Taming Past LTL and Flat Counter Systems

Transcription

1 Taming Past LTL and Flat Counter Systems Stéphane Demri 2, Amit Kumar Dhar 1, Arnaud Sangnier 1 1. LIAFA, Univ Paris Diderot, Sorbonne Paris Cité, CNRS, France 2. LSV, ENS Cachan, CNRS, INRIA, France June 26, 2012 IJCAR 2012, Manchester S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

2 Model Checking A System Satisfies A Property S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

3 Model Checking A System Satisfies A Property S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

4 Model Checking A System Satisfies A Property φ S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

5 Model Checking A System Satisfies A Property = φ S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

6 Model Checking A System Satisfies A Property = φ Decision Procedure ψ(x 1, x 2,, x n ) in decidable theory S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

7 Model Checking A System Satisfies A Property = φ Decision Procedure ψ(x 1, x 2,, x n ) in decidable theory Yes/No Tools & Solvers S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

8 Table of Contents 1 Introduction Counter Systems and Kripke Structures Linear-Time Temporal Logics with Arithmetical Constraints Existential Model-checking Problem 2 Path Schemas : Decomposing Flat Counter Systems even further 3 A new Stuttering Theorem for Past LTL 4 Our Main NP Algorithm 5 Conclusion S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

9 Table of Contents Introduction 1 Introduction Counter Systems and Kripke Structures Linear-Time Temporal Logics with Arithmetical Constraints Existential Model-checking Problem 2 Path Schemas : Decomposing Flat Counter Systems even further 3 A new Stuttering Theorem for Past LTL 4 Our Main NP Algorithm 5 Conclusion S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

10 Kripke Structures (KS) Introduction Counter Systems and Kripke Structures q 7 q 9 q 8 q 6 q 10 q 4 q 3 q 5 q 1 q 2 S = (Q,, l) l : Q 2 AP l(q 7 ) = {p, r} AP S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

11 Counter Systems (CS) Introduction Counter Systems and Kripke Structures q 7 q 9 q 8 q 6 q 10 q 4 q 3 q 5, (2, 3) q 1 q 2 2.x x 2 20, (5, 7) Guards : Boolean combination of linear constraints of the form Σ i a i.x i b and {=,,, <, >}. Updates : u Z n are translations. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

12 Introduction Counter Systems and Kripke Structures Runs in Counter Systems q 0, v 0 δ0 q 1, v 1 δ1 q 2, v 2 δ2 q 3, v 3 δ3 At each position i N, v i N n represents the counter values. δ i = q i, guard(δ i ), update(δ i ), q i+1 v i N n c 0 = q 0, v 0 is the initial configuration. i N, v i satisfies guard(δ i ) and v i+1 = v i + update(δ i ). Counter Systems are Turing-Complete. Thus most verification problems are undecidable. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

13 Introduction Example of Runs in Counter Systems Counter Systems and Kripke Structures q 2 q 7 c 1 > 1, (2, 4) c 1 5 c 2 < 25, (3, 1) c 1 = 0, (1, 0) q 1 c 1 0, (1, 0) q 5 q 6 c 1 2, (1, 5) q 8 q 1, (0, 0) q 1, (1, 0) q 5, (2, 0) q 6, (3, 5) q 7, (5, 9) q 6, (8, 10) q 7, (10, 14)..., 0 S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

14 Introduction Flat Counter Systems (CFS) Counter Systems and Kripke Structures q 2 q 7 c 1 > 1, (2, 4) c 1 5 c 2 < 25, (3, 1) c 1 = 0, (1, 0) q 1 c 1 0, (1, 0) q 5 q 6 c 1 2, (1, 5) q 8 Guards : Boolean combination of linear constraints of the form Σ i a i.x i b and {=,,, <, >}. Updates : u Z n are translations. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

15 Introduction Flat Counter Systems (CFS) Counter Systems and Kripke Structures q 7 Not Flat q 9 q 8 q 6 q 10 q 4 q 3 q 5, (2, 3) q 1 q 2 2.x x 2 20, (5, 7) Guards : Boolean combination of linear constraints of the form Σ i a i.x i b and {=,,, <, >}. Updates : u Z n are translations. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

16 Introduction Flat Counter Systems (CFS) Counter Systems and Kripke Structures q 7 Flat q 9 q 8 q 6 q 10 q 4 q 3 q 5, (2, 3) q 1 q 2 2.x x 2 20, (5, 7) Guards : Boolean combination of linear constraints of the form Σ i a i.x i b and {=,,, <, >}. Updates : u Z n are translations. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

