms�

x � C Concept constraint

xPy Role constraint

x �� y Inequalityconstraint

where C is a concept and P is a role name� Concepts are assumed tobe simple i�e� the only complements they contain are of the form �Awhere A is a concept name� Simple concepts are the analogous of rst�order formulae in negation normal form� Arbitrary ALCNR�concepts canbe rewritten into equivalent simple concepts in linear time �Donini et al��a�� A constraint system is a nite nonempty set of constraints�

We extend the interpretation of concepts to constraint systems as fol�lows� Given an interpretation I we dene an I�assignment � as a functionthat maps every variable of V to an element of �I�

��


logic and rene it operations to get a reasoning technique with optimalworst�case behavior�

�� Constraint Systems Semantics�Based Reasoning

The reasoning technique we present is based on a notational variant of therst�order tableaux calculus� The bulk of the work is the adaptation ofthe calculus to complexity analyses� In fact to obtain optimal algorithmsthe calculus is not blindly applied�as it is in a general�purpose rst�ordertheorem prover� Instead the structures of tableaux obtained when reason�ing within a given concept language are carefully analyzed and redundantverications in the tableaux are eliminated in order to give a strict upperbound on the complexity of the method�

We give an outline how one can devise a reasoning technique based onthe tableaux calculus �for a general introduction to tableaux in rst�orderlogic see e�g� Bell and Machover �� Fitting �� The key idea is� Tocheck whether a given formula F is a logical consequence of a given theoryT try to build with suitable propagation rules the most generic model of Twhere F is false� If you succeed the answer is NO �since F is not a logicalconsequence of T hence the method is complete�� if you fail the answer isYES �since looking for the most generic model ensures that in no possiblemodel of T is F false hence F is indeed a logical consequence of T hencethe method is sound�� The propagation rules come straightforwardly fromthe semantics of constructors� We give an example of how the semanticsof a given constructor can be used to devise a propagation rule�

Consider the constructor �R�C� If in a given interpretation I whosedomain contains the element a we have that a � ��R�C�I then from thesemantics we know that there must be an element b �not necessarily distinctfrom a� such that �a� b� � RI and b � CI� Since this must be true for anyinterpretation we can abstract from interpretations and their elementsand say that if in a generic interpretation we have a generic element x thatis in the interpretation of the concept �R�C �denote this by x � �R�C� thenthere must be a generic element y such that x and y are in relation throughR �denote it xRy� and y belongs to the interpretation of C �denoted asy � C��

Suppose now we want to construct a generic interpretation S such thatthe set corresponding to the concept �R�C contains at least one element�We can state this initial requirement as the constraint x � �R�C� Follow�ing the semantics we know that S must have a generic element y suchthat the constraints xRy and y � C must hold hence we can add thesenew constraints to S knowing that if S will ever satisfy them then it willalso satisfy the rst constraint� These considerations lead to the followingpropagation rule�

��


Example �� Consider the language FL obtained from FL� by addingrole restriction� as the role�forming constructor�

One can verify that the following Subsumption holds�

��P j�Q��A u ��P j�Q�B��A v �P �A

even if the concept �P �A subsumes neither of the two conjuncts ��P j�Q��Aand ��P j�Q�B��A if they are considered separately� This is becausetheir conjunction is equivalent to ��P j ��Q t �Q�B��A� But the concept�Q t �Q�B is equivalent to � hence the left�hand�side concept is in factequivalent to �P �A� Hence structural comparison would fail to recognizethis subsumption relation�

Subsumption in FL is proved coNP�hard by Brachman and Levesque�� and the result has been recently sharpened by Donini et al� �� where Subsumption in FL is proved PSPACE�complete�

Example �� Consider the language of the back system �called ALNRfollowing our convention� obtained from FL� by adding role intersectionnumber restrictions and negation of atomic concepts�

One can verify that the following Subsumption holds�

� � �CHILDu SON�� u � � �CHILDu DAUGHTER��u��SON�Male� u ��DAUGHTER��Male�

v � � CHILD�

although the concept � � CHILD� subsumes neither of the four left�hand�side conjuncts if they are considered separately� Therefore also i thiscase structural comparison would not reveal the subsumption relation�The intuitive reason is that the two value restrictions �SON�Male and�DAUGHTER��Male on the subroles CHILD u SON and CHILD u DAUGHTER ofthe role CHILD force a case analysis�

Subsumption between back�concepts is proved coNP�hard by Nebel�� and the result has been sharpened by Donini et al� ��a� whereSubsumption is proved PSPACE�complete�

These examples show that structural Subsumption is incomplete w�r�t�the rst�order semantics if particular combinations of constructors are al�lowed in the language� Of course structural algorithms can be enhancedby taking into account also particular combinations of constructors �e�g�see the algorithms proposed by Patel�Schneider �� Nebel ��b�� How�ever the di�cult part for structural algorithms is to prove completeness�i�e� prove that enough comparisons have been considered to capture allpossible interactions��

We now describe a di�erent approach for developing Subsumption al�gorithms� We start from the well�known tableaux calculus for rst�order

�Given a role R and a concept C� the role restriction of R to C is usually denoted asR jC and its semantics is �R jC�I � f�d�� d�� j �d�� d�� RI � d� � CIg�

��


�� Structural Comparison Syntax�Based Reasoning

The rst studies of the computational properties of terminological lan�guages are due to Brachman and Levesque �� Patel�Schneider et al�� and Nebel �� b�� The algorithms they give for computingSubsumption are based on structural comparisons between concept expres�sions�

At the heart of structural comparison is the idea that if the two conceptexpressions to be compared are made of subexpressions one can compareseparately one subexpression of a concept with all those of the others� Wesketch the structural algorithm of Brachman and Levesque �� forSubsumption checking in the language FL��

The algorithm works in two phases� rst concepts are rewritten in anormal form then their structures are compared� More specically

�� All nested conjunctions are �attened i�e� A u �B u C� � A u B uC� All conjunctions of universal quantications are factorized i�e��P �C u �P �D � �P ��C uD�� The rewritten concepts are logicallyequivalent to the previous ones hence Subsumption is preserved bythis transformation�

�� Let C � C� u � � � uCm and D � D� u � � �uDn� Then D subsumes Cif and only if for all Di�

a� if Di is either an atomic concept or is a concept of the form �P then there exists a Cj such that Di � Cj�

b� if Di is a concept of the form �P �D� then there exists a Cj ofthe form �P �C� �same atomic role P � such that D� subsumesC��

By induction on the nesting of ��quantiers one can prove that thecomplexity of the above algorithm is O�jCj jDj� �Levesque and Brach�man �� If subexpressions of each concept are ordered �e�g� lexicograph�ically� then it can be shown that the complexity is only linear so the dom�inant factor becomes the complexity of ordering concepts �Salomone ��The same algorithm can be extended up to the language ALN with thesame complexity�

Similar normalize�and�compare algorithms are actually used in the sys�tems classic �Borgida and Patel�Schneider �� back �Quantz and Kin�dermann �� and loom �MacGregor �� However all these algo�rithms are incomplete if a standard semantics of concepts is assumed�This is due to interactions between constructors which are not taken intoaccount by structural comparisons� An obvious example is the conceptA t �A which is equivalent to � and therefore subsumes every concept�However there are more subtle cases of interaction� We highlight thisthrough two examples�

��


Consistency written as � �j� the problem of checking whether � issatisable i�e� whether it has a model�

Instance Checking written as � j� C�a� the problem of checkingwhether the assertion C�a� is satised in every model of ��

