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A Preferential Tableau for Circumscriptive ALCO

A Preferential Tableau for Circumscriptive ALCO. RR 2009 Stephan Grimm Pascal Hitzler. Outline. Circumscriptive Description Logics (DLs) Preferential Tableau Example of calculating preferred models Conclusion. Circumscriptive DLs. DLs with circumscription

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A Preferential Tableau for Circumscriptive ALCO

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  1. A Preferential Tableau forCircumscriptive ALCO RR 2009 Stephan Grimm Pascal Hitzler

  2. Outline • Circumscriptive Description Logics (DLs) • Preferential Tableau • Exampleofcalculatingpreferredmodels • Conclusion

  3. Circumscriptive DLs • DLs withcircumscription • Circumscription (minimisingextensionsofpredicates) [McCarthy] • Combinationwith DLs (minimisingextensionsofconcepts/roles) [Bonatti,Lutz,Wolter] • Nospecificreasoningalgorithmsexist • Minimisationofpredicates • Keep extensionsofselectedpredicatesassmallaspossible • Allowsfornonmonotonicreasoninganddefeasibleinference • Appearanceofcircumscriptive DLs • CircumscriptionPattern CPfor a knowledgebaseKBCP = (M, V, F) circCP(KB)

  4. comparing interpretations by their extensions for minimized predicates Semantics of Circumscriptive DL • Preference relation <CP on Interpretations I = (I, I) • models of circCP(KB) are <CP-minimal models of KB,i.e. the preferred models of KB w.r.t. CP.

  5. Reasoning with Circumscribed KBs • Various forms of defeasible reasoning • defined with respect to (preferred) models of circCP(KB) • Concept Satisfiability A concept C is satisfiable w.r.t. circCP(KB)if some model of circCP(KB) satisfies CI • Subsumption C⊑ D holds w.r.t. circCP(KB) if CIDI holdsfor all models I of circCP(KB) • Entailment circCP(KB) ⊨C(a) holds if a  CI holdsfor all models I of circCP(KB)

  6. ExampleforCircumscriptiveReasoning • Nonmonotonicreasoningexample • Default behaviour due toconceptminimisation

  7. Preferential Tableau • Tableau toconstructpreferredmodels • Formalismconsidered: parallel conceptcircumscription in generalALCO knowledgebases • Extension ofclassicaltableaux • Additional check forpreferenceclashes • A tableaubranchcontains a preferenceclashifitrepresents non-preferredmodels • Implementationofpreferenceclash check • Reduce check toclassicalreasoningproblem (KB satisfiability in ALCO) • Constructtemporaryknowledgebase KB´ out of original KB andassertions in tableaubranch B, such that • Models of KB´ arepreferredoverthoserepresentedby B

  8. Algorithm for Constructing KB´ • Constructing KB´ forpreferenceclash check

  9. CP=( M={AbEUCity}, F=, V={EUCity} ) x :EUCity x :cur.{Euro} x:  EUCity x : AbEUCity  ⇜ x :cur.{Euro}   Example Preferential Tableau KB = { EUCity ⊑cur.{Euro} ⊔AbEUCity } • tableaux algorithm constructs a model for KB • tableaux branches represent (potential) models of KB • clashes represent contradictions in KB • eliminate non-preferred models by introducing additional preference clashes • preference clashes indicate non-minimality KB ⊨ EUCity ⊑cur.{Euro} ?

  10. ℬ x :EUCity x :cur.{Euro}  KB ’ = { EUCity ⊑cur.{Euro} ⊔AbEUCity,AbEUCity ⊑ {x} , ( AbEUCity ⊓{x}) () } x: AbEUCity AbEUCity x consistent Example Preference Clash Detection • collect positive assertions to minimised concepts • freeze extensions of minimised conceptsKB’ = KB { AbEUCity ⊑ {x} } • ensure minimalitycondition in KB’ KB’ ( AbEUCity ⊓{x}) ()new individual • test KB’ for consistencyKB’ is consistent  ℬ has a preference clash

