A Preferential Tableau forCircumscriptive 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) circCP(KB)

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.

comparing interpretations by their extensions for minimized predicates

Reasoning with Circumscribed KBs

Various forms of defeasible reasoning

• defined with respect to (preferred) models of circCP(KB) o Concept Satisfiability

A concept C is satisfiable w.r.t. circCP(KB)if some model of circCP(KB) satisfies CI

o SubsumptionC ⊑ D holds w.r.t. circCP(KB) if CI DI holdsfor all models I of circCP(KB)

o EntailmentcircCP(KB) ⊨ C(a) holds if a CI holdsfor all models I of circCP(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 : EUCityx : 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 minimalitycondition in KB’ KB’ ( AbEUCity ⊓ {x}) ()

new individual

test KB’ for consistency

KB’ is consistent ℬ has a preference clash

xAbEUCity 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 completeo Extension of classical DL tableau by preference clash

• Criterion for preference clash check on tableau brancheso Can be applied to open and closed tableau brancheso 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

tKB⊨ {fa,fb}

KB {fc}

⊨ {fa,fb,fc,fd}

KB {fc,fd}

⊨ {fa,fb,fc,fd}

SemanticWeb

Agent

KB {fa,fb} {fc} . . .

AgentKB ⊨ {fa, fb, fc, fx, fy, ... }

non-monotonic

KB {fc,fd,fe}

⊨ {fc,fd}. . .

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

susancs324

Role

susancs324 enrolled

I

susancs324

enrolled

Course I

Student I

I is a model of KB if it satisfies ist axioms

Student Graduate susanStudent enrolled

susancs324

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 : EUCityBerlin : 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 minimalitycondition in KB’ KB’ ( AbEUCity ⊓ {Berlin}) ()

new individual

test KB’ for consistency

KB’ is consistent ℬ has a preference clash

BerlinAbEUCity Berlin : EUCity

Berlin : cur.{Euro}

Berlin : AbEUCity

ℬ

KB ’ = { EUCity ⊑ cur.{Euro} ⊔ AbEUCity, EUCity(Berlin) ,

AbEUCity ⊑ {Berlin} ,( AbEUCity ⊓ {Berlin}) () }

consistent