Hybrid Logics and Ontology Languages

1. Hybrid Logics and Ontology Languages Ian Horrocks <horrocks@cs.man.ac.uk> Information Management Group School of Computer Science University of Manchester

2. Introduction to Description Logics

3. 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 C2) • Closely related to Propositional Modal, Hybrid & Dynamic Logics • Closely related to Guarded Fragment • Provision of inference services • Decision procedures for key problems (satisfiability, subsumption, etc) • Implemented systems (highly optimised)

4. DL Basics • Concepts (formulae) • E.g., Person, Doctor, HappyParent, (Doctor t Lawyer) • Roles (modalities) • E.g., hasChild, loves • Individuals (nominals) • E.g., John, Mary, Italy • Operators (for forming concepts and roles) restricted so that: • Satisfiability/subsumption is decidable and, if possible, of low complexity • No need for explicit use of variables • Restricted form of 9 and 8 (direct correspondence with hii and [i]) • Features such as counting (graded modalities) succinctly expressed

5. The DL Family (1) • Smallest propositionally closed DL is ALC (equivalent to 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 u8hasChild.(Doctor t 9hasChild.Doctor)

6. The DL Family (1) • Smallest propositionally closed DL is ALC (equivalent to 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 Æ [hasChild](Doctor ÇhhasChildiDoctor)

7. The DL Family (2) • S often used for ALC extended with transitive roles • i.e., the union of K(m) and K4(m) • Additional letters indicate other extensions, e.g.: • H for role hierarchy (e.g., hasDaughter v hasChild) • O for nominals/singleton classes (e.g., {Italy}) • I for inverse roles (converse modalities) • Q for qualified number restrictions (graded modalities, e.g., hiim) • N for number restrictions (graded modalities, e.g., hiim>) • S + role hierarchy (H) + nominals (O) + inverse (I) + NR (N) = SHOIN • SHOIN is the basis for W3C’s OWL Web Ontology Language

8. DL Knowledge Base • A TBox is a set of “schema” axioms (sentences), e.g.: {Doctor v Person, HappyParent´Person u8hasChild.(Doctor t 9hasChild.Doctor)} • i.e., a background theory (a set of non-logical axioms) • An ABox is a set of “data” axioms (ground facts), e.g.: {John:HappyParent, John hasChild Mary} • i.e., non-logical axioms including (restricted) use of nominals

9. DL Knowledge Base • A TBox is a set of “schema” axioms (sentences), e.g.: {Doctor ! Person, HappyParent$Person Æ [hasChild](Doctor ÇhhasChildiDoctor)} • i.e., a background theory (a set of non-logical axioms) • An ABox is a set of “data” axioms (ground facts), e.g.: {John!HappyParent, John!hhasChildiMary} • i.e., non-logical axioms including (restricted) use of nominals • A Knowledge Base (KB) is just a TBox plus an Abox

10. Ontologies and OWL

11. The Web Ontology Language OWL • Semantic Web led to requirement for a “web ontology language” • set up Web-Ontology (WebOnt) Working Group • WebOnt developed OWL language • OWL based on earlier languages OIL and DAML+OIL • OWL now a W3C recommendation(i.e., a standard) • OIL, DAML+OIL and OWL based on Description Logics • OWL effectively a “Web-friendly” syntax for SHOIN

12. OWL RDF/XML Exchange Syntax <owl:Class> <owl:intersectionOf rdf:parseType=" collection"> <owl:Class rdf:about="#Person"/> <owl:Restriction> <owl:onProperty rdf:resource="#hasChild"/> <owl:allValuesFrom> <owl:unionOf rdf:parseType=" collection"> <owl:Class rdf:about="#Doctor"/> <owl:Restriction> <owl:onProperty rdf:resource="#hasChild"/> <owl:someValuesFrom rdf:resource="#Doctor"/> </owl:Restriction> </owl:unionOf> </owl:allValuesFrom> </owl:Restriction> </owl:intersectionOf> </owl:Class> E.g., Person u8hasChild.(Doctor t 9hasChild.Doctor):

13. Class/Concept Constructors • C is a concept (class); P is a role (property); xi is an individual/nominal • XMLS datatypes as well as classes in 8P.C and 9P.C • Restricted form of DL concrete domains

14. Ontology Axioms • OWL ontology equivalent to DL KB (Tbox + Abox)

15. Why (Description) Logic? • OWL exploits results of 15+ years of DL research • Well defined (model theoretic) semantics

16. Why (Description) Logic? • OWL exploits results of 15+ years of DL research • Well defined (model theoretic) semantics • Formal properties well understood (complexity, decidability) I can’t find an efficient algorithm, but neither can all these famous people. [Garey & Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, 1979.]

