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
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)
comparing interpretations by their extensions for minimized predicates
 models of circCP(KB) are <CP-minimal models of KB,
i.e. the preferred models of KB w.r.t. CP.
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
Subsumption
C ⊑ D holds w.r.t. circCP(KB) if CI  DI holds
for all models I of circCP(KB)
o
Entailment
circCP(KB) ⊨ C(a) holds if a  CI holds
for all models I of circCP(KB)
Example for Circumscriptive Reasoning
 Nonmonotonic reasoning example
• Default behaviour due to concept minimisation
Preferential Tableau
 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 nonpreferred models
 Implementation of preference clash check
• Reduce check to classical reasoning problem (KB satisfiability in
•
•
7
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
Algorithm for Constructing KB´
 Constructing KB´ for preference clash check
Example Preferential Tableau
KB = { EUCity ⊑ cur.{Euro} ⊔ AbEUCity }
CP = ( M={AbEUCity}, F=, V={EUCity} )
KB ⊨ EUCity ⊑ cur.{Euro} ?
x : EUCity
x : cur.{Euro}
x:  EUCity
x : cur.{Euro}
x : 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
Example Preference Clash Detection
 collect positive assertions to
minimised concepts
x
ℬ
x : EUCity
x : cur.{Euro}
AbEUCity
 freeze extensions of minimised
concepts
KB’ = KB { AbEUCity ⊑ {x} }
 ensure minimality
condition in KB’
KB’  ( AbEUCity ⊓ {x}) ()
new individual 
x: AbEUCity
KB ’ = { EUCity ⊑ cur.{Euro} ⊔ AbEUCity,
AbEUCity ⊑ {x} ,
( AbEUCity ⊓ {x}) ()
}
 test KB’ for consistency
KB’ is consistent  ℬ has a preference clash
consistent
Conclusion
 Results
• Tableau calculus for circumscriptive ALCO
Proofed sound and complete
o Extension of classical DL tableau by preference clash
o
• Criterion for preference clash check on tableau branches
Can be applied to open and closed tableau branches
o Can be integrated into existing (optimised) tableau implementations
o
 Future work
• Extension to more expressive DLs
• Integration into open-source tableau implementations for testing
• Optimisations to cope with high complexity
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
KB  {fa,fb}  {fc}  . . .
Agent
Semantic
Web
 Reasoning allows to derive implicit knowledge
Agent
KB ⊨ {fa, fb, fc, fx, fy, ... }
 Reasoning is monotonic if the derived knowledge
monotonically grows
non-monotonic
KB
⊨ {fa,fb}
KB  {fc}
⊨ {fa,fb,fc,fd}
KB  {fc,fd}
⊨ {fa,fb,fc,fd}
KB  {fc,fd,fe} t
⊨ {fc,fd}
...
Non-Monotonicity for Common-Sense
 Situations of incomplete knowledge
KB = {Pizza(vesufo), hasTopping(vesufo,salami)}
?
KB
⊨
 SpicyDish(vesufo)
Agent
 Pragmatic conclusions by default assumptions
KB ⊭ {SpicyDish(vesufo), hasTopping(vesufo,chili)}
 Admit the jumping to conclusions
 KB ⊨  SpicyDish(vesufo)
KB  {x : hasTopping(x,salami)  SpicyDish(x)}
⊨ SpicyDish(vesufo)
Interpretations and Models in DL

I = (I, · I )
I
Individual
cs324
susan
Student I
Concept
Student
susan
Course
Role
Course I
cs324 enrolled susan

cs324
I is a model of KB if it satisfies ist axioms
Student
Graduate
Student
susan
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
CP = ( M={AbEUCity}, F=, V={EUCity} )
KB ⊨ cur.{Euro}(Berlin) ?
Berlin : EUCity
Berlin : cur.{Euro}
Berlin :  EUCity
Berlin : AbEUCity
Berlin : cur.{Euro}
⇜
 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)
}
Example Preference Clash Detection
 collect positive assertions to
minimised concepts
ℬ
Berlin : EUCity
Berlin : cur.{Euro}
AbEUCity
Berlin
 freeze extensions of minimised
concepts
KB’ = KB { AbEUCity ⊑ {Berlin} }
 ensure minimality
condition in KB’
KB’  ( AbEUCity ⊓ {Berlin}) ()
new individual 
Berlin : AbEUCity
KB ’ = { EUCity ⊑ cur.{Euro} ⊔ AbEUCity,
EUCity(Berlin) ,
AbEUCity ⊑ {Berlin} ,
( AbEUCity ⊓ {Berlin}) ()
}
 test KB’ for consistency
KB’ is consistent  ℬ has a preference clash
consistent