Ontology Reasoning

1. Ontology Reasoning Majid Sazvar sazvar@stu-mail.um.ac.ir Advisor: Dr. Naghibzadeh Knowledge Engineering Research Group Ferdowsi University of Mashhad 2010

2. Ontology Reasoning - Majid Sazvar - 2010 Outline • Introduction • Description Logics • Horn Logic • Ontology Reasoning • Why do we want it? • How do we do it? • Reasoning Tasks • Overview of DL Reasoners • Current Researches

3. Introduction

4. Ontology Reasoning - Majid Sazvar - 2010 Introduction KR + Reasoning = Knowledge Based System • KR: Providing high-level descriptions of the world that can be effectively used by computer. • Reasoning (Knowledge Processing): Finding implicit consequences from explicitly represented knowledge.

5. Ontology Reasoning - Majid Sazvar - 2010 Knowledge Representation Methods • Logic-Based Methods (particularly in the form of predicate logic First Order Logic) • Non-Logic-Based Methods • Rule-Based • Semantic Networks • Frame-Based

6. Ontology Reasoning - Majid Sazvar - 2010 The Importance of Logic • High-level language for expressing knowledge • High expressive power • Well-understood formal semantics • Precise notion of logical consequence • Proof systems that can automatically derive statements syntactically from a set of premises

7. Ontology Reasoning - Majid Sazvar - 2010 The Importance of Logic (2) • There exist proof systems for which semantic logical consequence coincides with syntactic derivation within the proof system • Soundness & completeness • Predicate logic is unique in the sense that sound and complete proof systems do exist. • Not for more expressive logics (higher-order logics) • Logic can provide explanations for answers • By tracing a proof

8. Description Logics

9. Ontology Reasoning - Majid Sazvar - 2010 Specializations of Predicate Logic: RDF and OWL • RDF/S and OWL (Lite and DL) are specializations of predicate logic • correspond roughly to a description logic • They define reasonable subsets of logic • Trade-off between the expressive power and the computational complexity: • The more expressive the language, the less efficient the corresponding proof systems

10. Ontology Reasoning - Majid Sazvar - 2010 What Are Description Logics? • A family of logic based Knowledge Representation formalisms • Descendants of semantic networks • Describe domain in terms of concepts (classes), roles (properties, relationships) and individuals • Distinguished by: • Decidable fragments of FOL • Provision of inference services • Decision procedures for key problems (satisfiability, subsumption, etc) • Implemented systems (highly optimised)

11. Ontology Reasoning - Majid Sazvar - 2010 DL Basics • Concept names are equivalent to unary predicates • In general, concepts equiv to formulae with one free variable • Role names are equivalent to binary predicates • In general, roles equiv to formulae with two free variables • Individual names are equivalent to constants • Operators 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 • Features such as counting can be succinctly expressed

12. Ontology Reasoning - Majid Sazvar - 2010 DL System Architecture Knowledge Base Tbox (schema) Man ´ Human u Male Happy-Father ´ Man u9 has-child Female u … Interface Inference System Abox (data) John : Happy-Father hJohn, Maryi : has-child John: 6 1 has-child

13. Ontology Reasoning - Majid Sazvar - 2010 The DL Family • Given DL defined by set of concept and role forming operators • Smallest propositionally closed DL is ALC • Concepts constructed using u, t, :, 9 and 8 • S often used for ALC with transitive roles (R+) • Additional letters indicate other extension, e.g.: • H for role inclusion axioms (role hierarchy) • O for nominals (singleton classes, written {x}) • I for inverse roles • N for number restrictions (of form 6nR, >nR) • Q for qualified number restrictions (of form 6nR.C, >nR.C) • E.g., ALC + R+ + role hierarchy + inverse roles + QNR = SHIQ • Have been extended in many directions

14. Ontology Reasoning - Majid Sazvar - 2010 DL Semantics • Semantics defined by interpretations • An interpretation I = (DI, ¢I), where • DI is the domain (a non-empty set) • ¢I is an interpretation function that maps: • Concept (class) name A! subset AI of DI • Role (property) name R! binary relation RI over DI • Individual name i!iI element of DI

15. Ontology Reasoning - Majid Sazvar - 2010 DL Semantics (cont.) • Interpretation function ¢I extends to concept (and role) expressionsin the obvious way, e.g.:

16. Ontology Reasoning - Majid Sazvar - 2010 DL Knowledge Base • A DL Knowledge baseK is a pair hT ,Ai where • T is a set of “terminological” axioms (the Tbox) • A is a set of “assertional” axioms (the Abox) • Tbox axioms are of the form: CvD, C´D, RvS, R´S and R+vR where C,D concepts, R, S roles, and R+ set of transitive roles • Abox axioms are of the form: x:D, hx,yi:R where x,y are individual names, D a concept and R a role

