A short introduction to ATL-like logics with resources St ephane - PowerPoint PPT Presentation

A short introduction to ATL-like logics with resources St´ ephane Demri CNRS LIMSI, November 2018

Logics for resource-bounded agents ◮ ATL-like logics with models where transitions have costs/rewards and resource requirements are expressed in the syntax. ◮ Model-checking problems for such logics are often undecidable as games on VASS are often undecidable. ◮ Many existing resource logics: ◮ RBTL ∗ [Bulling & Farwer, CLIMA X ’09] ◮ QATL ∗ [Bulling & Goranko, EPTCS 2013] ◮ RB ± ATL [Alechina et al., ECAI’14] ◮ etc. ◮ Other logics for resource-bounded agents: step logic, justification logic, etc.

Concurrent game structures p q ( a , a ) , ( a , b ) s 2 s 3 Agt = { 1 , 2 } ( b , a ) ( c , c ) S = { s 1 , s 2 , s 3 , s 4 } ( a , b ) , ( a , a ) ( b , b ) ( c , c ) Act = { a , b , c } ( b , b ) ( b , a ) s 1 s 4 p ◮ Action manager act : Agt × S → P ( Act ) \ {∅} . act ( 1 , s 3 ) = { c } . ◮ Transition function δ : S × ( Agt → Act ) → S . δ ( s 4 , [ 1 �→ c , 2 �→ c ]) = s 3 . ◮ Labelling L : S → P ( PROP ) .

Basic concepts: joint actions and computations ◮ f : A → Act : joint action by A ⊆ Agt in s . Proviso: for all a ∈ A , we have f ( a ) ∈ act ( a , s ) . ◮ D A ( s ) : set of joint actions by A in s . = { s ′ ∈ S | ∃ g ∈ D Agt ( s ) s . t . f ⊑ g & s ′ = δ ( s , g ) } def out ( s , f ) f 0 f 1 ◮ Computation λ = s 0 − → s 1 − → s 2 . . . such that for all i , we have s i + 1 ∈ δ ( s i , f i ) . ◮ Linear model L ( s 0 ) − → L ( s 1 ) − → L ( s 2 ) · · · .

Basic concepts: strategies ◮ A strategy F A for A is a map from the set of finite computations to the set of joint actions by A such that f 0 f n − 1 F A ( s 0 − → s 1 · · · − → s n ) ∈ D A ( s n ) . f 0 f 1 def ◮ λ = s 0 − → s 1 − → s 2 · · · respects F A ⇔ ∀ i < | λ | , f i − 1 f 0 s i + 1 ∈ out ( s i , F A ( s 0 − → s 1 . . . − → s i )) ◮ λ respecting F A is maximal whenever λ cannot be extended further while respecting the strategy. ◮ comp ( s , F A ) : max. computations from s respecting F A .

The logic ATL φ ::= p | ¬ φ | φ ∧ φ | �� A �� X φ | �� A �� G φ | �� A �� φ U φ p ∈ PROP A ⊆ Agt def M , s | ⇔ s ∈ L ( p ) = p def M , s | = �� A �� X φ ⇔ there is a strategy F A s.t. f 0 for all s 0 − → s 1 . . . ∈ comp ( s , F A ) , we have M , s 1 | = φ def M , s | = �� A �� φ 1 U φ 2 ⇔ there is a strategy F A s.t. for all f 0 − → s 1 . . . ∈ comp ( s , F A ) , λ = s 0 there is some i < | λ | s.t. M , s i | = φ 2 and for all j ∈ [ 0 , i − 1 ] , we have M , s j | = φ 1 .

Model-checking problem ◮ Model-checking problem for ATL: Input: φ in ATL, a finite CGS M and a state s , Question: M , s | = φ ? ◮ Model-checking problem for ATL is P -complete. Labeling algorithm. [Alur & Henzinger & Kupferman, JACM 2002] ◮ ATL ∗ = ATL + all path formulae ` a la CTL ∗ . ◮ Model-checking problem for ATL ∗ is 2 EXPTIME -complete.

Resource-bounded concurrent game structures Concurrent game structures + resources (counters) ◮ Number r of resources/counters. ◮ Partial cost function cost : S × Agt × Act → Z r . ◮ Action idle ∈ act ( a , s ) with no cost. ◮ Given a joint action f : A → Act , def � cost A ( s , f ) = cost ( s , a , f ( a )) a ∈ A

p q ( a , a ) , ( a , idle ) s 2 s 3 ( idle , a ) ( idle , idle ) ( a , idle ) , ( a , a ) ( idle , idle ) ( idle , idle ) ( idle , idle ) ( idle , a ) s 1 s 4 p cost ( s 2 , 1 , a ) = ( 1 , 1 , 1 , 1 ) cost ( s 2 , 2 , a ) = ( − 2 , 1 , − 3 , 1 ) cost { 1 , 2 } ( s 2 , [ 1 �→ a , 2 �→ a ]) = ( − 1 , 2 , − 2 , 2 )

b -strategies ◮ Initial budget b ∈ ( N ∪ { ω } ) r . f 0 f 1 ◮ λ = s 0 − → s 1 − → s 2 . . . in comp ( s , F A ) is b -consistent: def ◮ v 0 = b , f i − 1 f 0 def ◮ v i + 1 − → s 1 . . . − → s i )) , = v i + cost A ( s i , F A ( s 0 ◮ for all i , 0 � v i . Asymmetry between A and ( Agt \ A ) ◮ comp ( s , F A , b ) : set of all the b -consistent computations. def ◮ F A is a b -strategy w.r.t. s ⇔ comp ( s , F A ) = comp ( s , F A , b )