The importance of Concept Satisability and Subsumption has beenstressed by several authors �see for example Nebel ��a�� In particu�lar Subsumption is central to the more complex service of Classi�cation�Given a concept C and a TBox T for all concepts D of T determinewhether D subsumes C or D is subsumed by C �intuitively this amountsto nding the �right place for C in the taxonomy implicitly present inT �� Consistency is used for verifying whether the information contained ina knowledge base is coherent� Finally Instance Checking is used to checkwhether the knowledge base entails that an individual is an instance of aconcept and it can be considered the central reasoning task for retrievinginformation on individuals in the knowledge�base �Donini et al� ��b�� Aswe already mentioned Instance Checking is a also basic tool for more com�plex reasoning problems� For example Retrieval �or Query Answering��can be formulated in the following way� �given a knowledge base � and aconcept D nd the set fa j � j� D�a�g and it can be performed simplyby iterating Instance Checking for all the individuals in ��

Notice that for languages with the constructor C for expressing thenegation of concepts we have that � j� C v D if and only if � j� Cu�D �� and � j� C�a� if and only if � � f�C�a�gi j�� Therefore Subsumptioncan be reduced to Concept Satisability and Instance Checking to Consis�tency�

� Reasoning with Concept Expressions

The goal of this section is to overview the reasoning techniques and thecomplexity results for reasoning with concept expressions� In particularwe consider Subsumption and Concept Satisability in the case in whichthe knowledge base is empty� In order to simplify the notation in this casewe write those problems as C �� and C v D instead of h�� i �j� C � �and h�� i j� C v D�

We rst describe the approach based on structural comparison and sub�sequently we discuss the approach based on tableaux� Finally we discussthe complexity results for various languages and we highlight the corre�sponding sources of complexity�

We concentrate here on the so�called AL�languages i�e� languages inthe family AL#U $#E $#C$#N $#R$� Complexity results for Satisability andSubsumption including other constructors has been found among otherby Donini et al� ��b� Patel�Schneider �� Schild �� Schmidt�Schau% ��

��

Reasoning in Description Logics � �

satisfactory in all cases �Baader �� Nebel �� De Giacomo and Lenz�erini ��b Schild �� We refer to Section for a discussion on cyclicstatements and di�erent semantics for the TBox�

An interpretation I satis�es the statement A �� C if AI � CI thestatement A

�� C if AI � CI the statement C �� D if CI � DI and the

statement C�� D if CI � DI � An interpretation I is a model for a TBox

T if I satises all the statements in T �

�� Extensional Knowledge

The construction of the extensional component of a knowledge base theABox is realized by permitting concepts and roles to be used in assertionson individuals� Given a concept language L an ABox�statement in L hasone of the forms�

C�a� Concept Membership Assertion

R�a� b� Role Membership Assertion

where C is an L�concept R is an L�role and a� b are individuals�The semantics of the above assertions is as follows� If I � ��I� �I� is

an interpretation C�a� is satised by I if aI � CI and R�a� b� is satisedby I if �aI � bI� � RI�

A set A of assertions is called an ABox� An interpretation I is said tobe a model of A if every assertion of A is satised by I� A is said to besatis�able if it admits a model�

�� Knowledge Bases

Given a concept language L a knowledge base � in L is a pair � � hT �Aiwhere T is a TBox in L and A is an ABox in L� An interpretation I is amodel for � if it is both a model for T and a model for A� A knowledgebase � logically implies � where � is either a TBox� or an ABox�statementwritten � j� � if � is true in every model of ��

�� Reasoning Services

There are several reasoning services to be provided by a DL�system� Aswe already said in Section � we concentrate on the following basic oneswhich we can now formally dene�

De�nition �� Given a knowledge base � � hT �Ai two concepts C andD and an individual a we call�

Concept Satis�ability written as � �j� C � � the problem of checkingwhether C is satisable w�r�t� � i�e� whether there exists a model Iof � such that CI ��

Subsumption written as � j� C v D the problem of checkingwhether C is subsumed by D w�r�t� � i�e� whether CI � DI inevery model I of ��

�

� F� M� Donini� M� Lenzerini� D� Nardi� A� Schaerf

implies that di�erent strings may identify the same language� For exampleALEU is the same language as ALC �and ALEUC��

�� DL�systems

A knowledge base built using Description Logics is formed by two compo�nents� The intensional one called TBox and the extensional one calledABox�

�� Intensional Knowledge

We rst focus our attention on the intensional component of a knowledgebase i�e� the TBox� Di�erent proposals for a DL�system provide di�erentmechanisms for expressing the TBox� We list here the types of TBox�statements considered in the literature di�erent kinds of TBoxes di�erfrom each other by the type of TBox�statements they allow�

Given a language L a TBox�statement in L has one of the forms�

A �� C Primitive Concept Specication��

A�� C Concept Denition��

C �� D Concept Inclusion��

C�� D Concept Equation��

where A is a concept name and C�D are L�concepts�We call primitive�TBox a TBox composed only by statements of type ��

Primitive�TBoxes o�er nice computational properties �see e�g� Buchheit etal� ��b Buchheit et al� ��a�� For this reason a mechanism of thiskind is employed in other elds such as Object�Oriented Databases andSemantic Data Models �see Calvanese et al� ��

We call simple�TBox a set of statements of type � and �� Simple�TBoxesare the most commonly considered mechanism and it is the one employedby almost all the implemented systems �e�g� classic back��

In a simple�TBox a concept name appearing on the left�hand side of aconcept denition is called a de�ned concept a concept name appearing onthe left�hand side of a primitive concept specication is called a primitive

concept and a concept name not appearing on the left�hand side of anyTBox�statement is called an atomic concept�

An more general kind of TBox called free�TBox is obtained admittinggeneral concepts in the left�hand side of a TBox�statement� Free TBox�statements are of the form � and � �and of type � and � as particularcases�� When using free TBoxes the distinction between dened primitiveand atomic concepts does not make sense anymore�

Di�erent semantics have been proposed for the various TBox mecha�nisms depending also on the fact whether cyclic statements are allowedor not within the TBox� We consider below the so�called descriptive se�

mantics which seems to be the most appropriate one although it is not


tential quantication� ��R�� and AL �as named by Schmidt�Schau% andSmolka �� the language obtained from FL� adding �� and negationof concept names i�e� negation only of the form �A�

Following the convention the superlanguages of AL will be identiedwith a string of the form

AL#E $#U $#C$#R$#N $#O$

indicating which constructors are allowed in the language besides those ofAL� In a similar way languages that are superlanguages of FL� �but notof AL� will be identied by a string of the form FL#E $#U $#R$#N $#O$��

All the languages we consider in this paper are superlanguages of FL�

�and most of them are superlanguages of AL��

An interpretation I � ��I� �I� consists of a nonempty set �I �thedomain� and a function �I �the interpretation function� that maps everyconcept to a subset of �I every role to a subset of �I �I and everyindividual to an element of �I� Complex concepts and roles are inter�preted according to the semantics given in Tables � and � respectively� Inevery interpretation di�erent individuals are assumed to denote di�erentelements i�e� for every pair of individuals a� b and for every interpretationI if a �� b then aI �� bI � This is called the Unique Name Assumption andis usually assumed in database applications�

An interpretation I is a model for a concept C if CI is nonempty� Aconcept is satis�able if it has a model and unsatis�able otherwise� We saythat C is subsumed by D if CI � DI for every interpretation I and C isequivalent to D written C � D if CI � DI for every interpretation I�

It is important to observe that according to the given semantics theabove constructors are not all independent of each other� More preciselygiven a language L� and a constructor & we say that L� simulates & iffor every concept expressible in the language obtained by adding & to L�there exists an equivalent one expressible in L��

For example in su�ciently powerful languages any concept containinggeneral negation can be rewritten into an equivalent one containing onlynegation of concept names� In fact the � sign can be �pushed inside untilwe obtain only negation of the form �A� For this reason the constructorC can be simulated by the language ALEU � In a similar way we can alsoshow that ALC simulates E and U �

From this point on we assume �for the sake of simplicity� that in alanguage L all the constructors simulated by L are available in L� This

�The constructor �R�� also written �R� denotes the set of objects d� such that thereexists an object d� related to d� by means of the role R� The existential quanticationis unqualied in the sense that no condition is stated to d� other than its existence�

�


Constructor Name Syntax Semantics

concept name A AI � I

top � I

bottom � �

conjunction C uD CI DI

disjunction �U� C tD CI DI

negation �C� �C I n CI

universal quantication �R�C fd� j �d� � �d�� d�� RI d� � CIgexistential quantication �E� �R�C fd� j �d� � �d�� d�� RI � d� � CIg

number restrictions �N � �� nR� fd� j �fd� j �d�� d�� RIg � ng�� nR� fd� j �fd� j �d�� d�� RIg � ng

collection of individuals �O� fa�� ang faI�� aI

ng

TABLE � Syntax and semantics of concept�forming constructors�

Constructor Name Syntax Semantics

role name P PI � I �I

role conjunction �R� Q uR QI RI

TABLE � Syntax and semantics of the role�forming constructors�

�� Syntax and Semantics of Concept Languages

A concept language is composed by the symbols taken from the set ofConcept Names Role Names and Individual Names �or Individuals��

Besides concept role and individual names the alphabet of conceptlanguages includes a number of constructors that permit the formationof concept expressions and role expressions � The set of constructors forconcepts expressions and role expressions considered in this work are listedin Tables � and � respectively� The description of other constructors canbe found in �Baader et al� �� Woods and Schmolze �� Patel�Schneiderand Swartout �� Schaerf ��a��

Concept names are denoted with the letters A�B role names with P and individuals names with a� b possibly with subscripts� Concept ex�pressions and role expressions �or simply concepts and roles� are denotedrespectively with the letters C�D and Q�R�

The various languages di�er from one another based on the set of con�structors they allow� A concept language L is uniquely identied by its setof constructors� A concept �resp� role� obtained using the constructors ofL is called an L�concept �resp� L�role��

For naming concept languages we adopt the terminology introducedby Schmidt�Schau% and Smolka �� and Donini et al� ��a�� follow�ing such terminology an identifying letter is associated to a subset of theconstructors as shown in Tables � and ��

Moreover after Brachman and Levesque �� we call FL� the lan�guage including universal quantication conjunction and unquali�ed exis�

�


from the technique are not practical� However for some languages theyare the only known complete deduction procedures and can lead to ac�tual implementations by the adoption of clever control strategies �see forexample Baader et al� ��

Finally the analysis of pathological cases has lead to the discovery offorms of incompleteness of the algorithms developed for implemented sys�tems and can be used in the denition of suitable test sets for verifyingimplementations� For example the comparison of implemented systems de�scribed by Baader et al� �� beneted from the results of the complexityanalysis�

The survey is organized as follows�

Section � provides a formal introduction to concept languages theiruse in the construction of the ABox and the TBox and the associatedreasoning services�

Section � deals with reasoning on concept expressions� We start byrecalling the notion of structural Subsumption and show that thereare inferences that it does not capture� We then present the tableaux�based calculus and the complexity results for Satisability and Sub�sumption of concept expressions that were obtained by exploitingsuch a calculus�

Section � deals with reasoning on the ABox� We consider the problemof checking if a set of membership assertions logically implies thata given individual is an instance of a given concept� Further weintroduce an epistemic operator that provides query facilities that gobeyond the rst�order setting of concept languages� We show thatepistemic queries may have both a natural interpretation and goodcomputational properties�

Section deals with reasoning on the TBox� We discuss di�erentforms of concept denitions available in the TBox their semanticsand the complexity of reasoning with them�

Section � brie�y revises the problems related to reasoning in the gen�eral setting where both the TBox and the ABox are present�

Conclusions are drawn in Section � where several research issuesrelated to DL�systems and not addressed in this paper are discussed�

� Basic notions

In this section we present the basic notions regarding concept languagesknowledge bases built up using concept languages and reasoning servicesthat must be provided for inferring information from such knowledge bases�

�


guage and the di�culty of reasoning on the representations built using thatlanguage� In other words the more expressive the language the harder thereasoning� They also provide a rst example of this trade�o� by analyzingthe language FL� and showing that for such a language the Subsumptionproblem can be solved in polynomial time while adding the constructorcalled role restriction to the language makes Subsumption a coNP�hardproblem� Their work started a large body of research whose aim is to an�alyze the complexity of reasoning in di�erent concept languages relyingon standard notions from complexity theory and in particular worst�casecomplexity analysis�

