States and Time in Modal Action Logic Camilla Schwind Laboratoire - PowerPoint PPT Presentation

Introduction The logic D al D al -based Action Systems Example Conclusion States and Time in Modal Action Logic Camilla Schwind Laboratoire d’Informatique Fondamentale Marseille June 27, 2007 Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Contents Introduction 1 The logic D al 2 D al -based Action Systems 3 Example 4 Conclusion 5 Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Motivation States and Time Actions frequently describe state transitions. But those take place in time. Actions and Agents The same action performed by several agents. Other aspects Knowledge, Communication Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Language: Idea action terms a ( t ) or a ( x ) : x variable of the langauge, modalities [ a ( t )] or [ a ( x )] , formulas [ a ( t )] A ( t ) , ∀ x ∃ y [ a ( x , y )] φ ( x , y ) , application: describe states and time, modality ✷ (characterizes any succeeding state) Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Language I: FO predicate logic L 0 a set of variables x , y , x 1 , y 1 , . . . , a set F F of function symbols F , P of predicate symbols P , including ⊤ and ⊥ , a set P the logical symbols ¬ , ∧ , ∀ , other symbols ( ∨ , . . . ) are defined as usual Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Language II: Action Terms Action symbols are special sysmbols set A A of action symbols a 1 , a 2 , . . . where A A ∩ P P = ∅ Action terms are built from action symbols and terms of L 0 . if a is an action symbol of arity n ≥ 0 and t 1 , . . . t n are terms of L 0 , then a ( t 1 , . . . t n ) is an action term. a is a constant for n = 0. An action term is called grounded if no variable occurs free in it. Example: a , a 1 ( c 1 , c 2 , c 3 ) are grounded, a 1 ( x , c 2 , y ) is not grounded. Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Language III: Formulas If a is an action term then [ a ] is an action operator. [ ε ] is an action operator (empty action operator). If φ is a formula and [ A ] is an action operator, then [ A ] φ is a formula. If φ is a formula, then ✷ φ is a formula. If φ is a formula and x is a variable, then ∀ x φ is a formula. If φ and ψ are formulas then ¬ φ and φ ∧ ψ are formulas Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Semantics of Action Logic = ( W , {S w : w ∈ W} , A , R , τ, τ ′ ) , where M W is a set of worlds ∀ w ∈ W , S w = ( O , F , P ) classical structure, O set of individual objects, F set functions over O P set predicates over O . → 2 W , n ∈ ω A set of functions f : W × O × . . . × O − � �� � n Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Semantics of Action Logic II R ⊆ W × W is a binary accessability relation (characterizing the modal operator). We set R ( x ) = { y : ( x , y ) ∈ R } τ is an interpretation function assigning, for every world w ∈ W , objects from O to terms of L 0 , functions (from F ) to function symbols (from F F ) and predicates to predicate symbols. In order to speak about objects from O , we introduce into the language, for every o ∈ O , a 0 − place function symbol (which we call o , for simplicity) τ ′ is a function assigning action functions to action symbols, such that arity ( τ ′ ( a )) = arity ( a ) + 1 Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Truth Values for Action Logic A valuation is defined as follows: Let t 1 , t 2 , . . . , t n be grounded terms and φ , ψ grounded formulas. if P is an n-ary predicate symbol then τ ( w , Pt 1 , t 2 , . . . , t n ) = ⊤ iff ( τ ( w , t 1 ) , . . . τ ( w , t n )) ∈ τ ( w , P ) τ ( w , ¬ φ ) = ⊤ iff τ ( w , φ ) = ⊥ τ ( w , φ ∧ ψ ) = ⊤ iff τ ( w , φ ) = τ ( w , ψ ) = ⊤ τ ( w , ∀ x φ ) = ⊤ iff ∀ o ∈ O , τ ( w , φ x o ) = ⊤ τ ( w , ✷ φ ) = ⊤ iff ∀ w ′ ∈ R ( w ) , τ ( w ′ , φ ) = ⊤ τ ( w , [ a ( t 1 , t 2 , . . . , t n )] φ ) = ⊤ iff ∀ w ′ ∈ τ ′ ( a )( w , τ ( w , t 1 ) , . . . , τ ( w , t n )) , τ ( w ′ , φ ) = ⊤ Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Axioms and inference rules of first - order Action Logic [ A 0 ] all of classical logic [ A 1 ] For any action operator [ X ] all of modal logic K [ A 2 ] For the modal operator ✷ all of modal logic S 4 [ A 3 ] ✷ φ → [ a ] φ [ A 4 ] [ ε ] φ → φ ∀ x φ → φ x [ A 5 ] c for any term c [ A 6 ] ∀ x [ X ] φ ↔ [ X ] ∀ x φ for any modal operator X, with no occurrence of x Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Soundness and Completeness of D al The D al -logic is sound and complete: Theorem ⊢D al φ if and only if | = D al φ Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Application to Action Systems Decidable subset of D al : formulas ∀ x . . . Hybrid Representation Modellizing state transition aspects of actions Modellizing temporal aspects of actions Modellizing spatial aspects of actions Modellizing agent aspects of actions . . . Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Ordering States Time axis T , linearly ordered (dense or continuous or discrete). D al -structure M determines an “ordering”-relation on the set of its states W , which will be related to the order on T . Definition Let M , be a D al -model. Then w ≺ 0 w ′ iff ∃ a ∈ A of arity n and there are terms t 1 , . . . , t n , such that w ′ ∈ f ( w , t 1 , . . . , t n ) . Let be � the reflexive and transitive closure of ≺ 0 . Intuitively w ≺ w ′ if we can “reach” w ′ from w by performing actions a 1 , a 2 , . . . , a n . � is transitive and reflexive. Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Linking States to Time → T , where w � w ′ implies time ( w ) ≤ time ( w ′ ) time : W − actions operators with complex temporal structures, beginning and ending and duration of actions, define preconditions and results of actions to occur at freely determinable time instances before or after the instance when the action occurs. When an action a occurs in the state w , time ( w ) gives us the time point at which a occurs. If the duration of the action is ∆ , the time point of the resulting state w ′ is time ( w ′ ) = time ( w ) + ∆ . Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Action Laws Action terms as action predicates a ( t , d , − → x ) , where t denotes the instance on which a occurs, d denotes the duration of a , − → x sequence of the other variables involved move ( t , 3 , TGV , Marseille , Paris ) is the action “train TGV goes from Marseille to Paris, the duration being 3 hours”. Action axioms at ( t , x , y ) → [ move ( t , d , x , y , z )] at ( t + d , x , z ) can be instantiated to at ( 6 , TGV , Marseille ) → [ move ( 6 , 3 , TGV , Marseille , Paris )] at ( 9 , TGV , Paris ) ), Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Action Laws suite General form of an action law π ( t 1 , − → x 1 ) → [ a ( t , d , − → x 2 )] l ( t 2 , − x 3 ) , w here − → → x 1 ∪ − → x 2 ⊆ − → x 3 π ( t 1 , − → x 1 ) is any FO formula and l ( t 2 , − → x 3 ) is a literal Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Frame Laws Idea: fluent f is true either as the result of an action or by persisting over the execution of an action. Two possibilities: Abductive construction. Extension E s at state s Add laws 1 α → [ a ] α to E s as longs as [ a ] ¬ α �∈ E s Completion construction (as in Reiter’s situation calculus). 2 Camilla Schwind States and Time in Modal Action Logic

Introduction The logic D al D al -based Action Systems Example Conclusion Example Billy and Suzanne throw rocks at a bottle. Suzanne throws first and her rock arrives first. The bottle shatters. When Billy’s rock gets to where the bottle used to be, there is nothing there but flying shards of glass. Without Suzanne’s throw, the impact of Billy’s rock on the intact bottle would have been one of the final steps in the causal chain from Billy’s throw to the shattering of the bottle. But, thanks to Suzanne’s preempting throw, that impact never happens. Camilla Schwind States and Time in Modal Action Logic