Towards a model-checker for counter systems S. Demri 1 A. Finkel 1 V. Goranko 2 G. van Drimmelen 2 1 LSV, CNRS & ENS Cachan & INRIA Futurs 2 University of Witwatersrand, Johannesburg ATVA, October 2006, Beijing
Motivations Presburger temporal logic Decision procedure Procedure Concluding remarks Overview Motivations Decision procedure Counter systems (CS) Admissible CS Fast success story Translation into PA Presburger temporal logic Procedure Presburger counter systems Flattening Specification language Completeness Problems Concluding remarks S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Counter systems (CS) Decision procedure Fast success story Procedure Concluding remarks Counter systems ◮ Model-checking of infinite-state systems needed for formal verification. ◮ Ubiquity of counter systems (CS) ◮ Embedded systems/protocols, Petri nets, . . . ◮ Programs with pointer variables. [Bardin et al, AVIS 06; Bouajjani et al, CAV 06] ◮ Broadcast protocols. [Leroux & Finkel, FSTTCS 02] ◮ Logics for data words. [Boja´ nczyk et al, LICS 06] ◮ (High) undecidability ◮ Checking safety properties for CS is undecidable. ◮ Checking liveness properties for CS is Σ 1 1 -hard. S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Counter systems (CS) Decision procedure Fast success story Procedure Concluding remarks Taming counter systems ◮ Classes with decidable reachability problems ◮ Reversal-bounded CS. [Ibarra, JACM 78] ◮ Flat relational CS. [Comon & Jurski, CAV 98] ◮ Flat linear CS. [Boigelot, PhD 98; Finkel & Leroux, FSTTCS 02] ◮ Petri nets. [Kosaraju, STOC 82] ◮ Verification techniques ◮ Acceleration method, . . . [Boigelot & Wolper, CAV 94; Finkel & Leroux, FSTTCS 02] ◮ Flatness is central in the verification of CS. [Leroux & Sutre, ATVA 05; Bardin et al, ATVA 05] ◮ Tools: Fast , Lash , TReX , . . . S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Counter systems (CS) Decision procedure Fast success story Procedure Concluding remarks Fast success story ◮ Verification of standard examples from Petri nets to TTP protocol and broadcast protocols. ◮ Cornerstones: ◮ Flat CS with Presburger-definable reachability sets. ◮ Homomorphisms between CS and flat CS preserving the reachability sets. ◮ Complete procedure in Fast to enumerate flattenings. ◮ Fast Extended Release. [Bardin & Leroux & Point, CAV 06] S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Counter systems (CS) Decision procedure Fast success story Procedure Concluding remarks Our motivations Theoretical ground to verify richer properties within Fast ◮ To design classes of counter systems with decidable temporal a la CTL ⋆ , . . . ). properties richer than reachability (` S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Counter systems (CS) Decision procedure Fast success story Procedure Concluding remarks Our motivations Theoretical ground to verify richer properties within Fast ◮ To design classes of counter systems with decidable temporal a la CTL ⋆ , . . . ). properties richer than reachability (` ◮ To provide the adequate notion of trace-flattening for such richer properties (preservation of traces, bisimulation, . . . ). S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Counter systems (CS) Decision procedure Fast success story Procedure Concluding remarks Our motivations Theoretical ground to verify richer properties within Fast ◮ To design classes of counter systems with decidable temporal a la CTL ⋆ , . . . ). properties richer than reachability (` ◮ To provide the adequate notion of trace-flattening for such richer properties (preservation of traces, bisimulation, . . . ). ◮ To design a procedure to enumerate trace-flattenings and then check the temporal properties. S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks Presburger counter systems (PCS) � Σ , Q , T � q 0 q 1 q 2 S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks Presburger counter systems (PCS) � Σ , Q , T � x ′ x ′ x ′ 1 = x 1 + 1 2 = x 2 + 1 3 = x 3 + 1 ψ ( x , x ′ ) ψ ′ ( x , x ′ ) q 0 q 1 q 2 x ′ 1 = x ′ 2 = x ′ 3 = 0 ◮ Labels: Presburger formulae over ◮ x = � x 1 , x 2 , x 3 � (current values). ◮ x ′ = � x ′ 1 , x ′ 2 , x ′ 3 � (next values). S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks Presburger transition systems (PTS) Presburger CS − → Presburger TS C = � Σ , Q , T � �→ S C = � S , →� ◮ S = Q × N n . ψ ( x , x ′ ) → q ′ ∈ T s.t. a , a ′ | ◮ � q , a � → � q ′ , a ′ � iff ∃ q = ψ ( x , x ′ ). − − − ◮ Configuration path π : infinite path in � S , →� . S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks FOCTL ⋆ (Pr) formulae Pr CTL ⋆ FO ���� � �� � � �� � ϕ ::= ψ ( t ) | ¬ ϕ | ϕ ∧ ϕ | X ϕ | ϕ U ϕ | A ϕ | ∃ y ϕ . ◮ Variables: x 0 : control state. x 1 , . . . , x n : counters. y , z , t , . . . : auxiliary variables. ◮ ψ ( t ): Presburger formula with free variables from tuple t . S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks Satisfaction relation π, i | = env ϕ ◮ π : infinite configuration path of some transition system S C . ϕ U ϕ ′ , ϕ ϕ ϕ ϕ ϕ ′ ◮ i : position along π . ◮ env : environment VAR → N . ◮ ϕ : FOCTL ⋆ (Pr) formula. S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks Main clauses of | = env ◮ π, i | = env ψ ( t ) iff π ( i ) , env | = ψ ( t ) in PA, S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks Main clauses of | = env ◮ π, i | = env ψ ( t ) iff π ( i ) , env | = ψ ( t ) in PA, ◮ π, i | = X ϕ iff π, i + 1 | = ϕ , S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks Main clauses of | = env ◮ π, i | = env ψ ( t ) iff π ( i ) , env | = ψ ( t ) in PA, ◮ π, i | = X ϕ iff π, i + 1 | = ϕ , ◮ π, i | = env ∃ y ϕ iff there is m ∈ N such that π, i | = env [ y ← m ] ϕ , S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks Main clauses of | = env ◮ π, i | = env ψ ( t ) iff π ( i ) , env | = ψ ( t ) in PA, ◮ π, i | = X ϕ iff π, i + 1 | = ϕ , ◮ π, i | = env ∃ y ϕ iff there is m ∈ N such that π, i | = env [ y ← m ] ϕ , = ϕ U ϕ ′ iff there is some j ≥ i s.t. π, j | = ϕ ′ and for ◮ π, i | i ≤ k < j , we have π, k | = ϕ , = A ϕ iff for every infinite configuration path π ′ s.t. ◮ π, i | π ′ ≤ i = π ≤ i we have π ′ , i | = ϕ . S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
Motivations Presburger temporal logic Presburger counter systems Decision procedure Specification language Procedure Problems Concluding remarks Examples of properties Determinism : The reachability graph is deterministic: � ¬∃ y ( EX ( x i = y ) ∧ EX ( x i � = y )) . AG 0 ≤ i ≤ n S. Demri, A. Finkel, V. Goranko, G. van Drimmelen Towards a model-checker for counter systems
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