Event calculus Problem Event Calculus I Constraint Logic Fritz - PowerPoint PPT Presentation

Event calculus Hamm & Schlotterbeck The Frame Event calculus Problem Event Calculus I Constraint Logic Fritz Hamm ∗ Programming Event Calculus Fabian Schlotterbeck ∗∗ II Applications ∗ Seminar für Sprachwissenschaft Integrity ∗∗ Sonderforschungsbereich 833 constraints Universität Tübingen Feferman theories August 23, 2010 References

Situation Calculus I Event calculus Ontology : situation, fluents, actions Hamm & Schlotterbeck Basic predicates : Holds ( f , s ) , Result ( a , s ) The Frame Problem Event Calculus I Constraint Logic Programming Event Calculus II A Applications B Integrity s 0 : constraints C D Feferman table theories References On ( x , y ) , Clear ( x )

Situation Calculus II Event calculus Hamm & Σ Schlotterbeck (1) Holds ( On ( C , table ) , s 0 ) The Frame a. Problem Holds ( On ( B , C ) , s 0 ) b. Event Calculus I Holds ( On ( A , B ) , s 0 ) c. Constraint Logic Programming Holds ( On ( D , table ) , s 0 ) d. Event Calculus Holds ( Clear ( A ) , s 0 ) e. II Holds ( Clear ( D ) , s 0 ) f. Applications Holds ( Clear ( table ) , s 0 ) g. Integrity constraints Feferman theories References ¬ Holds ( Clear ( B ) , s 0 )

Situation Calculus III Event calculus Hamm & Schlotterbeck Move ( x , y ) The Frame Problem Event Calculus I Constraint Logic Programming Event Calculus II Applications B A Integrity constraints Result ( Move ( A , D ) , s 0 ) C D Feferman table theories References

Effect axioms I Event calculus Hamm & Schlotterbeck ∆ (2) The Frame ( Holds ( Clear ( x ) , s ) ∧ Holds ( Clear ( y ) , s ) a. Problem ∧ x � = y ∧ x � = table ) → Event Calculus I Holds ( On ( x , y ) , Result ( Move ( x , y ) , s )) Constraint Logic Programming b. ( Holds ( Clear ( x ) , s ) ∧ Holds ( Clear ( y ) , s ) Event Calculus II ∧ Holds ( On ( x , z ) , s ) ∧ y � = z ∧ x � = y )) Applications → Holds ( Clear ( z ) , Result ( Move ( x , y ) , s )) Integrity constraints ∆ ∧ Σ ⊢ (3) a. Feferman Holds ( On ( A , D ) , Result ( Move ( A , D ) , s 0 )) theories ∆ ∧ Σ ⊢ b. References Holds ( Clear ( B ) , Result ( Move ( A , D ) , s 0 ))

Effect axioms II Event calculus Hamm & Schlotterbeck The Frame Problem Event Calculus I Constraint Logic Programming ∆ ∧ Σ �⊢ Holds ( On ( B , C ) , Result ( Move ( A , D ) , s 0 )) (4) Event Calculus II Applications Integrity constraints Feferman theories References

The frame problem I Event calculus Hamm & Schlotterbeck The Frame Problem (5) Holds ( On ( v , w ) , s ) ∧ x � = v → Event Calculus I Holds ( On ( v , w ) , Result ( Move ( x , y ) , s )) Constraint Logic Programming Event Calculus II Applications Integrity constraints Holds ( On ( B , C ) , Result ( Move ( A , D ) , s 0 )) Feferman theories References

The frame problem II Event calculus Hamm & Schlotterbeck Holds ( Clear ( x ) , s ) ∧ x � = z → (6) Holds ( Clear ( x ) , Result ( Move ( y , z ) , s )) The Frame Problem Event Calculus I Constraint Logic Programming Event Calculus II Holds ( Colour ( x , c ) , Result ( Paint ( x , c ) , s )) (7) Applications Integrity constraints Feferman theories Holds ( Colour ( x , Red ) , s 0 ) (8) References

The frame problem III Event calculus Hamm & Schlotterbeck The Frame Problem Event Calculus I Constraint Logic ∆ ′ ∧ Σ ′ �⊢ Programming (9) Event Calculus Holds ( Colour ( A , Red ) , Result ( Move ( A , D ) , s 0 )) II Applications Integrity constraints Feferman theories References

The frame problem IV Event calculus Hamm & Schlotterbeck The Frame Holds ( Colour ( x , c ) , s ) → Problem (10) a. Event Calculus I Holds ( Colour ( x , c ) , Result ( Move ( y , z ) , s )) Constraint Logic Holds ( Colour ( x , c 1 ) , s ) ∧ x � = y → b. Programming Holds ( Colour ( x , c 1 ) , Result ( Paint ( y , c 2 ) , s )) Event Calculus II Holds ( On ( x , y ) , s ) → (11) a. Applications Holds ( On ( x , y ) , Result ( Paint ( z , c ) , s )) Integrity constraints Holds ( Clear ( x ) , s ) → b. Feferman Holds ( Clear ( x ) , Result ( Paint ( y , c ) , s )) theories References

Event calculus I: Ontology Event calculus Hamm & Schlotterbeck The Frame Problem Event Calculus I EC formalises two types of change Constraint Logic Programming • momentanous change Event Calculus II • continous change Applications Integrity constraints • Ontology : eventtypes, fluents, real numbers, individuals. Feferman theories References