17 Introduction Flat Kripke Structures (KFS) Counter Systems and Kripke Structures q 7 q 9 q 8 q 6 q 10 q 4 q 3 q 5 q 1 q 2 S = (Q,, l) S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

18 Related Works Introduction Counter Systems and Kripke Structures Flat Counter Systems are extensively studied. See e.g. [Boigelot 98, Comon and Jurski - CAV 98]. Flatness is a useful property. See e.g. [Comon and Cortier - CSL 00, Leroux and Sutre - ATVA 05] Flatness leads to decidable safety and reachability property. See e.g.[ Finkel and Leroux - FSTTCS 02, Bozga et al. - CAV 10] Local model checking of Presburger-CTL* is decidable in flat counter systems. See [Demri et al. - JANCL 10] S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

19 Introduction Linear-Time Temporal Logics with Arithmetical Constraints Temporal Logic PLTL[ ] (Standard version) φ ::= p φ φ φ φ φ Xφ φuφ X 1 φ φsφ where p AT. For a model σ Σ ω and Σ = 2 AP σ, i = p def def σ, i = Xφ def σ, i = φ 1 Uφ 2 def σ, i = X 1 φ def σ, i = φ 1 Sφ 2 p σ(i) σ, i + 1 = φ σ, j = φ 2 for some i j such that σ, k = φ 1 for all i k < j i > 0 and σ, i 1 = φ σ, j = φ 2 for some 0 j i such that σ, k = φ 1 for all j < k i S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

20 Introduction Linear-Time Temporal Logics with Arithmetical Constraints Logic PLTL[C] (with arithmetical constraints) φ ::= p g φ φ φ φ φ Xφ φuφ X 1 φ φsφ where g is defined as : t ::= a.x t + t g ::= t b g g g g where x C n (counters) for some n, a Z, b Z and {=,,, <, >}. The model are essentially runs of Counter Systems σ Σ ω and Σ = 2 AP N n σ, i = g def v i = g where v i (x j ) is the value of counter x j in σ(i) S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

21 Introduction Existential Model-checking Problem Existential Model Checking: MC(L,C): Input: A system S C, a configuration c 0 and a formula φ L; Output: Does there exist a run ρ starting from c 0 in S such that ρ, 0 = φ? Known Results: MC(PLTL[C],CFS) is decidable by translation into Presburger Arithmetic. [Demri et al. - JANCL 10] MC(PLTL[ ],KFS). (MC(LTL,KFS) is NP-complete [Khutz and Finkbeiner - CONCUR 11]). S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

22 Introduction Existential Model-checking Problem Existential Model Checking: MC(L,C): Input: A system S C, a configuration c 0 and a formula φ L; Output: Does there exist a run ρ starting from c 0 in S such that ρ, 0 = φ? Known Results: MC(PLTL[C],CFS) is decidable by translation into Presburger Arithmetic. [Demri et al. - JANCL 10] Can we have better complexity? MC(PLTL[ ],KFS). (MC(LTL,KFS) is NP-complete [Khutz and Finkbeiner - CONCUR 11]). S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

23 Introduction Existential Model-checking Problem Existential Model Checking: MC(L,C): Input: A system S C, a configuration c 0 and a formula φ L; Output: Does there exist a run ρ starting from c 0 in S such that ρ, 0 = φ? Known Results: MC(PLTL[C],CFS) is decidable by translation into Presburger Arithmetic. [Demri et al. - JANCL 10] Can we have better complexity? MC(PLTL[ ],KFS). (MC(LTL,KFS) is NP-complete [Khutz and Finkbeiner - CONCUR 11]). Is the same possible with Past and Counters? S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

24 Introduction Existential Model-checking Problem Existential Model Checking: MC(L,C): Input: A system S C, a configuration c 0 and a formula φ L; Output: Does there exist a run ρ starting from c 0 in S such that ρ, 0 = φ? Known Results: MC(PLTL[C],CFS) is decidable by translation into Presburger Arithmetic. [Demri et al. - JANCL 10] Can we have better complexity? Yes MC(PLTL[ ],KFS). (MC(LTL,KFS) is NP-complete [Khutz and Finkbeiner - CONCUR 11]). Is the same possible with Past and Counters? Yes S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

