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1 Evaluating Reasoning Systems: Ontology Languages Michael Grunginger, U. Toronto Conrad Bock, U.S. NIST February 22 nd, 2007

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2 Ontologies An ontology includes a declaration of terminology together with a specification of the meaning (semantics) of the terms. Ontologies differ on the language used for the specification of meaning

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3 Semantic Spectrum Common Logic MeSH RuleML, SWRLRDF Description Logics (OWL) ISO WordNet Dublin Core IUPAC Terms Topic Maps SyntacticMetaDataFormal OntologiesThesauri FOL

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4 Languages What do we mean by ontology language? –Syntax logical lexicon (e.g. connectives, quantifiers) nonlogical lexicon (vocabulary for concepts) grammar –Semantics (model theory) –Proof theory Syntactic methods that can be used to infer new sentences

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5 First Order Logic Syntax –Logical lexicon : forall, exists, and, or, not, if, iff Proof theory (inference rules) –modus ponens Given (if P Q), P, infer Q –resolution Given (or (not P) Q), (or P R), infer (or Q R)

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6 Semantics A minimal formal description of those aspects of a world which is sufficient to establish the truth or falsity of any expression Interpretations –Mapping from expressions (syntax) to elements in the interpretation Names denote objects in the domain Predicates have associated extensions –Since the syntax is often specified recursively, interpretations are also often compositional. Some way of assigning a truth value to atomic expressions in terms of the semantic values of its constituent names and predicates Connectives are used to compose expressions, and we specify rules to define the truth of expressions in terms of the truth of its sub- expressions.

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7 Key Semantic Notions Satisfiability –A sentence true in some interpretation Validity –A sentence is satisfied in all interpretations Entailment –A sentence is satisfied in all models (satisfying interpretations) of a theory (set of sentences). Logical equivalence –Two theories have the same set of models Logical independence –A sentence is independent of a theory if there exists a model of the theory that satisfies the sentence and a model of the theory that falsifies the sentence.

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8 Examples from FOL The theory (forall (?x ?y ?z) (if(and(before ?x ?y) (before ?y ?z)) (before ?x ?z))) (forall (?x ?y) (if (before ?x ?y) (not (before ?y ?x)))) is satisfiable -- any partial ordering is a satisfying interpretation.

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9 Examples from FOL The sentence (forall (?x) (not (before ?x ?x))) is entailed by the theory. The sentence (forall (?x ?y) (or(before ?x ?y) (before ?y ?x) (= ?x ?y))) is independent of the theory -- it is satisfied by any linear ordering but not by an arbitrary partial ordering.

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10 Definability We identify the intended semantics of a concept with a class of models (satisfying interpretations). A class of interpretations M is definable in a language if there is a set of sentences T in the language such that all and only interpretations in M are models of T. –equivalently, we say that T axiomatizes M

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11 First-Order Ontologies Time –Hayes 1996 PSL (Process Specification Language) –Gruninger 2003 Geometry –Hilbert 1906, Tarski 1954 Mereotopology –Asher & Vieu 1995, Pratt 2004, Cohn 2005 Liquids –Hayes 85 SUMO (Suggested Upper Merged Ontology)

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12 Expressiveness Definability gives us a way of comparing languages: –A language L 1 is as expressive as a language L 2 if for every sentence S in L 2 there exists a sentence P in L 1 such that the set of models of S is equal to the set of models of P. The languages in the Semantic Spectrum are ordered by expressiveness -- languages to the right are more expressive than languages to the left. Why dont we just use the most expressive language possible?

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13 Decidability A language is decidable if there is an algorithm for checking for the validity of all sentences in the language. –This is equivalent to saying that there is an effective way of listing the set of valid formulas in the language and the set of invalid formulas. A language is semidecidable if there exists a procedure that eventually terminates if the formula is valid. –This is equivalent to saying that there is an effective way of listing the set of valid formulas in the language

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14 Decidability and FOL First-order logic is only semidecidable –There do exist first-order ontologies that are decidable Hilberts and Tarskis axiomatizations of geometry are decidable Real arithmetic is decidable Can we identify decidable restrictions of first- order logic that are still expressive enough for realistic problems?

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15 Restrictions of First-Order Logic Restrictions on quantifiers –Existential –Universal Clauses Horn clauses Datalog –Universal-existential Restrictions on predicates –Monadic FOL –Description logics

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16 Examples from FOL The sentences (forall (?x ?y ?z) (if(and(before ?x ?y) (before ?y ?z)) (before ?x ?z))) (forall (?x ?y) (if (before ?x ?y) (not (before ?y ?x)))) are Horn clauses without function symbols (i.e. Datalog). The sentence (forall (?x ?y) (or(before ?x ?y) (before ?y ?x) (= ?x ?y))) is a clause, but it is not Horn.

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17 Rules Syntactically, SWRL (Semantic Web Rules Language) is restricted to Datalog clauses. –There are extensions based on RuleML that include arbitrary clauses.

