Taming Past LTL and Flat Counter Systems

Transcription

1 Taming Past LTL and Flat Counter Systems Amit Kumar Dhar LIAFA, Univ Paris Diderot, Paris Cité Sorbonne, CNRS, France April 2, 2012 Joint work with : Stéphane Demri(LSV) and Arnaud Sangnier(LIAFA) LIAFA Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

2 Model Checking A System Satisfies A Property Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

3 Model Checking A System Satisfies A Property Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

4 Model Checking A System Satisfies A Property φ Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

5 Model Checking A System Satisfies A Property = φ Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

6 Model Checking A System Satisfies A Property = φ Decision Procedure ψ(x 1, x 2,, x n ) in decidable theory Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

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 Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

8 Table of Contents 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

9 Table of Contents Introduction Models 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

10 Kripke Structures (KS) Introduction Models 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. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

11 Counter Systems (CS) Introduction Models 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) S = (Q, C n,, l) Guards : Boolean combination of linear constraints of the form Σ i a i.x i b and {=,,, <, >}. Updates : u Z n are translations. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

12 Introduction Models 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 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 ). Main Drawback: Most of the verification problems are undecidable for counter systems. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

13 Introduction Flat Counter Systems (CFS) Models 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 S = (Q, C n,, l) Guards : Boolean combination of linear constraints of the form Σ i a i.x i b and {=,,, <, >}. Updates : u Z n are translations. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

14 Introduction Flat Counter Systems (CFS) Models 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) S = (Q, C n,, l) Guards : Boolean combination of linear constraints of the form Σ i a i.x i b and {=,,, <, >}. Updates : u Z n are translations. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

15 Introduction Flat Counter Systems (CFS) Models 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) S = (Q, C n,, l) Guards : Boolean combination of linear constraints of the form Σ i a i.x i b and {=,,, <, >}. Updates : u Z n are translations. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

16 Introduction Flat Kripke Structures (KFS) Models q 7 q 9 q 8 q 6 q 10 q 4 q 3 q 5 q 1 q 2 S = (Q,, l) Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

17 Table of Contents Introduction Logic 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

18 Introduction Temporal Logic PLTL[ ] (Standard version) Logic 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 j Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

19 Introduction Logic PLTL[C] (with arithmetical constraints) Logic 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 here is defined as σ Σ ω and Σ = 2 AP N n σ, i = g def v i = g where v i (x j ) = π 2 (σ(i))(x j ) Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

20 Example of PLTL[C] Introduction Logic 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 = U(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 Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

21 Table of Contents Introduction Related Works 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

22 Introduction Related Works Related Works 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] Flateness 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. See [Demri et al. - JANCL 10] Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

23 Table of Contents Introduction Problem 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

24 Introduction Problem 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? Problems Studied: 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]). Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

25 Table of Contents Path Schemas 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

26 P = p 1 l 1 + p 2l 2 + p 3l3 ω p 1 = (e 10 ) l 1 = (e 1, e 2, e 3, e 4 ) p 2 = (e 1, e 5, e 6 ) l 2 = (e 7, e 8, e 86 ) p 3 = (e 7, e 9 ) l 3 = (e 0 ) Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52 Path Schemas 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

27 Path Schemas 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 3l3 ω p 1 = (e 10 ) l 1 = (e 1, e 2, e 3, e 4 ) p 2 = (e 1, e 5, e 6 ) l 2 = (e 7, e 8, e 86 ) p 3 = (e 7, e 9 ) l 3 = (e 0 ) Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

28 Path Schemas 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 3l3 ω p 1 = (e 10 ) l 1 = (e 1, e 2, e 3, e 4 ) p 2 = (e 1, e 5, e 6 ) l 2 = (e 7, e 8, e 86 ) p 3 = (e 7, e 9 ) l 3 = (e 0 ) Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

29 Why Path Schemas? Path Schemas Exponentially many minimal path schemas in a flat system. Every run in a flat system respects one of the minimal path schemas of the system. Simpler structure Easy to study. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

30 Why Path Schemas? Path Schemas Exponentially many minimal path schemas in a flat system. Any transition occurs at most twice. Every run in a flat system respects one of the minimal path schemas of the system. Simpler structure Easy to study. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

31 Path Schemas Why Path Schemas? Exponentially many minimal path schemas in a flat system. 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. Simpler structure Easy to study. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

32 Table of Contents PLTL[ ] over KPS,KFS 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

33 PLTL[ ] over KPS,KFS 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[ ]. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

34 Stuttering Theorem PLTL[ ] over KPS,KFS Stuttering Theorem PLTL[ ] formula: φ = ((q 5 Xq 6 ) q 6 )Uq 7 Temporal Depth (td): q 5 q 6 q 5 q 6... q 5 q 6 q 5 q 6 q 7... = φ q 5 q 6 q 7... = φ Formula (φ) td(φ) U(q 7 Xq 6 c 1 > 5) 2 ((q 5 Xq 6 ) (q 6 Xq 5 ))Uq 7 2 q 2 U(q 3 U(Xq 4 )) 3 Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

35 PLTL[ ] over KPS,KFS Stuttering Theorem Stuttering Theorem Theorem Given two models σ, σ such that σ = σ 1 s M σ 2, σ = σ 1 s M σ 2 (2 AT ) ω and M, M 2N + 1,N 2 then, for every PLTL[ ] formula φ with td(φ) N, we have σ, 0 = φ iff σ, 0 = φ. Proof. The proof proceeds by induction on the structure of the formula: For each temporal operator show that the satifiability does not change by changing the number of repetitions of s by 1. Easy to see that 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). Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

36 PLTL[ ] over KPS,KFS Stuttering Theorem PLTL[ ] over Kripke Path Schema Theorem MC(PLTL[ ], KPS) is in NP. Proof. 1 Guess the number of times each loop will be taken - at most 2.td(φ) Unfold the loops to obtain an ultimately periodic path of at most polynomial size. 3 Check for the satisfiability of the formula over the ultimately periodic path in polynomial time [Laroussinie, Markey and Schnoebelen - LICS 02]. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

37 PLTL[ ] over KPS PLTL[ ] over KPS,KFS Stuttering Theorem 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 )]. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

38 PLTL[ ] over KPS PLTL[ ] over KPS,KFS Stuttering Theorem 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) Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

39 PLTL[ ] over KPS,KFS Stuttering Theorem PLTL[ ] over KFS Theorem MC(PLTL[ ], KFS) is NP-complete. Proof. 1 Guess a minimal path schema in the Kripke structure. 2 Apply the algorithm for model checking of KPS. Note: Finite number of minimal path schemas of at most polynomial length in a given Kripke structure. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

40 PLTL[ ] over KPS,KFS Stuttering Theorem PLTL[ ] over KPS(n) Fixing the number of the loops makes the problem easier Lemma MC(PLTL[ ], KPS(n)) is in PTime. Proof. Enumerate all possible paths that can be obtained by repeating each loop at most (2.td(φ) + 5) times, and check for the satisfiability of the formula. Number of different possible paths = (2.td(φ) + 5) n (It is polynomial for a fixed n). Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

41 Table of Contents PLTL[C] over CPS, CFS 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

42 PLTL[C] over CPS, CFS 3 main ingredients for proving MC(PLTL[C], CFS) in NP 1 Characterize all valid runs in a counter system (without disjunction) by System of equations. Respecting the updates, guards and non-negative counter values. 2 Elimination of disjunction in guards and arithmetical constraints in formula. 3 Stuttering theorem for PLTL. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

43 Table of Contents PLTL[C] over CPS, CFS Characterizing runs by equations 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

44 PLTL[C] over CPS, CFS Characterizing runs by equations Characterizing runs by equations Relation between system of equations and path schemas Equivalence between : (n 1, n 2,..., n k 1 ) is a solution of the system of equation. There exists a run in the path schema in which loop l i is taken n i times. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

45 PLTL[C] over CPS, CFS Characterizing runs by equations Characterizing runs by equations Relation between system of equations and path schemas Equivalence between : (n 1, n 2,..., n k 1 ) is a solution of the system of equation. There exists a run in the path schema in which loop l i is taken n i times. Number of variables in equation = Number of loops in path schema Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

46 PLTL[C] over CPS, CFS Characterizing runs by equations Characterizing runs by equations Relation between system of equations and path schemas Equivalence between : (n 1, n 2,..., n k 1 ) is a solution of the system of equation. There exists a run in the path schema in which loop l i is taken n i times. Number of variables in equation = Number of loops in path schema Solution of equations signifies the number of times each loop in the path schema is taken. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

47 PLTL[C] over CPS, CFS Characterizing runs by equations 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 (for counter c 2 ): Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

48 PLTL[C] over CPS, CFS Characterizing runs by equations 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 (for counter c 2 ): To ensure that it is taken at least once: 0.X < 25. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

49 PLTL[C] over CPS, CFS Characterizing runs by equations 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 (for counter c 2 ): To ensure that it is taken at least once: To ensure that it is taken X 2 times: 0.X < X (X 2 1) < 25 Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