17. Ontology Reasoning - Majid Sazvar - 2010 Knowledge Base Semantics • An interpretationI satisfies (models) a Tbox axiom A (I²A): I²CvD iffCIµDII²C´D iffCI = DI I²RvS iffRIµSII²R´S iffRI = SI I²R+vR iff (RI)+µRI • Isatisfiesa TboxT (I²T ) iffI satisfies every axiom A in T • An interpretationI satisfies (models) an Abox axiom A (I²A): I²x:D iffxI2DII²hx,yi:R iff (xI,yI) 2RI • Isatisfies an AboxA (I²A) iffI satisfies every axiom A in A • Isatisfies an KBK (I²K) iffI satisfies both T and A

18. Ontology Reasoning - Majid Sazvar - 2010 Short History of Description Logics Phase 1: • Incomplete systems (Back, Classic, Loom, . . . ) • Based on structural algorithms Phase 2: • Development of tableau algorithms and complexity results • Investigation of optimisation techniques Phase 3: • Tableau algorithms for very expressive DLs • Highly optimised tableau systems for ExpTime logics (e.g., FaCT, DLP, Racer) • Relationship to modal logic and decidable fragments of FOL

19. Ontology Reasoning - Majid Sazvar - 2010 Recent Developments Phase 4: • Mainstream applications and tools • Databases • Consistency of conceptual schemata (EER, UML etc.) • Schema integration • Query subsumption (w.r.t. a conceptual schema) • Ontologies, e-Science and Semantic Web/Grid • Ontology engineering (schema design, maintenance, integration) • Reasoning with ontology-based annotations (data) • Mature implementations • Research implementations • FaCT, FaCT++, Racer, Pellet, … • Commercial implementations • Cerebra system from Network Inference (and now Racer)

20. Horn Logic

21. Ontology Reasoning - Majid Sazvar - 2010 Specializations of Predicate Logic: Horn Logic • A rule has the form: A1, . . ., An B • Ai and B are atomic formulas • There are 2 ways of reading such a rule: • Deductive rules: If A1,..., An are known to be true, then B is also true • Reactive rules: If the conditions A1,..., An are true, then carry out the action B

22. Ontology Reasoning - Majid Sazvar - 2010 Description Logic vs. Horn Logic • Neither of them is a subset of the other • It is impossible to assert that a person X who is brother of Y is uncle of Z (where Z is child of Y) in OWL • This can be done easily using rules: brother(X,Y), childOf(Z,Y)  uncle(X,Z) • Rules cannot assert the information that a person is either a man or a woman • This information is easily expressed in OWL using disjoint union

23. Ontology Reasoning - Majid Sazvar - 2010 Monotonic vs. Non-monotonic Rules • Example: An online vendor wants to give a special discount if it is a customer’s birthday Solution 1 R1: If birthday, then special discount R2: If not birthday, then not special discount • But what happens if a customer refuses to provide his birthday due to privacy concerns?

24. Ontology Reasoning - Majid Sazvar - 2010 Monotonic vs. Non-monotonic Rules (2) Solution 2 R1: If birthday, then special discount R2’: If birthday is not known, then not special discount • Solves the problem but: • The premise of rule R2' is not within the expressive power of predicate logic • We need a new kind of rule system

25. Ontology Reasoning - Majid Sazvar - 2010 Monotonic vs. Non-monotonic Rules (3) • The solution with rules R1 and R2 works in case we have complete information about the situation • The new kind of rule system will find application in cases where the available information is incomplete • R2’ is a nonmonotonic rule

26. Ontology Reasoning - Majid Sazvar - 2010 Example: Family Relations • Facts in a database about relations: • mother(X,Y), X is the mother of Y • father(X,Y), X is the father of Y • male(X), X is male • female(X), X is female • Inferred relation parent: A parent is either a father or a mother mother(X,Y)  parent(X,Y) father(X,Y)  parent(X,Y)

27. Ontology Reasoning - Majid Sazvar - 2010 Example: Family Relations (2) • male(X), parent(P,X), parent(P,Y), notSame(X,Y)  brother(X,Y) • female(X), parent(P,X), parent(P,Y), notSame(X,Y) sister(X,Y) • brother(X,P), parent(P,Y)  uncle(X,Y) • mother(X,P), parent(P,Y)  grandmother(X,Y) • parent(X,Y)  ancestor(X,Y) • ancestor(X,P), parent(P,Y)  ancestor(X,Y)