17. Why (Description) Logic? • OWL exploits results of 15+ years of DL research • Well defined (model theoretic) semantics • Formal properties well understood (complexity, decidability) • Known reasoning algorithms

18. Why (Description) Logic? • OWL exploits results of 15+ years of DL research • Well defined (model theoretic) semantics • Formal properties well understood (complexity, decidability) • Known reasoning algorithms • Implemented systems (highly optimised) Pellet

19. Why (Description) Logic? • Foundational research was crucial to design of OWL • Informed Working Group decisions at every stage, e.g.: • “Why not extend the language with feature x, which is clearly harmless?” • “Adding x would lead to undecidability - see proof in […]”

20. Applications of Ontologies • e-Science, e.g., Bioinformatics • Open Biomedical Ontologies Consortium (GO, MGED) • Used e.g., for “in silico” investigations relating theory and data • E.g., relating data on phosphatases to (model of) biological knowledge

21. CentralSulcus Frontal Lobe Parietal Lobe Occipital Lobe Temporal Lobe Lateral Sulcus Applications of Ontologies • Medicine • Building/maintaining terminologies such as Snomed, NCI & Galen

22. Applications of Ontologies • Organising complex and semi-structured information • UN-FAO, NASA, Ordnance Survey, General Motors, Lockheed Martin, …

23. Nominals in Ontologies • Used in extensionally defined classes • e.g., class EU might be defined as {Austria, …, UnitedKingdom} • Written in OWL as oneOf(Austria … UnitedKingdom) • Equivalent to a disjunction of nominals: Austria Ç … Ç UnitedKingdom • Allows inferences such as: • EU contains 25 countries (assuming UNA/axioms) • If in the EU and not in oneOf(Austria … Sweden) ! in UnitedKingdom • Used in extended OWL Abox axioms • e.g., individual(Jim value(friend individual(value(friend Jane)))) • Equivalent to {Jim} v9 friend.(9 friend.{Jane}) • i.e., Jim !hfriendi(hfriendiJane) • Widely used in ontologies • e.g. in Wine ontology used for colours, grape types, regions, etc.

24. Ontology Reasoning:How do we do it?

25. Using Standard DL Techniques • 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

26. Tableaux Reasoning (1) • E.g., KB: {HappyParent´Person u8hasChild.(Doctor t 9hasChild.Doctor), John:HappyParent, John hasChild Mary, Mary:: Doctor Wendy hasChild Mary, Wendy marriedTo John} Person 8hasChild.(Doctor t 9hasChild.Doctor)

27. Tableaux Reasoning (2) • Tableau rules correspond to constructors in logic (u, 9 etc) • E.g., John:(Person u Doctor) --!John:Person andJohn: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: {Personv9hasParent.Person, John:Person}

28. Tableaux Reasoning (3) • In general, (representation of) model consists of: • Named individuals forming arbitrary directed graph • Trees of anonymous individuals rooted in named individuals

29. Decision Procedures • Algorithms are decision procedures, 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

30. Reasoning with Nominals:A Tableaux Algorithm for SHOIQ

31. Recall Motivation for OWL Design • Exploit results of DL research: • … • Known tableaux decision procedures and implemented systems But not for SHOIN(until recently)! So why is/was SHOIN so hard?

32. SHIQ is Already Tricky • Does not have finite model property, e.g.: {ITN v61edge–u9edge.ITN, R:(ITN u60edge–)} • Double blocking • Block interpreted as infinite repetition

33. SHIQ is Already Tricky • Does not have finite model property, e.g.: {ITN v61edge–u9edge.ITN, R:(ITN u60edge–)} • Double blocking • Block interpreted as infinite repetition • Termination problem due to > and 6, e.g.: {John:9hasChild.Doctor u>2 hasChild.Lawyer u62hasChild} • Add inequalities between nodes generated by > rule • Clash if 6 rule only applicable to  nodes

34. SHOIQ: Loss (almost) of TMP • Interactions between O, I, and Q lead to new termination problems • Anonymous branches can loop back to named individuals (O) • E.g., 9r.{Mary} • Number restrictions (Q) on incoming edges (I) lead to non-tree structure • E.g., Mary:61r– • 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

35. 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

36. Generate & Merge Problem is Back! E.g., KB: {VMP´Person u9loves.{Mary} u9hasFriend.VMP, John:9hasFriend.VMP Mary:62loves–} • Blocking prevented by N-nodes • Repeated generation and merging of nodes leads to non-termination

