1. Combining the strengths of UMIST and The Victoria University of Manchester A Tableaux Decision Procedure for SHOIQ Ian Horrocks and Ulrike Sattler University of Manchester Manchester, UK

2. Combining the strengths of UMIST and The Victoria University of Manchester SHOIQ : the Final Frontier Ian Horrocks and Ulrike Sattler University of Manchester Manchester, UK

3. Combining the strengths of UMIST and The Victoria University of Manchester Talk Outline Introduction to Description Logics Ontologies Ontology Reasoning Why do we want it? How do we do it? Tableaux Algorithms for Description Logic Reasoning Current Work and Research Challenges Summary

4. Combining the strengths of UMIST and The Victoria University of Manchester Introduction to Description Logics

5. Combining the strengths of UMIST and The Victoria University of Manchester What Are Description Logics? A family of logic based Knowledge Representation formalisms –Descendants of semantic networks and KL-ONE –Describe domain in terms of concepts (classes), roles (properties, relationships) and individuals Distinguished by: –Formal semantics (typically model theoretic) Decidable fragments of FOL (often contained in C 2 ) Closely related to Propositional Modal & Dynamic Logics Closely related to Guarded Fragment –Provision of inference services Decision procedures for key problems (satisfiability, subsumption, etc) Implemented systems (highly optimised)

6. Combining the strengths of UMIST and The Victoria University of Manchester Applications of DLs Databases Ontologies (knowledge bases) –OWL Lite Web Ontology Language based on SHIF –OWL DL Web Ontology Language based on SHOIN Motivation for OWL design was to exploit results of DL research: –Well defined semantics –Formal properties well understood (complexity, decidability) –Known tableaux decision procedures and implemented systems But not for SHOIN (up until now)

7. Combining the strengths of UMIST and The Victoria University of Manchester DL Basics Concepts (unary predicates/formulae with one free variable) –E.g., Person, Doctor, HappyParent, Doctor t Lawyer Roles (binary predicates/formulae with two free variables) –E.g., hasChild, loves, ( hasBrother ± hasDaughter ) Individual names (constants) –E.g., John, Mary, Italy Operators (for forming concepts and roles) restricted so that: –Language is decidable and, if possible, of low complexity –No need for explicit use of variables Restricted form of 9 and 8 (direct correspondence with ◊ and []) –Features such as counting can be succinctly expressed

8. Combining the strengths of UMIST and The Victoria University of Manchester The DL Family (1) Smallest propositionally closed DL is ALC (equiv modal K (m) ) –Concepts constructed using booleans u, t, :, plus restricted quantifiers 9, 8 –Only atomic roles E.g., Person all of whose children are either Doctors or have a child who is a Doctor: Person u 8 hasChild.(Doctor t 9 hasChild.Doctor)

9. Combining the strengths of UMIST and The Victoria University of Manchester The DL Family (2) S often used for ALC with transitive roles ( R + ) Additional letters indicate other extension, e.g.: – H for role hierarchy (e.g., hasDaughter v hasChild) – O for nominals/singleton classes (e.g., {Italy}) – I for inverse roles (e.g., isChildOf ´ hasChild – ) – N for number restrictions (e.g., > 2 hasChild, 6 3 hasChild) – Q for qualified number restrictions (e.g., > 2 hasChild.Doctor) ALC + R + + role hierarchy + nominals + inverse + QNR = SHOIQ

10. Combining the strengths of UMIST and The Victoria University of Manchester Knowledge Bases (Ontologies) A TBox is a set of “schema” axioms (sentences), e.g.: {Doctor v Person, HappyParent ´ Person u 8 hasChild.(Doctor t 9 hasChild.Doctor)} An ABox is a set of “data” axioms (ground facts), e.g.: {John:HappyParent, John hasChild Mary} A Knowledge Base (KB) is a TBox plus and ABox An ontology is usually taken to be equiv. to a TBox –But in OWL, an ontology is an arbitrary set of axioms (i.e., equiv. to a KB)

11. Combining the strengths of UMIST and The Victoria University of Manchester OWL Syntax (Abstract & RDF) intersectionOf(Person restriction(hasChild allValuesFrom( unionOf(Doctor restriction(hasChild someValuesFrom(Doctor)))))) E.g., Person u 8 hasChild.(Doctor t 9 hasChild.Doctor):

12. Combining the strengths of UMIST and The Victoria University of Manchester Description Logic Reasoning

13. Combining the strengths of UMIST and The Victoria University of Manchester Tableaux Reasoning (1) Key reasoning tasks reducible to KB (un)satisfiability –E.g., C v D w.r.t. KB K iff K [ {x:(C u : D)} is not satisfiable State of the art DL systems typically use (highly optimised) tableaux algorithms to decide satisfiability (consistency) of KB Tableaux algorithms work by trying to construct a concrete example (model) consistent with KB axioms: –Start from ground facts (ABox axioms) –Explicate structure implied by complex concepts and TBox axioms Syntactic decomposition using tableaux expansion rules Infer constraints on (elements of) model