28. Ontology Reasoning:Why do We Want It?

29. Ontology Reasoning - Majid Sazvar - 2010 Why Ontology Reasoning? • Given key role of ontologies in many applications, it is essential to provide tools and services to help users: • Design and maintain high quality ontologies, e.g.: • Meaningful— all named classes can have instances • Correct— captured intuitions of domain experts • Minimally redundant— no unintended synonyms • Richly axiomatised— (sufficiently) detailed descriptions • Answer queries over ontology classes and instances, e.g.: • Find more general/specific classes • Retrieve individuals/tuples matching a given query • Integrate and align multiple ontologies

30. Ontology Reasoning - Majid Sazvar - 2010 Why Decidable Reasoning? • OWL is an W3C standard DL based ontology language • OWL constructors/axioms restricted so reasoning is decidable • Consistent with Semantic Web's layered architecture • XML provides syntax transport layer • RDF(S) provides basic relational language and simple ontological primitives • OWL provides powerful but still decidable ontology language • Further layers (e.g. SWRL) will extend OWL • Will almost certainly be undecidable • W3C requirement for “implementation experience” • “Practical” decision procedures • Several implemented systems • Evidence of empirical tractability

31. Ontology Reasoning - Majid Sazvar - 2010 Why Correct Reasoning? • Need to have high level of confidence in reasoner • Most interesting/useful inferences are those that were unexpected • Likely to be ignored/dismissed if reasoner known to be unreliable • Many realistic web applications will be agent ↔ agent • No human intervention to spot glitches in reasoning

32. Ontology Reasoning:How do we do it?

33. Ontology Reasoning - Majid Sazvar - 2010 Use a (Description) Logic • OWL DL based on SHIQDescription Logic • In fact it is equivalent to SHOIN(Dn) DL • OWL DL Benefits from many years of DL research • Well defined semantics • Formal properties well understood (complexity, decidability) • Known reasoning algorithms • Implemented systems (highly optimised) • In fact there are three “species” of OWL (!) • OWL full is union of OWL syntax and RDF • OWL DL restricted to First Order fragment (¼ DAML+OIL) • OWL Lite is “simpler” subset of OWL DL (equiv to SHIF(Dn))

34. Ontology Reasoning - Majid Sazvar - 2010 Class/Concept Constructors • C is a concept (class); P is a role (property); x is an individual name

35. Ontology Reasoning - Majid Sazvar - 2010 Ontologies / Knowledge Bases • OWL ontology equivalent to a DL Knowledge Base • OWL ontology consists of a set of axioms and facts • Note: an ontology is usually thought of as containing only Tbox axioms (schema)---OWL is non-standard in this respect • Recall that a DL KB K is a pair hT ,Ai where • T is a set of “terminological” axioms (the Tbox) • A is a set of “assertional” axioms (the Abox)

36. Ontology Reasoning - Majid Sazvar - 2010 Ontology/Tbox Axioms • Obvious FO/Modal Logic equivalences • E.g.,DL: CvD FOL: x.C(x)!D(x) ML: C!D • Often distinguish two kinds of Tbox axioms • “Definitions” CvD or C´D where C is a concept name • General Concept Inclusion axioms (GCIs) where C may be complex

37. Ontology Reasoning - Majid Sazvar - 2010 Ontology Facts / Abox Axioms • Note: using nominals (e.g., in SHOIN), can reduce Abox axioms to concept inclusion axioms • equivalent to • equivalent to

38. Ontology Reasoning:Reasoning Tasks

39. Ontology Reasoning - Majid Sazvar - 2010 Reasoning Tasks • TBOX reasoning tasks • Satisfiability checks whether a class C can have instances according to the current ontology. • Subsumption checks whether a class D subsumes a class C according to the current ontology. Property subsumption is defined analogously. • Classification computing the complete subsumption hierarchy of the ontology.

40. Ontology Reasoning - Majid Sazvar - 2010 Reasoning Tasks • ABOX reasoning tasks • Consistency checks whether the Abox is consistent with respect to the TBox. • Instance checking checks whether an assertion is entailed by the Abox. • Retrieval problem retrieves all individuals that instantiate a class C, dually we can find all named classes C that an individual a belongs to. • Property fillers retrieves, given a property R and an individual i, all individuals x which are related with i via R. Similarly we can retrieve the set of all named properties R between two individuals i and j, ask whether the pair (i,j) is a filler of P or ask for all pairs (i,j) that are a filler of P.

41. Ontology Reasoning - Majid Sazvar - 2010 Reasoning Tasks • ABOX reasoning tasks • Conjunctive Queries are a popular query formalism capable of expressing the class of selection/projection/join/renaming relational queries. • Theoretically it is possible to reduce all reasoning tasks to the task of checking KB consistency. • However in practice this is not necessarily the fastest way of reasoning and various optimizations are taken for different tasks.