25 Example of PLTL[C] Introduction Existential Model-checking Problem q 2 q 7 c 1 > 1, (2, 4) c 1 5 c 2 < 25, (3, 1) c 1 = 0, (1, 0) q 1 c 1 0, (1, 0) q 5 q 6 c 1 2, (1, 5) q 8 φ = F(q 7 Xq 6 c 1 > 5) q 1, (0, 0) q 1, (1, 0) q 5, (2, 0) q 6, (3, 5) q 7, (5, 9) q 6, (8, 10) q 7, (10, 14)..., 0 = φ S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

26 Path Schemas : Decomposing Flat Counter Systems even further Table of Contents 1 Introduction Counter Systems and Kripke Structures Linear-Time Temporal Logics with Arithmetical Constraints Existential Model-checking Problem 2 Path Schemas : Decomposing Flat Counter Systems even further 3 A new Stuttering Theorem for Past LTL 4 Our Main NP Algorithm 5 Conclusion S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

27 Path Schemas : Decomposing Flat Counter Systems even further Path Schema in Flat Kripke Structures (KPS) e 0 e 9 q 7 e 8 e 7 q 9 q 8 e 86 q 6 q 10 e 10 q 4 e 3 e 6 q 3 e 5 e 4 e 2 e q 1 1 q 2 q 5 P = p 1 l + 1 p 2l + 2 p 3l ω 3 S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

28 Path Schemas : Decomposing Flat Counter Systems even further Path Schema in Flat Kripke Structures (KPS) q 3 e 8 q 4 e 3 e 4 e 2 e 1 q 2 q 8 e 86 e 7 q 7 e 0 q 10 q 1 e 10 q 2 q 5 q 6 e 1 e 5 e 6 q 7 q 9 e 7 e 9 P = p 1 l + 1 p 2l + 2 p 3l ω 3 S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

29 Path Schemas : Decomposing Flat Counter Systems even further Path Schema in Flat Counter System (CPS) q 3 e 3 e 2 q 4 e 4 e 1, (2, 3) e 10 q 10 q 1 e 8 q 2 q 8 2.x x 2 20 e 86 (5, 7) q 2 q 5 q 6 e 1 e 5 e 6 q 7 e 0 e 7 q 7 q 9 e 7 e 9 P = p 1 l + 1 p 2l + 2 p 3l ω 3 S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

30 Path Schemas : Decomposing Flat Counter Systems even further Why Path Schemas Are Nice Decompositions? Fundamental Structure: Minimal path schemas. Every run in a flat system respects one of the minimal path schemas of the system. Exponentially many minimal path schemas in a flat system. Simpler structure Easy to study. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

31 Path Schemas : Decomposing Flat Counter Systems even further Why Path Schemas Are Nice Decompositions? Fundamental Structure: Minimal path schemas. Any transition occurs at most twice. Every run in a flat system respects one of the minimal path schemas of the system. Exponentially many minimal path schemas in a flat system. Simpler structure Easy to study. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

32 Path Schemas : Decomposing Flat Counter Systems even further Why Path Schemas Are Nice Decompositions? Fundamental Structure: Minimal path schemas. Any transition occurs at most twice. Every run in a flat system respects one of the minimal path schemas of the system. run belongs to the language described by the minimal path schema. Exponentially many minimal path schemas in a flat system. Simpler structure Easy to study. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

33 Table of Contents A new Stuttering Theorem for Past LTL 1 Introduction Counter Systems and Kripke Structures Linear-Time Temporal Logics with Arithmetical Constraints Existential Model-checking Problem 2 Path Schemas : Decomposing Flat Counter Systems even further 3 A new Stuttering Theorem for Past LTL 4 Our Main NP Algorithm 5 Conclusion S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

34 A new Stuttering Theorem for Past LTL PLTL[ ] over KPS Theorem MC(PLTL[ ], KPS) is NP-complete. Stuttering Theorem for LTL (no past time operator) proposed earlier by [Kučera and Strejček - Acta Informatica 05] Translating PLTL to LTL [Gabbay - TLS 87] Exponential blowup of temporal depth. We need to extend the Stuttering Theorem for PLTL[ ]. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

35 Stuttering Theorem A new Stuttering Theorem for Past LTL PLTL[ ] formula: φ = ((p Xq) q)ur (pq) 15 r... = φ (pq)r... = φ PLTL[ ] formula: φ = F(r F(q Fp))Us (pqr) 15 s... = φ (pqrpqrpqr)s... = φ S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