14. Combining the strengths of UMIST and The Victoria University of Manchester Tableaux Reasoning (2) E.g., KB: { HappyParent ´ Person u 8 hasChild.(Doctor t 9 hasChild.Doctor), John:HappyParent, John hasChild Mary, Mary: : Doctor Wendy hasChild Mary, Wendy marriedTo John} Person 8 hasChild.(Doctor t 9 hasChild.Doctor)

15. Combining the strengths of UMIST and The Victoria University of Manchester Tableaux Reasoning (3) Tableau rules correspond to constructors in logic ( u, 9 etc) –E.g., John: ( Person u Doctor) -- ! John:Person and John:Doctor Stop when no more rules applicable or clash occurs –Clash is an obvious contradiction, e.g., A(x), : A(x) Some rules are nondeterministic (e.g., t, 6 ) –In practice, this means search Cycle check (blocking) often needed to ensure termination –E.g., KB: { Person v 9 hasParent.Person, John:Person}

16. Combining the strengths of UMIST and The Victoria University of Manchester Tableaux Reasoning (4) In general, (representation of) model consists of: –Named individuals forming arbitrary directed graph –Trees of anonymous individuals rooted in named individuals

17. Combining the strengths of UMIST and The Victoria University of Manchester Decision Procedure Algorithm is a decision procedure, i.e., KB is satisfiable iff rules can be applied such that fully expanded clash free graph is constructed: Sound –Given a fully expanded and clash-free graph, we can trivially construct a model Complete –Given a model, we can use it to guide application of non-deterministic rules in such a way as to construct a clash-free graph Terminating –Bounds on number of named individuals, out-degree of trees (rule applications per node), and depth of trees (blocking) Crucially depends on (some form of) tree model property

18. Combining the strengths of UMIST and The Victoria University of Manchester SHOIQ : Why is it Hard?

19. Combining the strengths of UMIST and The Victoria University of Manchester SHIQ is Already Tricky Does not have finite model property, e.g.: { ITN v 6 1 edge – u 8 edge.ITN u 9 edge.ITN, R:(ITN u 6 0 edge – )} –Double blocking –Block interpreted as infinite repetition

20. Combining the strengths of UMIST and The Victoria University of Manchester SHIQ is Already Tricky Does not have finite model property, e.g.: { ITN v 6 1 edge – u 8 edge.ITN u 9 edge.ITN, R:ITN u 6 0 edge – u 9 edge.ITN } –Double blocking –Block interpreted as infinite repetition Yo-yo problem due to > and 6, e.g.: { John: 9 hasChild.Doctor u > 2 hasChild.Lawyer u 6 2 hasChild } –Add inequalities between nodes generated by > rule –Clash if 6 rule only applicable to  nodes

21. Combining the strengths of UMIST and The Victoria University of Manchester SHOIQ : ExpTime ! NExpTime Interactions between O, I, and Q lead to termination problems –Anonymous branches can loop back to named individuals ( O ) E.g., 9 r.{Mary} –Number restrictions ( Q ) on incoming edges ( I ) lead to non-tree structure E.g., Mary: 6 1 r – –Result is anonymous nodes that act like named individual nodes –Blocking sequence cannot include such nodes Don’t know how to build a model from a graph including such a block

22. Combining the strengths of UMIST and The Victoria University of Manchester Intuition: Nominal Nodes Nominal nodes (N-nodes) include: –Named individual nodes –Nodes affected by number restriction via outgoing edge to N-node Blocking sequence cannot include N-nodes Bound on number of N-nodes –Must initially have been on a path between named individual nodes –Length of such paths bounded by blocking –Number of incoming edges at an N-node is limited by number restrictions

23. Combining the strengths of UMIST and The Victoria University of Manchester SHOIQ : Yo-Yo Problem is Back! E.g., KB: { VMP ´ Person u 9 loves.{Mary} u 9 hasFriend.VMP, John: 9 hasFriend.VMP Mary: 6 2 loves – } Blocking prevented by N-nodes Repeated creation and merging of nodes leads to non-termination

24. Combining the strengths of UMIST and The Victoria University of Manchester Intuition: Guess Exact Cardinality New Ro?-rule guesses exact cardinality constraint on N-nodes { VMP ´ Person u 9 loves.{Mary} u 9 hasFriend.VMP, John: 9 hasFriend.VMP Mary: 6 2 loves – } Inequality between resulting N-nodes fixes yo-yo problem Introduces new source of non-determinism –But only if nominals used in a “nasty” way Usage in ontologies typically “harmless” –Otherwise behaves as for SHIQ

25. Combining the strengths of UMIST and The Victoria University of Manchester Summary DLs are a family of logic based KR formalisms –Well known as basis of ontology languages such as OWL Key motivation for the design of OWL was the existence of DL tableaux decision procedures and implementations –But, no procedure/implementation for OWL DL/ SHOIN (up to now) SHOIQ algorithm solves this (very embarrassing) problem –Ro?-rule introduces new source of non-determinism But good “pay as you go” characteristics Implementation already underway in FaCT++ and Pellet systems –Should work well in realistic ontology applications

26. Combining the strengths of UMIST and The Victoria University of Manchester Questions?