A Preferential Tableau for Circumscriptive ALCO RR 2009 Stephan Grimm Pascal Hitzler

 Circumscriptive Description Logics (DLs)  Preferential Tableau  Example of calculating preferred models  Conclusion Outline 2

Circumscriptive DLs  DLs with circumscription Circumscription (minimising extensions of predicates) [McCarthy] Combination with DLs (minimising extensions of concepts/roles) [Bonatti,Lutz,Wolter] No specific reasoning algorithms exist  Minimisation of predicates Keep extensions of selected predicates as small as possible Allows for nonmonotonic reasoning and defeasible inference  Appearance of circumscriptive DLs Circumscription Pattern CP for a knowledge base KB CP = (M, V, F)circ CP (KB)

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

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

Example for Circumscriptive Reasoning  Nonmonotonic reasoning example Default behaviour due to concept minimisation

 Tableau to construct preferred models Formalism considered: parallel concept circumscription in general ALCO knowledge bases  Extension of classical tableaux Additional check for preference clashes A tableau branch contains a preference clash if it represents non- preferred models  Implementation of preference clash check Reduce check to classical reasoning problem (KB satisfiability in ALCO) Construct temporary knowledge base KB´ out of original KB and assertions in tableau branch B, such that Models of KB´ are preferred over those represented by B Preferential Tableau 7

Algorithm for Constructing KB´  Constructing KB´ for preference clash check

Example Preferential Tableau  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} ⊔ AbEUCity } KB ⊨ EUCity ⊑  cur.{Euro} ? x : EUCity x :  cur.  {Euro} x:  EUCity x :  cur.{Euro} x : AbEUCity   ⇜  CP = ( M={AbEUCity}, F= , V={EUCity} )

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

 Results Tableau calculus for circumscriptive ALCO o Proofed sound and complete o Extension of classical DL tableau by preference clash Criterion for preference clash check on tableau branches o Can be applied to open and closed tableau branches o Can be integrated into existing (optimised) tableau implementations  Future work Extension to more expressive DLs Integration into open-source tableau implementations for testing Optimisations to cope with high complexity Conclusion 11

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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 Assumption: Pizzas with non-chili toppings only are typically non-spicy

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

(Non-)Monotonicity of Reasoning  Agent collects knowledge in the web  Reasoning allows to derive implicit knowledge  Reasoning is monotonic if the derived knowledge monotonically grows t KB ⊨ { f a,f b } KB  {f c } ⊨ {f a,f b,f c,f d } KB  {f c,f d } ⊨ {f a,f b,f c,f d } Semantic Web Agent KB  {f a,f b }  {f c } ... Agent KB ⊨ {f a, f b, f c, f x, f y,... } non-monotonic KB  {f c,f d,f e } ⊨ {f c,f d }...

Non-Monotonicity for Common-Sense  Situations of incomplete knowledge  Pragmatic conclusions by default assumptions  Admit the jumping to conclusions Agent KB = { Pizza(vesufo), hasTopping(vesufo,salami) } KB ⊨  SpicyDish(vesufo) ? KB ⊭ { SpicyDish(vesufo), hasTopping(vesufo,chili) }  KB ⊨  SpicyDish(vesufo) KB  {  x : hasTopping(x,salami)  SpicyDish(x) } ⊨ SpicyDish(vesufo)

Interpretations and Models in DL  I = (  I, · I ) Concept Student Course Individual susan cs324 Role susan cs324 enrolled II susan cs324 enrolled Course I Student I I is a model of KB if it satisfies ist axioms Student Graduate susan Student enrolled susan cs324

Concept Minimisation  Trade models for conclusions the less models the more conclusion nonmonotonicity: regain models by learning new knowledge  Example models of KB...

Example Preferential Tableau  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} ⊔ AbEUCity, EUCity(Berlin) } KB ⊨  cur.{Euro}(Berlin) ? Berlin : EUCity Berlin :  cur.  {Euro} Berlin :  EUCity Berlin :  cur.{Euro} Berlin : AbEUCity   ⇜  CP = ( M={AbEUCity}, F= , V={EUCity} )

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