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The Model Evolution Calculus and an Application to Ontological Reasoning Peter Baumgartner Max-Planck-Institute for Informatics Saarbrücken,Germany Joint work with Cesare Tinelli, U Iowa Background – Instance Based Methods • Model Evolution is related to Instance Based Methods – Ordered Semantic Hyper Linking [Plaisted et al] – Primal Partial Instantiation [Hooker et al] – Disconnection Method [Billon], DCTP [Letz&Stenz] – Inst-Gen [Ganzinger&Korovin] – First-Order DPLL [B.] • Principle: Reduce proof search in first-order (clausal) logic to propositional logic in an „intelligent“ way • Different to Resolution, Model Elimination,… (Pro‘s and Con‘s) A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 2 Background - DPLL • The best modern SAT solvers (satz, MiniSat, zChaff, Berkmin,…) are based on the Davis-Putnam-Logemann-Loveland procedure [DPLL 1960-1963] • Can DPLL be lifted to the first-order level? Can we combine – successful SAT techniques (unit propagation, backjumping, learning,…) – successful first-order techniques? (unification, subsumption, ...)? • Model Evolution (ME) and its predecessor First-Order DPLL do so • ME different to Resolution, Tableaux and Model Elimination but related to "Instance Based Methods" A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 3 DPLL procedure Input: Propositional clause set Output: Model or „unsatisfiable” Algorithm components: - Propositional semantic tree enumerates interpretations - Simplification - Split - Backtracking Lifting to first-order logic? A B C : A : B : C ? f A, Bg j= : A _ : B _ C _ D, : : : No, split on C: f A, B, Cg j= : A _ : B _ C _ D, : : : A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 4 Model Evolution as First-Order DPLL Lifing of semantic tree data structure and derivation rules to first-order Input: First-order clause set Output: Model or „unsatisfiable” if termination Algorithm components: - First-order semantic tree enumerates interpretations - Simplification - Split - Backtracking v is a "parameter" P(a, v) : P(a, b) : P(a, v) P(a, b) ? f P(a, v), : P(a, b)g j= Q(x, y) _ P(x, y) Interpretation induced by a branch? A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 5 Interpretation Induced by a Branch A branch literal specifies the truth value of its ground instances unless there is a more specific branch literal with opposite sign : v Branch: f : v, P(a, z), : P(a, b)g P(a, v) Induced Interpretation true: P(a, a) false: P(a, b), Q(a, b) : P(a, b) How to determine Split literal? Calculus? Q(a, b) : P(a, v) P(a, b) : Q(a, b) ? f : v, P(a, v), : P(a, b)g j= Q(x, y) _ P(x, y) No, because ) f : v, P(a, v), : P(a, b)g 6 j= Q(a, b) _ P(a, b) Split with Q(a, b) to satisfy P(a, b) _ Q(a, b) A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 6 Model Evolution Calculus • Branches and clause sets may shrink as the derivation proceeds • Such dynamics is best modeled with a sequent style calculus: ¤ ` © Current Clause Set Context: A set of literals (the „current branch“) • Derivation Rules – To simplify the clause set , to simplify the context – Splitting – Close A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 7 Derivation Rules – Simplified (1) Split ¤ ` ©, C _ L ¤, L¾ ` ©, C _ L ¤, L¾ ` ©, C _ L if 1. ¾ is a simultaneous mgu of C _ L against ¤, 2. neither L¾ nor L¾ is contained in ¤, and 3. L¾ contains no variables (parameters OK) ¤: P(u, u) Q(v, b) C _ L : : P(x, y) _ : Q(a, z) ¾ = f x !7 u, y 7 ! u, v7 ! a, z !7 b g (C _ L)¾ : : P(x, x) _ : Q(a, b) 2. violated 2. satisfied L¾ = : Q(a, b) is admissible for Split A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 8 Derivation Rules – Simplified (2) Close ¤ ¤ ` ` ©, C ? if 1. © 6 = ; or C 6 = ? 2. there is a simultaneous mgu ¾ of C against ¤ such that ¤ contains the complement of each literal of C¾ ¤: P(u, u) Q(a, b) C : : P(x, y) _ : Q(a, z) ¾= f x 7 ! u, y 7 ! u, z 7 ! bg C¾ : : P(x, x) _ : Q(a, b) 2. satisfied 2. satisfied Close is applicable A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 9 Derivation Rules – Simplification Rules (1) Propositional level: Subsume ¤, L ` ¤, L ` ©, L _ C © First-order level ¼ unit subsumption: - All variables in context literal L must be universally quantified - Replace equality by matching A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 10 Derivation Rules – Simplification Rules (2) Propositional level: Resolve ¤, L ` ¤, L ` ©, L _ C ©, C First-order level ¼ restricted unit resolution - All variables in context literal L must be universally quantified - Replace equality by unification - The unifier must not modify C A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 11 Derivation Rules – Simplification Rules (3) Compact ¤, K, L ` ¤, K ` © © if 1. all variables in K are universally quantified 2. K¾ = L, for some substitution ¾ A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 12 Derivations and Completeness : v ` Input clause set Fairness Closed tree or open limit tree, with some branch satisfying: ¤ 1 ` ©1 ¤ 2 ` ©2 closed ¤ n ` ©n ¤ 1 := ©1 := S S T i¸ 0 i¸ 