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18 Monadic FOL In monadic FOL, all predicates are unary. Taxonomies are definable in monadic FOL (forall (?x) (if(human ?x) (mammal ?x))) Monadic FOL is decidable.

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19 Description Logics The basic elements of any description logic are concepts (unary relations to represent classes) and roles (binary relations between elements of classes). Different description logics are characterized by the constructors, which can be used to specify arbitrary concept and role expressions from atomic concepts and roles.

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20 Constructors The following constructors specify the description logic AL

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21 Constructors OWL-Lite corresponds to AL OWL-DL includes all of the following constructors:

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22 Description Logics and FOL Description logics are less expressive than first- order logic Menzel and Hayes provide an embedding of OWL-DL into Common Logic –They specify a set of axioms in Common Logic whose interpretations provably correspond in a meaning- preserving way to the intended interpretations of the source languages relative to their semantics.

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23 Reified First-Order Logics Languages whose syntax extends that of first-order logic, but whose model theory is still first-order. Such languages are often referred to as reified logics, since the extension allows predicates and function symbols to be arguments of predicates and function symbols, so that in some sense there are relations and functions that are elements in the domain. Examples: –Common Logic –RDF –Hilog

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24 Common Logic The central intuition for the syntax of Common Logic is that all entities (individuals, properties, and relations alike) are first-class logical citizens that jointly constitute a single domain of quantification. Hence, such entities can themselves have properties, stand in relations, and serve as potential objects of reference. The resulting syntactic freedom allows a wide variety of alternative first-order axiomatic styles to co-exist within a common syntactic framework, with their meanings related by axioms, all expressed in a single uniform language. Common Logic without sequence variables is semantically equivalent to first-order logic.

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25 Limitations of First-Order Logic There are concepts that are not first-order definable: –Transitive closure –Connected graphs –Finite state automata

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26 Beyond First-Order Logic Transitive closure logic –extension of first-order logic that is closed under the transitive closure of all definable relations. Monadic second-order logic –Quantification over unary relations –Definable concepts: Connected graphs Finite state automata Second-order logic –Quantification over all possible relations –Definable concepts: Standard model of arithmetic Infinitary logic –extension of first-order logic that is closed under conjunction of arbitrary (possibly infinite) sets of formulae. –Common Logic with sequence variables is semantically equivalent to a weak infinitary logic

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27 Descriptive Complexity Theory Given a complexity class C, is there a logic L such that the queries definable in L are precisely the queries in C ? Each of the various extensions to first-order logic on the earlier slides each captures a different complexity class.

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28 Beyond First-Order Logic If there are concepts that are not definable in first- order logic, why not use these more expressive languages?

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29 Soundness Soundness (of a logic): A logic is sound if for any sentence S and theory T, if S is provable from T by some inference procedure, then S is satisfied by all models of T. – Equivalently, a logic is sound if for any sentence S and theory T, if there exists a model of T that satisfies S, then S is consistent with T

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30 Completeness Completeness (of a logic): A logic is complete if for any sentence S and theory T, if S is satisfied by all models of T, then S is provable from T by some finitary inference procedure –Equivalently, a logic is complete if for any sentence S and theory T, if S is consistent with T then there exists a model of T that satisfies S.

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31 Incompleteness First-order logic is sound and complete. The languages that extend FOL on the earlier slide are sound but not complete.

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32 Modal Logics A modal logic is a syntactic and semantic extension of propositional or first-order logic with new operators ( and ) on sentences. These operators have various intuitive interpretations: is necessity and is possibility; is knowledge or belief; is provability and is consistency. There are also modal operators for temporal concepts (e.g. Until, Eventually)

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33 Applications of Modal Logics Modal logics are used to analyze ontologies –Ontoclean Ontologies written using modal logics –DOLCE

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34 Monotonicity The classical logics we have seen in this talk are monotonic – any inferences drawn from a theory are preserved by any set of sentences containing the theory. However, many scenarios in commonsense reasoning do not possess this property, for example, when some inferences depend on the failure of other inferences. –If the particular statements that failed to be proved are in a set of sentences that contains the theory, the original inferences will not hold.

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35 Nonmonotonic Logics Semantics of logic programs Reiters default logic Model preference defaults Circumscription –Relationship to closed world assumptions and semantics of logic programs Autoepistemic logic

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36 Summary We have presented a taxonomy of representation languages, ordered by expressiveness. We use the following characteristics to describe each language in the taxonomy: model-theoretic properties soundness and completeness decidability / complexity monotonicity

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37 Acknowledgements Work funded under the U.S. NSAs High Performance Evidentiary Reasoning Program. Full report: Evaluating Reasoning Systems, Bock, Gruninger, Libes, Lubell, Subrahmanian, 2006, A review and categorization of reasoning systems. Covers: representation languages, inference procedures, user and software interfaces, with introduction to information metrology, model theory, and inference.

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