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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
12
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