0 T j¸ i ¤j j¸ i ©j 1. Close not applicable to any ¤ i 2. For all C 2 ©1 and subst. ° , ``if for some i, ¤ i 6 j= C° then there is j ¸ i such that ¤ j j= C° (Use Split to achieve this) Complet eness Suppose a fair derivation that is not a closed tree Show that ¤ 1 j= ©1 A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 13 Implementation: Darwin • „Serious“ Implementation Part of Master Thesis, will be continued in Ph.D. project • (Intended) Applications • detecting dependent variables in CSP problems • strong equivalence of logic programs • Bernays-Schoenfinkel fragment of autoepistemic logic • Currently extended: • Lemma learning • Equality inference rules • Written in OCaml, 14K LOC • User manual, proof tree output (GraphViz) A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 14 Darwin Performance Results of ATP system competition at IJCAR 2004 MIX: Clause logic with Equality EPR: function free clause logic (without Equality) A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 15 Application: Ontological Reasoning • Automated reasoning on formal ontologies is of growing interest • Description logics are a widely used logical formalism, e.g. OWL 9 partOf AuthoredChapter 9 hasPart AuthoredChapter v CollectionBook v CollectionBook 9 partOf : CollectionBook 9 hasPart : AuthoredChapter • Highly optimized reasoners for decidable DLs can cope with realistically sized ontologies (FaCT, Racer) • Can one also use Darwin/off-the-shelf provers? • And why? A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 16 Why? Going Beyond Description Logics DL + Rules: 9 partOf AuthoredChapter 9 hasPart AuthoredChapter CollectionBook v v ::: EditorIsAuthor(x) à CollectionBook 9 partOf : CollectionBook 9 hasPart : AuthoredChapter CollectionBook(x) ^ hasPart(x, y) ^ hasAuthor(y, z) ^ hasEditor(x, z) • Rules cause undecidability • Cannot use DL reasoner • Translate to first-order logic and use theorem prover • How? (Naive approach fails) A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 17 How? Our Approach DL First-Order Logic Facts Rules Query Clause Normalform Equality Needed for efficiency / termination Blocking Clause Logic Darwin Model Evolution 8 (L1 _ ¢¢¢_ Ln ), where Li (negated) atom Result - "Unsatisfiable" - "Model" A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning KRHyper Tableaux 18 Equality Work in collaboration with Master's student Equality comes in, e.g., in the translation of – nominals ("oneOf") WhiteLoire v 8 madeFromGrape : Sauvignon t Chenin t Pinot WhiteLoire(x) ^ madeFromGrape(x, y) ) y = Sauvignon _ y = Chenin _ y = Pinot – cardinality restrictions Cation v · 4 hasCharge Cation(x) ^ hasCharge(x, x1) ^ ¢¢¢^ hasCharge(x, x5) ) x1 = x2 _ x1 = x3 _ ¢¢¢_ x4 = x5 -> Need an (efficient) way to treat equality A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 19 Equality • Options: equality axioms - builtin in prover - by transformation • Our transformation: Given clause: c= d à Our trafo: c= d à f(x) = b ^ x = a + ref, sym, trans [Brand 1975]: c= d d= c + ref à à f(a) = b f(x) = y ^ x = a ^ y = b f(x) = y ^ x = a ^ y = b - Brand's transformation is theoretically more attractive - But advantages do not apply for "typical" ontologies - In practice, our transformation works much better A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 20 Blocking • Problem: Termination in case of satisfiable input. Caused by certain DL language constructs and cyclic definitions: 9 hasAuthor AuthoredChapter 9 hasPart CollectionBook • Solution: Idea: Re-use old individual to satisfy -quantifier. Technical: encode search for finite domain model in clause set: aC(hP(hA(a))) ^ hP(hA(a)) = a aC(a) ^ dom(a) Try this first aC(hP(hA(a))) ^ dom(hP(hA(a))) • Issue: Search space reduction: don't speculate all possible equalities A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 21 o Experimental Evaluation – OWL Test Cases from W3C Consist ent 56 problems Inconsist ent 72 problems Ent ailment 57 problems Darwin +blocking Darwin - blocking KRHyper +blocking KRHyper - blocking 89% 7% 86% 75% 92% 94% 89% 94% 89% 93% 93% 93% Darwin [ KRHyper Hoolet (Vampire) Surnia (Otter) Euler ("Prover") Fact (DL) Pellet (DL) OWLP (DL) Cerebra (DL) FOWL (OWL) ConsVISor (OWL-full) 93% 78% 0% 42% 96% 50% 90% 53% 77% 94% 94% 0% 98% 85% 98% 26% 59% 4% 65% 93% 72% 13% 100% 7% 86% 53% 61% 32% - Syst em A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 22 Conclusions • Objective: "robust" reasoning support beyond description logics: – Equality treatment – Blocking (decides standard services for cyclic ALC TBoxes) – It's not only "strategy hacking" – need theoretical results – Competitive with DL systems on common domain • "Rules" not benchmarked (no benchmarks available), but they turned out to be very useful in own application projects: – Reputational risk management – Document management (E-Learning) – Upper ontology for computational linguistic application • Nonmonotonic negation of KRHyper very useful How to integrate it in Model Evolution? A First-Order Davis-Putnam Procedure and its Application to Ontological Reasoning 23