42. Overview of DL Reasoners

43. Ontology Reasoning - Majid Sazvar - 2010 Overview of DL Reasoners • CEL • is a free (for non-commercial use) LISP-based reasoner for EL+. It implements a refined version of a known polynomial-time classification algorithm and supports new features like module extraction and axiom pinpointing. Currently, it accepts inputs in a small extension of the KRSS syntax and supports the DIG interface. • Cerebra Engine • is a commercial C++-based reasoner. It implements a tableau-based decision procedure for general TBoxes (subsumption, satisfiability, classification) and ABoxes (retrieval, tree-conjunctive query answering using a XQuery-like syntax). It supports the OWL-API and comes with numerous other features.

44. Ontology Reasoning - Majid Sazvar - 2010 Overview of DL Reasoners (2) • Bossam • an RETE-based rule engine with native supports for reasoning over OWL ontologies, SWRL ontologies, and RuleML rules. • FaCT++ • is a free (GPL/LGPL) open-source C++-based reasoner for SROIQ with simple datatypes (i.e., for OWL 2). It implements a tableau-based decision procedure for general TBoxes (subsumption, satisfiability, classification) and ABoxes (retrieval). It supports the OWL-API, the lisp-API and the DIG interface.

45. Ontology Reasoning - Majid Sazvar - 2010 Overview of DL Reasoners (3) • fuzzyDL • is a free Java/C++ based reasoner for fuzzy SHIF with concrete fuzzy concepts (explicit definition of fuzzy sets + modifiers). It implements a tableau + Mixed Integer Linear Programming optimization decision procedure to compute the maximal degree of subsumption and instance checking w.r.t. a general TBox and Abox. It supports Zadeh semantics, Lukasiewicz semantics and is backward compatible with classical description logic reasoning. • HermiT • is a free (under LGPL license) Java reasoner for SHIQ with description graphs. It implements a hypertableau-based decision procedure for general TBoxes and description graphs. It uses the KAON2 API.

46. Ontology Reasoning - Majid Sazvar - 2010 Overview of DL Reasoners (4) • Hoolet • reasons over OWL-DL ontologies by translating them to full first-order logic and then applying a first-order theorem prover. • KAON2 • is a free (free for non-commercial usage) Java reasoner for SHIQ extended with the DL-safe fragment of SWRL. It implements a resolution-based decision procedure for general TBoxes (subsumption, satisfiability, classification) and ABoxes (retrieval, conjunctive query answering). It comes with its own, Java-based interface, and supports the DIG interface.

47. Ontology Reasoning - Majid Sazvar - 2010 Overview of DL Reasoners (5) • MSPASS • is a free open-source C reasoner for numerous description logics. It implements a resolution-based decision procedure for extensions of ALB (which is ALC with inverse and Boolean operators on roles) with general TBoxes (satisfiability, subsumption) and ABoxes (instance checking, retrieval). It is an extension of the theorem prover SPASS, and can thus also be used to reason about arbitrary first-order statements. • AllegroGraph • is a disk-based RDF Database, that aims at providing a solid storage layer for powerful geotemporal reasoning, social network analytics and ontology modeling capabilities for today's Semantic Technology applications.

48. Ontology Reasoning - Majid Sazvar - 2010 Overview of DL Reasoners (6) • Jena • an open source semantic web framework for Java which includes a number of different semantic reasoning modules. • SweetRules • an integrated set of tools for Semantic web rules and ontologies.

49. Ontology Reasoning - Majid Sazvar - 2010 Overview of DL Reasoners (7) • OWLIM • a high-performance semantic repository developed in Java and available in two versions: free SwiftOWLIM and commercial BigOWLIM. Supports a subset of OWL-Lite semantics, which can be configured through rule-set definition and selection. • Pellet • is a free open-source Java-based reasoner for SROIQ with simple datatypes (i.e., for OWL 1.1). It implements a tableau-based decision procedure for general TBoxes (subsumption, satisfiability, classification) and ABoxes (retrieval, conjunctive query answering). It supports the OWL-API, the DIG interface, and the Jena interface and comes with numerous other features.

50. Ontology Reasoning - Majid Sazvar - 2010 Overview of DL Reasoners (8) • Owlgres • is a reasoner for the DL Lite sub-language of OWL 2. It is based on the PostgreSQL relational DBMS. Besides the open source license AGPL, Owlgres can also be obtained via a commercial licensing model. • QuOnto • is a free (for non-commercial use) Java-based reasoner for DL-lite with GCIs. It implements a query rewriting algorithm for both consistency checking and query answering for unions of conjunctive queries over DL-Lite knowledge bases, whose ABox is managed through relational database technology. It comes with its own Java-based interface.