50 PLTL[C] over CPS, CFS Characterizing runs by equations Characterizing runs by equations Lemma Let S = Q, C n,, l be a flat counter system without disjunctions in guards, P = p 1 l 1 + p 2l p klk ω be one of its valid path schemas and c 0 be a configuration. One can compute a constraint system E such that the set of solutions of E is the number of times each loop in P can be taken, E has k 1 variables, The greatest absolute value of constants in the equations is polynomial in the maximum constant appearing in the input. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

51 Table of Contents PLTL[C] over CPS, CFS Elimination of Disjunction and Constraints 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

52 PLTL[C] over CPS, CFS Elimination of Disjunction and Constraints Elimination of Disjunction and Constraints Counter Path Schema: x 3 x = 1, +2 1 ω q 0 q 1 q 2, +1, +1 PLTL[C] formula : φ = F((x 2)Uq 2 ) Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

53 PLTL[C] over CPS, CFS Elimination of Disjunction and Constraints Elimination of Disjunction and Constraints Intervals = {[, 1), [1, 1], (1, 2), [2, 2], (2, 3), [3, 3], (3, ]}. Terms = {x}. x 3 x = 1, +2 1 ω q 0 q 1 q 2, +1, +1 x + 2 > 3, +2 1 ω q 0, [, 1) q 1, [1, 1] q 1, [3, 3] q 1, (3, ] x + 1 = 1, +1 x + 2 = 3, +2 x + 2 3, +2 x + 1 3, +1 q 2, (3, ] Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

54 PLTL[C] over CPS, CFS Elimination of Disjunction and Constraints Elimination of Disjunction and Constraints Properties of the set of unfolded Path Schemas (Y P ): No path schema in Y P contains disjunction in guards. Every path schema in Y P is polynomial in the size of the input. Runs respecting the original path schema = Runs respecting all the path schemas in Y P. Checking whether a path schema is in Y P can be done in polynomial time. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

55 PLTL[C] over CPS, CFS Elimination of Disjunction and Constraints Elimination of Disjunction and Constraints Properties of the set of unfolded Path Schemas (Y P ): No path schema in Y P contains disjunction in guards. Every path schema in Y P is polynomial in the size of the input. Runs respecting the original path schema = Runs respecting all the path schemas in Y P. Checking whether a path schema is in Y P can be done in polynomial time. Y P is equivalent to P Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

56 Table of Contents PLTL[C] over CPS, CFS NP-Algorithm for PLTL[C] over CPS 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

57 PLTL[C] over CPS, CFS NP-Algorithm for PLTL[C] over CPS NP-Algorithm for PLTL[C] over CPS 1: guess a compatible unfolding and intervals for each term on each node and formula and construct P = p 1 l + 1 p 2l l + k 1 p kl ω k 2: guess y [1, 2td(φ) + 5] k 1 3: guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 4: check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] p k lk ω, 0 = symb φ k 1 Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

58 PLTL[C] over CPS, CFS NP-Algorithm for PLTL[C] over CPS NP-Algorithm for PLTL[C] over CPS 1: guess a compatible unfolding and intervals for each term on each node and formula and construct P = p 1 l + 1 p 2l l + k 1 p kl ω k 2: guess y [1, 2td(φ) + 5] k 1 3: guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 4: check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] k 1 p k lk ω, 0 = symb φ 5: build the constraint system E over the variables y 1,..., y k 1 for P with initial counter values v 0. 6: for i = 1 k 1 do 7: if y[i] = 2td(φ) + 5 then 8: ψ i y i 2td(φ) + 5 9: else 10: ψ i y i = y[i] 11: end if 12: end for 13: check that y = E ψ 1 ψ k 1 Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

59 Algorithm in NP PLTL[C] over CPS, CFS NP-Algorithm for PLTL[C] over CPS 1: guess a compatible unfolding and intervals for each term on each node and construct P = p 1 l + 1 p 2l l + k 1 p kl ω k 2: guess y [1, 2td(φ) + 5] k 1 3: guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 4: check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] k 1 p k lk ω = symb φ 5: build the constraint system E over the variables y 1,..., y k 1 for P with initial counter values v 0. 6: for i = 1 k 1 do 7: if y[i] = 2td(φ) + 5 then 8: ψ i y i 2td(φ) + 5 9: else 10: ψ i y i = y[i] 11: end if 12: end for 13: check that y = E ψ 1 ψ k 1 Polynomial size guesses (may be of exponential values) Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