The logic RB ± ATL ( Agt , r ) [Alechina et al., ECAI’14] φ ::= p | ¬ φ | φ ∧ φ | �� A b �� X φ | �� A b �� G φ | �� A b �� φ U φ b ∈ ( N ∪ { ω } ) r p ∈ PROP A ⊆ Agt def M , s | = p ⇔ s ∈ L ( p ) def = �� A b �� X φ M , s | ⇔ there is a b -strategy F A w.r.t. s f 0 s.t. for all s 0 − → s 1 . . . ∈ comp ( s , F A ) , we have M , s 1 | = φ def = �� A b �� φ 1 U φ 2 M , s | ⇔ there is a b -strategy F A w.r.t. s f 0 s.t. for all λ = s 0 − → s 1 . . . ∈ comp ( s , F A ) there is some i < | λ | s.t. M , s i | = φ 2 and for all j ∈ [ 0 , i − 1 ] , we have M , s j | = φ 1 .

Alternative semantics ◮ In RB ± ATL, comp ( s , F A ) = comp ( s , F A , b ) implies the maximal computations are infinite. ◮ Infinite semantics: arbitrary strategy but quantifications over infinite computations only. ◮ Finite semantics: arbitrary strategy but quantifications over maximal computations only.

Resource-bounded reasoners for AI ◮ RB ± ATL is one of the logics for reasoning about resources. See papers in AAAI, IJCAI, ECAI, etc. ◮ Relationships with counter machines known for establishing undecidability or complexity lower bounds. ◮ Various flavours of resource-bounded logics exist: RBCL, RAL, PRB-ATL, etc.

Alternating VASS [Courtois & Schmitz, MFCS’14] ◮ Alternating VASS A = ( Q , r , R 1 , R 2 ) : ◮ R 1 is a finite subset of Q × Z r × Q . (unary rules) ◮ R 2 is a finite subset of � β ≥ 2 Q β (fork rules) ◮ Proof: tree labelled by elements in Q × N r following the rules in A . . . . . . . . . ( q 3 , ( 4 , 8 )) ( q 0 , ( 0 , 8 )) ( q 2 , ( 1 , 5 )) ( q 1 , ( 1 , 5 )) ( q 0 , ( 1 , 5 )) ( q 1 , ( 2 , 2 )) ( − 1 , + 3 ) (+ 3 , + 3 ) − − − − → q 0 q 0 − → q 1 , q 2 − − − − → q 3 q 1 q 2

Decision problems ◮ State reachability problem for AVASS: Input: AVASS A , control states q 0 and q f , Question: is there a finite proof of AVASS with root ( q 0 , 0 ) and each leaf belongs to { q f } × N r ? ◮ Non-termination problem for AVASS: Input: A , q 0 , Question: is there a proof with root ( q 0 , 0 ) and all the maximal branches are infinite? VASS games with asymmetry between the two players

Main Correspondences RB ± ATL Alternating VASS Logic in AI Verification games proponent restriction condition updates in R 1 / no update in R 2 computation tree for F A proof formulae in the scope of �� A b �� monotone objectives ◮ From RB ± ATL model-checking to the state reachability and the non-termination problems for AVASS. ◮ From RB ± ATL ∗ model-checking to the parity games for AVASS. ◮ Parameters synthesis thanks to the computation of the Pareto frontier of parity games. See [Abdulla et al., CONCUR’13]

Complexity of RB ± ATL fragments r \ card ( Agt ) arbitrary 2 1 arbitrary 2 EXPTIME -c. 2 EXPTIME -c. EXPSPACE -c. ≥ 4 EXPTIME -c. EXPTIME -c. PSPACE -c. 2 , 3 PSPACE -h. PSPACE -h. PSPACE -c. in EXPTIME in EXPTIME 1 in PSPACE in PSPACE PTIME -c. Complexity characterisations established in [Alechina et al., JCSS 2017; Alechina et al., RP’16; etc.] based on the relationships with (A)VASS and results from [Habermehl, ICATPN’97; Courtois & Schmitz, MFCS’14; Colcombet et al., LICS’17]

Parameterized RB ± ATL ∗ : ParRB ± ATL ∗ ◮ b ∈ ( N ∪ { ω } ) r replaced by tuples of variables. ��{ 1 } ( x 1 , x 2 ) ��⊤ U q f ∧ ��{ 2 } ( x 2 , x 3 ) ��⊤ U q ′ f ◮ MC problem for ParRB ± ATL ∗ : compute the maps v : { x 1 , . . . , x n } → ( N ∪ { ω } ) such that M , s | = v ( φ ) . ◮ Symbolic representation for such maps are computable.

Other temporal logics for AI ◮ TIME: International Symposium on Temporal Representation and Reasoning ◮ Artificial Intelligence ◮ Temporal Databases ◮ Logic ◮ Interval temporal logics, ATL-like logics, temporal logics over concrete domains, etc.

Concluding remarks ◮ Formal relationships between resource-bounded logics and games on alternating VASS. ◮ Open problems: ◮ Parameter synthesis. ◮ Complexity for small fragments by bounding further the syntactic resources. ◮ Alternative semantics for applications.