  11. Conclusion • Results • Tableau calculusforcircumscriptive ALCO • Proofedsoundandcomplete • Extension ofclassical DL tableaubypreferenceclash • Criterionforpreferenceclash check on tableaubranches • Can beappliedto open andclosedtableaubranches • Can beintegratedintoexisting (optimised) tableauimplementations • Future work • Extension tomore expressive DLs • Integration into open-sourcetableauimplementationsfortesting • Optimisationstocopewithhighcomplexity

  12. Assumption: Pizzas with non-chili toppings only are typically non-spicy Defeasible Inference • Inferences in OWL are universally true • based on description logics (monotonic) • conclusions only drawn from ensured evidence (OWA) • Defeasible Inferences are based on common-sense conjectures • conclusions drawn based on assumptions about what typically holds • retracted in the presence of counter-evidence • Example

  13. Circumscriptive DLs • DLs with circumscription • minimising extensions of DL-predicates [Bonatti,Lutz] • Circumscription Pattern CP for a knowledge base KB • Model-theoretic semantics • Preference relation <CP on Interpretations • only models minimal w.r.t. <CP remain models of

  14. KB  {fa,fb}  {fc}  . . . Semantic Web KB ⊨ {fa, fb, fc, fx, fy, ... } KB ⊨{fa,fb} KB  {fc,fd} ⊨{fa,fb,fc,fd} KB  {fc} ⊨{fa,fb,fc,fd} non-monotonic Agent Agent KB  {fc,fd,fe} ⊨{fc,fd} . . . (Non-)MonotonicityofReasoning • Agent collects knowledge in the web • Reasoning allows to derive implicit knowledge • Reasoning is monotonic if the derived knowledge monotonically grows t

  15. KB = {Pizza(vesufo), hasTopping(vesufo,salami)} ? KB ⊨ SpicyDish(vesufo) KB ⊭ {SpicyDish(vesufo), hasTopping(vesufo,chili)}  KB ⊨ SpicyDish(vesufo) Agent Non-Monotonicityfor Common-Sense • Situationsofincompleteknowledge • Pragmatic conclusionsbydefaultassumptions • Admitthejumpingtoconclusions KB {x : hasTopping(x,salami)  SpicyDish(x)} ⊨ SpicyDish(vesufo)

  16. I Concept Individual Course Student susan cs324 enrolled susan Student I cs324 Role enrolled cs324 susan CourseI Student Student Graduate enrolled susan susan cs324 Interpretations and Models in DL • I= (I,·I) • Iis a model ofKBif it satisfies ist axioms

  17. . . . models ofKB Concept Minimisation • Trade models for conclusions • the less models the more conclusion • nonmonotonicity: regain models by learning new knowledge • Example

  18. CP=( M={AbEUCity}, F=, V={EUCity} ) Berlin :EUCity Berlin :cur.{Euro} Berlin :  EUCity Berlin : AbEUCity  ⇜ Berlin :cur.{Euro}   Example Preferential Tableau KB = { EUCity ⊑cur.{Euro} ⊔AbEUCity , EUCity(Berlin)} • tableaux algorithm constructs a model for KB • tableaux branches represent (potential) models of KB • clashes represent contradictions in KB • eliminate non-preferred models by introducing additional preference clashes • preference clashes indicate non-minimality KB ⊨ cur.{Euro}(Berlin) ?

  19. ℬ Berlin :EUCity Berlin :cur.{Euro}  KB ’ = { EUCity ⊑cur.{Euro} ⊔AbEUCity, EUCity(Berlin) ,AbEUCity ⊑ {Berlin} , ( AbEUCity ⊓{Berlin}) () } Berlin : AbEUCity AbEUCity Berlin consistent Example Preference Clash Detection • collect positive assertions to minimised concepts • freeze extensions of minimised conceptsKB’ = KB { AbEUCity ⊑ {Berlin} } • ensure minimalitycondition in KB’ KB’ ( AbEUCity ⊓{Berlin}) ()new individual • test KB’ for consistencyKB’ is consistent  ℬ has a preference clash

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