36 A new Stuttering Theorem for Past LTL Stuttering Theorem Theorem Given two models σ, σ and a formula φ such that σ = σ 1 s M σ 2, σ = σ 1 s M σ 2 (2 AT ) ω and M, M 2td(φ) + 5 then, we have σ, 0 = φ iff σ, 0 = φ. proof sketch. By induction on the structure of the formula: For each temporal operator the satifiability does not change by changing the number of repetitions of s by 1. Boolean combination does not need to change the number of repetitions. Hence, maximum number of repetitions that can be distinguished is dependent on td(φ). S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

37 A new Stuttering Theorem for Past LTL A NP Algorithm for MC(PLTL[ ], KPS) 1 Guess the number of times each loop in the path schema is taken. at most polynomial times by Stuttering Theorem. 2 Unfold the loops, the guessed number of times. polynomial size path schema with a single loop at the end. 3 Perform PTime model checking for PLTL[ ] from [Laroussinie, Markey and Schnoebelen - LICS 02] S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

38 PLTL[ ] over KPS A new Stuttering Theorem for Past LTL Lemma (Khutz and Finkbeiner - CONCUR 11) MC(PLTL[ ], KPS) is NP-hard. Proof. p 1 p 2 p 3 p n start φ = ψ 1 2 ψ truth ψ 1 2 = [ i (G(q i XXq i XXXG q i ))] ψ truth = φ[p i F(q i XXq i )]. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

39 PLTL[ ] over KPS A new Stuttering Theorem for Past LTL Lemma (Khutz and Finkbeiner - CONCUR 11) MC(PLTL[ ], KPS) is NP-hard. Proof. p 1 p 2 p 3 p n start φ = ψ 1 2 ψ truth ψ 1 2 = [ i (G(q i XXq i XXXG q i ))] ψ truth = φ[p i F(q i XXq i )]. Same proof for MC(PLTL[ ], CPS), MC(PLTL[ ], KFS),MC(PLTL[C], CFS) S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

40 A new Stuttering Theorem for Past LTL PLTL[ ] over Subclasses of Kripke Structures Theorem (Path Schema) MC(PLTL[ ], KPS) is NP-complete. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

41 A new Stuttering Theorem for Past LTL PLTL[ ] over Subclasses of Kripke Structures Theorem (Path Schema) MC(PLTL[ ], KPS) is NP-complete. Theorem (Flat Kripke Structure) MC(PLTL[ ], KFS) is NP-complete. Guess a path schema and apply the previous result. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

42 A new Stuttering Theorem for Past LTL PLTL[ ] over Subclasses of Kripke Structures Theorem (Path Schema) MC(PLTL[ ], KPS) is NP-complete. Theorem (Flat Kripke Structure) MC(PLTL[ ], KFS) is NP-complete. Guess a path schema and apply the previous result. Theorem (Bounded Loop PS) MC(PLTL[ ], KPS(n)) is in PTime. (KPS(n) denotes path schemas in flat Kripke structure with exactly n loops). S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

43 Table of Contents Our Main NP Algorithm 1 Introduction Counter Systems and Kripke Structures Linear-Time Temporal Logics with Arithmetical Constraints Existential Model-checking Problem 2 Path Schemas : Decomposing Flat Counter Systems even further 3 A new Stuttering Theorem for Past LTL 4 Our Main NP Algorithm 5 Conclusion S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

44 Our Main NP Algorithm 3 main ingredients for proving MC(PLTL[C], CFS) in NP 1 Elimination of disjunction in guards and arithmetical constraints in formula. 2 Characterize all valid runs in a counter system without disjunction as System of equations. Respecting the updates, guards and non-negative counter values. 3 Stuttering theorem for PLTL. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

45 Our Main NP Algorithm Characterizing runs by equations q 2 q 7 c 1 > 1, (2, 4) e e 5 4 c1 5 c2 < 25, (3, 1) c 1 0, (1, 0) e q 2 e 3 e 1 1 c 1 = 1, (1, 0) c 1 2, (1, 5) q 5 q 6 e 6 q 8 e 7 Consider the path schema e + 1 (e 2, e 3 ).(e 4, e 5 ) +.e 6.e ω 7 Consider the equations for the edge e 5 and counter c 2 : (X 1 and X 2 variables for the 2 loops) S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

46 Our Main NP Algorithm Characterizing runs by equations q 7 c 1 > 1, (2, 4) e e 5 4 c1 5 c2 < 25, (3, 1) c 1 0, (1, 0) e 2 e q 1 q 3 5 q e 6 1 c 1 = 1, (1, 0) c 1 2, (1, 5) e 6 q 8 e 7 Consider the path schema e + 1 (e 2, e 3 ).(e 4, e 5 ) +.e 6.e ω 7 Consider the equations for the edge e 5 and counter c 2 : (X 1 and X 2 variables for the 2 loops) To ensure that it is taken at least once (first iteration): 0.X < 25. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