The goal of this paper is to present the progress made both in thedesign of reasoning techniques and in the computational characterizationof the reasoning problems arising in DL�systems� For this purpose weconsider a rather general framework which takes into account the actualreasoning services that need to be available in DL�systems� As Brachmanand Levesque �� we start from the computational problems arisingwhen reasoning on concept expressions but then look at the mechanisms fordening the knowledge base namely the ABox �which requires reasoning onindividuals� and the TBox �which requires to deal with axioms for deningproperties of concepts��

One of the assumptions underlying this research is to use worst�caseanalysis as a measure of the complexity of reasoning tasks� Although suchan assumption has sometimes been criticized �see for example Doyle andPatil �� we consider the outcomes of this research rather signicant�In particular we outline three main contributions to the area of conceptlanguages and more generally to knowledge representation�

First of all the study of the computational behavior of concept lan�guages has led to a clear understanding of the properties of the languageconstructors and their interaction� This is not only valuable from a theo�retical viewpoint but gives insight to the designer of deduction procedureswith clear indications of which language constructors are di�cult to dealwith and general methods to cope with the computational problems� Forexample the system classic �Borgida et al� �� was designed takinginto account the indications of the complexity analysis in the choice of theconstructors included in the language�

Secondly the complexity results have been obtained by exploiting ageneral technique for Satisability checking in Description Logics� Thetechnique relies on tableaux calculus for rst�order logic and is a simpli�cation of the one used by Schmidt�Schau% and Smolka �� It has provedextremely useful for studying both the correctness and the complexity ofthe algorithms� More specically it provides an algorithmic frameworkthat is parametric with respect to language constructors� In general thealgorithms for Concept Satisability and Subsumption obtained directly

�


concepts are used to represent classes as sets of individuals and roles arebinary relations used to specify their properties or attributes�

Concepts are denoted by expressions formed by means of special con�structors� For example the concept expression ParentuMaleu�CHILD�Maledenotes the class of fathers �male parents� all of whose children are male�The symbol u denotes concept conjunction and is interpreted as set inter�section� The expression �CHILD�Male denotes the set of individuals whosechildren are all male thus specifying a property which relates to other in�dividuals through the role CHILD� Expressions of the form �R�C are calleduniversal role quantications� Instead �CHILD�Male is an example of ex�istential role quantication denoting the set of individuals with a malechild� Existential role quantication is sometimes unqualied and written�CHILD in which case it denotes the set of individuals with a child� Thebasic language FL� includes concept conjunction universal role quan�tication and unqualied existential role quantication� More powerfullanguages are then dened by adding other constructors to this basic lan�guage�

A general characteristic of DL�systems is that the knowledge base ismade up of two di�erent components� Informally speaking the rst isa general schema concerning the classes of individuals to be representedtheir general properties and mutual relationships and it is usually referredto as TBox while the second is a �partial� instantiation of this schemacontaining assertions relating either individuals to classes or individuals toeach other and is usually called ABox� This setting is inherited from kl�

one �Brachman and Schmolze �� and is also shared by several proposalsof Database models �see Abrial �� Beck et al� �� Mylopoulos et al��

Retrieving information in DL�systems is a deductive process involvingboth the schema �TBox� and its instantiation �ABox�� In fact the TBoxis not just a set of constraints on possible ABoxes but contains intensionalinformation about classes and this information is taken into account whenanswering queries to the knowledge base�

�� Goal and Organization of the Paper

One of the main challenges in the study of Description Logics is to identifythe fragment of formal logic that both captures the features needed for rep�resentation purposes and still allows for the design of e�ective and e�cientreasoning methods� To this aim several concept languages were consid�ered and their expressiveness and computational complexity were studiedfollowing the work of Brachman and Levesque �� who formally studiedsuch properties for a simple language called FL� �Frame Language��

Brachman and Levesque �see also Levesque and Brachman �� arguethat there is a trade�o� between the expressivity of a representation lan�

�


identied two main types of arcs� Those representing intensional knowl�edge and those representing extensional knowledge�

A Frame �Minsky �� usually represents a concept �or a class� andis dened by an identier and a number of elements called slots� Eachslot corresponds to an attribute that members of the class can have� Itcontains information about the corresponding attribute such as defaultvalues restrictions on possible llers attached procedures or methods forcomputing values when needed and procedures for propagating side e�ectswhen the slot is lled� The values of the attributes are either elements ofa concrete domain �e�g� integers strings� or identiers of other frames� Aframe can also represent a single individual in this case it is related withthe attribute instance�of to the frame representing the class of which theindividual is an instance� Early frame�based systems su�ered from the sameproblem highlighted above for Semantic Networks� In fact the semantics offrames was not formally dened and in particular the distinction betweenthe di�erent types of knowledge embedded in a frame system was not clear�

One attempt to providing a formal ground to Semantic Networks andFrames led to the development of the system kl�one �Brachman andSchmolze �� Subsequently the research has gone further in this direc�tion by providing systems with an explicit model�theoretic semantics �seefor example the system omega Attardi and Simi �� More recently KRsystems based on Description Logics �or terminological systems� have beenproposed as successors of kl�one with the Tarskian semantics for rst or�der logic� The family of these systems includes krypton �Brachman etal� �� nikl �Kaczmarek et al� �� back �Quantz and Kindermann�� loom �MacGregor and Bates �� classic �Borgida et al� ��kris �Baader and Hollunder �� and others �see Rich �� Woods andSchmolze ��

Description Logics systems are the subject of this survey which focusesin particular on the foundations of the techniques needed for reasoning inthese systems and the computational properties of the reasoning mech�anisms� It is important to note that such foundations have been proveduseful not only in Articial Intelligence but also in areas such as Databasesand Programming Systems where structured formalisms for the represen�tation of knowledge have been investigated�

Description Logics systems have been used for several application ar�eas including conguration natural language processing and software en�gineering �see e�g� Wright et al� ��

�� Representing Knowledge with Description Logics

The core of a KR system based on Description Logics �DL�system fromnow on� is its concept language which can be viewed as a set of constructsfor denoting classes and relationships among classes� In concept languages

�

�

Reasoning in Description Logics

F� M� Donini� M� Lenzerini� D� Nardi� A� Schaerf

abstract�

Description Logics stem from Semantic Networks and Frames�They deal with the representation of structured concepts� and rea�soning with them� The structure of a concept is described usinga language� called concept language� comprising boolean operators�conjunction� disjunction� negation� and various forms of quanti�ca�tion over the attributes �or� roles� of the given concept� A noticeabledi�erence with other structuring formalisms is that also roles canbe given some internal structure� e�g� a role can be expressed as aconjunction of two other roles�

We survey techniques for reasoning� and computational complex�ity of reasoning problems in Description Logics� referring to fourdi�erent settings �� reasoning with plain concept expressions! ��reasoning with instances of concepts! �� reasoning with axioms ex�pressing properties of concepts! �� reasoning with both instances ofconcepts and axioms�

In the end of the paper� we mention important aspects of De�scription Logics not dealt within this survey�

Introduction

The idea of developing knowledge representation systems based on a struc�tured representation of knowledge was rst pursued with Semantic Net�

works and Frames�Semantic Networks �Quillian �� Lehmann �� represent knowledge

under the form of a labeled directed graph where nodes denote conceptsand arcs represent the various relations between concepts� As pointed outby Woods �� and Brachman �� despite their name early SemanticNetworks su�ered from the drawback that they did not have clear seman�tics� Di�culties arise from the fact that in Semantic Networks arcs canrepresent di�erent kinds of relations between nodes� In particular Woods

A Great Collection

U� Gnowho and U� Gnowho�Else� eds�Copyright c� �� CSLI Publications�

�

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