Event calculus II: Language I Event calculus Hamm & Schlotterbeck The Frame Problem • Primitive predicates 1: Event Calculus I Constraint Logic Initially ( f ) Programming Happens ( e , t ) Event Calculus II Initiates ( e , f , t ) Applications Integrity Terminates ( e , f , t ) constraints Feferman theories References

Event calculus II: Language II Event calculus Hamm & Schlotterbeck The Frame Problem Event Calculus I • Primitive predicates 2: changing partial objects Constraint Logic Programming Releases ( e , f , t ) Event Calculus II Trajectory ( f 1 , t , f 2 , d ) Applications Integrity constraints Feferman theories References

Event calculus II: Language III Event calculus Hamm & Schlotterbeck The Frame Problem • Primitive predicates 3: no f -relevant events between t 1 and t 2 Event Calculus I Clipped ( t 1 , f , t 2 ) Constraint Logic Programming Event Calculus II Applications • Primitive predicates 4: truth predicate Integrity HoldsAt ( f , t ) constraints Feferman theories References

Axiomatisation : Inertia Event calculus Hamm & Schlotterbeck The Frame Problem Event Calculus I Constraint Logic If a fluent holds initially or has been initiated by some event Programming occurring at time t and no event terminating f has occurred Event Calculus between t and t ′ > t , then f holds at t ′ . II Applications Integrity constraints Feferman theories References

Axiomatisation I: Shanahan I Event calculus Hamm & Schlotterbeck The Frame Problem Definition Event Calculus I Clipped ( t ′ , f , t ′′ ) := Constraint Logic ∃ e , t ( Happens ( e , t ) ∧ ( Terminates ( e , f , t ) ∨ Programming Releases ( e , f , t )) ∧ t ′ < t ∧ t < t ′′ ) Event Calculus II Definition Applications Declipped ( t ′ , f , t ′′ ) := Integrity ∃ e , t ( Happens ( e , t ) ∧ ( Initiates ( e , f , t ) ∨ constraints Releases ( e , f , t )) ∧ t ′ < t ∧ t < t ′′ ) Feferman theories References

Axiomatisation I: Shanahan II Event calculus Hamm & Schlotterbeck Axiom The Frame Initially ( f ) ∧¬ Clipped ( 0 , f , t ) → HoldsAt ( f , t ) Problem Event Calculus I Axiom Constraint Logic Happens ( e , t ) ∧ Initiates ( e , f , t ) ∧ t < t ′ ∧¬ Clipped ( t , f , t ′ ) → HoldsAt ( f , t ′ ) Programming Event Calculus II Axiom Happens ( e , t ) ∧ Terminates ( e , f , t ) ∧ t < t ′ ∧¬ Declipped ( t , f , t ′ ) → ¬ HoldsAt ( f , t ′ ) Applications Integrity constraints Axiom Feferman Happens ( e , t ) ∧ Initiates ( e , f 1 , t ) ∧ t < t ′ ∧ t ′ = theories t + d ∧ Trajectory ( f 1 , t , f 2 , d ) ∧¬ Clipped ( t , f 1 , t ′ ) → HoldsAt ( f 2 , t ′ ) References

Axiomatisation II: Constraint logic programming Event calculus Hamm & Axiom Schlotterbeck Initially ( f ) → HoldsAt ( f , 0 ) The Frame Problem Axiom Event Calculus I HoldsAt ( f , r ) ∧ r < t ∧¬∃ s < rHoldsAt ( f , s ) ∧¬ Clipped ( r , f , t ) → HoldsAt ( f , t ) Constraint Logic Programming Axiom Event Calculus Happens ( e , t ) ∧ Initiates ( e , f , t ) ∧ t < t ′ ∧¬ Clipped ( t , f , t ′ ) → HoldsAt ( f , t ′ ) II Applications Axiom Integrity constraints Happens ( e , t ) ∧ Initiates ( e , f 1 , t ) ∧ t < t ′ ∧ t ′ = t + d ∧ Trajectory ( f 1 , t , f 2 , d ) ∧¬ Clipped ( t , f 1 , t ′ ) → HoldsAt ( f 2 , t ′ ) Feferman theories References Axiom Happens ( e , s ) ∧ t < s < t ′ ∧ ( Terminates ( e , f , s ) ∨ Releases ( e , f , s )) → Clipped ( t , f , t ′ )

Constraint System I Event calculus Hamm & Definition Schlotterbeck Let Σ = ( S , F , R ) be a signature where R contains at least = s for The Frame Problem each sort s ∈ S .Let X be a set of Σ variables. Further let D be a Σ Event Calculus I structure with equality und T a Σ theory. Constraint Logic Programming A constraint (over Σ ) is a formula r ( t 1 ,..., t m ) with r ∈ R is a Event Calculus predicate symbol and the t i are terms of the respective sorts. Let II C be the set of all constraints (over Σ ). C contains the constraints Applications Integrity true and false with constraints Feferman D | = true and D �| = false theories References A 5-tupel ζ = (Σ , D , T , X , CS ) with { true , false } ⊆ CS ⊆ C is a constraint system.

Constraint System II Event calculus Hamm & Schlotterbeck The Frame Problem Definition Event Calculus I Let ζ = (Σ , D , T , X , CS ) be a constraint system. A structure D and Constraint Logic Programming a theory T correspond with respect to constraints from CS if Event Calculus II D is a model of T , and Applications for each c ∈ CS : D | = ∃ c iff T | = ∃ c Integrity constraints Feferman theories References