47 Our Main NP Algorithm Characterizing runs by equations q 7 c 1 > 1, (2, 4) e e 5 4 c1 5 c2 < 25, (3, 1) c 1 0, (1, 0) e 2 e q 1 q 3 5 q e 6 1 c 1 = 1, (1, 0) c 1 2, (1, 5) e 6 q 8 e 7 Consider the path schema e + 1 (e 2, e 3 ).(e 4, e 5 ) +.e 6.e ω 7 Consider the equations for the edge e 5 and counter c 2 : (X 1 and X 2 variables for the 2 loops) To ensure that it is taken at least once (first iteration): 0.X < 25. To ensure that it is taken X 2 times (last iteration): 0.X (X 2 1) < 25 S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

48 Our Main NP Algorithm NP Algorithm for PLTL[C] over CPS x 3 x = 1, +2 φ = F((x 2)Uq 2 ) q 0 q 1 q 2, +1 x 2 15, +1 S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

49 Our Main NP Algorithm NP Algorithm for PLTL[C] over CPS x 3 x = 1, +2 φ = F((x 2)Uq 2 ) q 0 q 1 q 2, +1 x 2 15, +1 x + 2 > 3 x , +2 q 0, (, 1) q 1, [1, 1] q 1, [3, 3] q 1, (3, 2 15 ) q 2, [2 15, + ) x + 1 = 1, +1 x + 2 = 3, +2 x + 2 3, +2 x , +1 S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

50 Our Main NP Algorithm NP Algorithm for PLTL[C] over CPS x 3 x = 1, +2 φ = F((x 2)Uq 2 ) q 0 q 1 q 2, +1 x 2 15, +1 x + 2 > 3 x , +2 q 0, (, 1) q 1, [1, 1] q 1, [3, 3] q 1, (3, 2 15 ) q 2, [2 15, + ) x + 1 = 1, +1 x + 2 = 3, +2 x + 2 3, +2 x , +1 Construct E with k variables Guess Solution y [Borosh and Treybig - AMS 76] S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

51 Our Main NP Algorithm NP Algorithm for PLTL[C] over CPS x 3 x = 1, +2 φ = F((x 2)Uq 2 ) q 0 q 1 q 2, +1 x 2 15, +1 x + 2 > 3 x , +2 q 0, (, 1) q 1, [1, 1] q 1, [3, 3] q 1, (3, 2 15 ) q 2, [2 15, + ) x + 1 = 1, +1 x + 2 = 3, +2 x + 2 3, +2 x , +1 Construct E with k variables Guess Solution y [Borosh and Treybig - AMS 76] Truncate y using Stuttering Theorem and check φ. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

52 Our Main NP Algorithm PLTL[C] over subclasses of CFS Theorem (Flat Counter Systems) MC(PLTL[C], CFS) is NP-complete. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

53 Our Main NP Algorithm PLTL[C] over subclasses of CFS Theorem (Flat Counter Systems) MC(PLTL[C], CFS) is NP-complete. Theorem (Bounded Loop 2) MC(PLTL[C], CPS(n)) is NP-complete for n 2. Different complexity bound than KPS(n)! S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

54 Our Main NP Algorithm PLTL[C] over subclasses of CFS Theorem (Flat Counter Systems) MC(PLTL[C], CFS) is NP-complete. Theorem (Bounded Loop 2) MC(PLTL[C], CPS(n)) is NP-complete for n 2. Different complexity bound than KPS(n)! Theorem (One Loop) MC(PLTL[C], CPS(1)) is in PTime. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

55 Conclusion Conclusion Classes of Systems PLTL[ ] PLTL[C] KPS NP-complete CPS NP-complete NP-complete KPS(n) PTime CPS(n), n > 1?? NP-complete CPS(1) PTime PTime KFS NP-complete CFS NP-complete NP-complete S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

56 Conclusion Conclusion On Going Work: Classes of Systems PLTL[ ] PLTL[C] KPS NP-complete CPS NP-complete NP-complete KPS(n) PTime CPS(n), n > 1?? NP-complete CPS(1) PTime PTime KFS NP-complete CFS NP-complete NP-complete Extending to linear mu-calculus, ETL,... Implementation using SMT Solvers. S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34

57 Conclusion Thank You! S. Demri, A.K. Dhar, A. Sangnier (LIAFA, LSV) Taming Past LTL and Flat Counter Systems June 26, / 34