37. Intuition: Guess Exact Cardinality • New Ro?-rule guesses exact cardinality constraint on N-nodes {VMP´Person u9loves.{Mary} u9hasFriend.VMP, John:9hasFriend.VMP Mary:62loves–} • Inequality between resulting N-nodes fixes generate & merge problem • Introduces new source of non-determinism • But only if nominals used in a “nasty” way • Usage in ontologies typically “harmless” • Otherwise behaves as forSHIQ

38. Conjunctive Query Answering:Using binders (maybe)

39. Conjunctive Queries • Want to query KB using DB style conjunctive query language • e.g., hx,zià WinehxiÆ drunkWithhx,yiÆ DishhyiÆ fromRegionhy,zi • How to answer such queries? • Reduce to boolean queries w.r.t. candidate answer tuples • e.g., hià WinehChiantiiÆ drunkWithhChianti,yiÆ DishhyiÆ fromRegionhy,Venetoi • Transform query into concept Cq by “rolling up” • e.g., Cq = {Chianti} u9 drunkWith.(Dish u9 fromRegion.{Veneto}) • such that query can be reduced to KB satisfiability test • hT,Ai² q iff hT[ {>v:Cq},Ai is not satisfiable

40. B u9P.C u9S.D A u9R.(B u9P.C u9S.D) B u9P.C Rolling Up (1) • View query as a labeled graph and “roll up” from leaves to root • e.g., hià AhwiÆ Rhw,xiÆ BhxiÆ Phx,yiÆ ChyiÆ Shx,ziÆ Chyi

41. Cyclical Queries • Problems arise when trying to roll up cyclical queries • e.g., hià AhwiÆ Rhw,xiÆ BhxiÆ Phx,yiÆ ChyiÆ Shx,ziÆ ChyiÆ Rhy,zi

42. C u9P-..x A u9R.(x.(B u9S.(D u9R-.(C u9P-..x)))) x.B x.B x.(B u9S.(D u9R-.(C u9P-..x))) D u9R-.(C u9P-..x) Rolling Up with Binders (1) • Problem could be solved by extending DL with binder: • e.g., hià AhwiÆ Rhw,xiÆ BhxiÆ Phx,yiÆ ChyiÆ Shx,ziÆ ChyiÆ Rhy,zi

43. Rolling Up with Binders (2) • Unfortunately, already known that ALC + binder is undecidable [Blackburn and Seligman] • But, when used in rolling up, only occurs in very restricted form: • Only intersection, existential and positive state variables • and when negated (in sat test), only union, universal and negated vars • in form 8R.:x • Now known that SHIQ conjunctive query answering is decidable • Binders would potentially lead to a more “practical” algorithm

44. SHIQ Conjunctive Queries

45. Rolling Up with Binders (2) • Unfortunately, already known that ALC + binder is undecidable [Blackburn and Seligman] • But, when used in rolling up, only occurs in very restricted form: • Only intersection, existential and positive state variables • and when negated (in sat test), only union, universal and negated vars • in form 8R.:x • Now known that SHIQ conjunctive query answering is decidable • Binders would potentially lead to a more “practical” algorithm • But not trivial to extend tableaux algorithm to SHIQ + binder • Blocking is difficult because binder introduces new concepts • Decidability of SHOIQ conjunctive query answering still open • Although believe we now have a solution

46. Summary • DLs are a family of logic basedKR formalisms • Describe domain in terms of concepts, roles and individuals • Closely related to Modal & Hybrid Logics • DLs are the basis for ontology languages such as OWL • Nominals widely used in ontologies • Reasoning with SHOIQ is tricky, but now reasonalby well understood • Binders potentially useful for conjunctive query answering • Allow for rolling up of arbitrary queries • Required extensions known to be decidable • But reasoning with extended languages still an open problem

47. Acknowledgements Thanks to: • Birte Glimm • Uli Sattler

48. Resources • Slides from this talk • http://www.cs.man.ac.uk/~horrocks/Slides/HyLo06.ppt • FaCT++ system (open source) • http://owl.man.ac.uk/factplusplus/ • Protégé • http://protege.stanford.edu/plugins/owl/ • W3C Web-Ontology (WebOnt) working group (OWL) • http://www.w3.org/2001/sw/WebOnt/ • DL Handbook, Cambridge University Press • http://books.cambridge.org/0521781760.htm

49. Thank you for listening Any questions?