60 Algorithm in NP PLTL[C] over CPS, CFS NP-Algorithm for PLTL[C] over CPS 1: guess a compatible unfolding and intervals for each term on each node and construct P = p 1 l + 1 p 2l l + k 1 p kl ω k 2: guess y [1, 2td(φ) + 5] k 1 3: guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 4: check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] k 1 p k lk ω = symb φ 5: build the constraint system E over the variables y 1,..., y k 1 for P with initial counter values v 0. 6: for i = 1 k 1 do 7: if y[i] = 2td(φ) + 5 then 8: ψ i y i 2td(φ) + 5 9: else 10: ψ i y i = y[i] 11: end if 12: end for 13: check that y = E ψ 1 ψ k 1 Polynomial size path schema, symbolic model checking and constraint system Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

61 Algorithm in NP PLTL[C] over CPS, CFS NP-Algorithm for PLTL[C] over CPS 1: guess a compatible unfolding and intervals for each term on each node and construct P = p 1 l + 1 p 2l l + k 1 p kl ω k 2: guess y [1, 2td(φ) + 5] k 1 3: guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 4: check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] k 1 p k lk ω = symb φ 5: build the constraint system E over the variables y 1,..., y k 1 for P with initial counter values v 0. 6: for i = 1 k 1 do 7: if y[i] = 2td(φ) + 5 then 8: ψ i y i 2td(φ) + 5 9: else 10: ψ i y i = y[i] 11: end if 12: end for 13: check that y = E ψ 1 ψ k 1 Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

62 Algorithm in NP PLTL[C] over CPS, CFS NP-Algorithm for PLTL[C] over CPS 1: guess a compatible unfolding and intervals for each term on each node and construct P = p 1 l + 1 p 2l l + k 1 p kl ω k 2: guess y [1, 2td(φ) + 5] k 1 3: guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 4: check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] k 1 p k lk ω = symb φ 5: build the constraint system E over the variables y 1,..., y k 1 for P with initial counter values v 0. 6: for i = 1 k 1 do 7: if y[i] = 2td(φ) + 5 then 8: ψ i y i 2td(φ) + 5 9: else 10: ψ i y i = y[i] 11: end if 12: end for 13: check that y = E ψ 1 ψ k 1 Arithmetic operations on polynomial bits Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

63 PLTL[C] over CPS, CFS Correctness of Algorithm NP-Algorithm for PLTL[C] over CPS guess a compatible unfolding and intervals for each term on each node and construct P = p 1 l 1 + p 2l l + k 1 p klk ω guess y [1, 2td(φ) + 5] k 1 guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] k 1 p k lk ω = symb φ build the constraint system E over the variables y 1,..., y k 1 for P with initial counter values v 0. for i = 1 k 1 do if y[i] = 2td(φ) + 5 then ψ i y i 2td(φ) + 5 else ψ i y i = y[i] end if end for check that y = E ψ 1 ψ k 1 Stuttering Invariant Property Ensures the satisfiability of the formula by the path schema Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

64 PLTL[C] over CPS, CFS Correctness of Algorithm NP-Algorithm for PLTL[C] over CPS guess a compatible unfolding and intervals for each term on each node and construct P = p 1 l 1 + p 2l l + k 1 p klk ω guess y [1, 2td(φ) + 5] k 1 guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] k 1 p k lk ω = symb φ build the constraint system E over the variables y 1,..., y k 1 for P with initial counter values v 0. for i = 1 k 1 do if y[i] = 2td(φ) + 5 then ψ i y i 2td(φ) + 5 else ψ i y i = y[i] end if end for check that y = E ψ 1 ψ k 1 Small Solution Property [Borosh and Treybig - AMS 76] exists an exponential solution Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

65 PLTL[C] over CPS, CFS Correctness of Algorithm NP-Algorithm for PLTL[C] over CPS guess a compatible unfolding and intervals for each term on each node and construct P = p 1 l 1 + p 2l l + k 1 p klk ω guess y [1, 2td(φ) + 5] k 1 guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] k 1 p k lk ω = symb φ build the constraint system E over the variables y 1,..., y k 1 for P with initial counter values v 0. for i = 1 k 1 do if y[i] = 2td(φ) + 5 then ψ i y i 2td(φ) + 5 else ψ i y i = y[i] end if end for check that y = E ψ 1 ψ k 1 Combining Two Properties Adding constraints to ensure the previous satisfiability Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

66 PLTL[C] over CPS, CFS Correctness of Algorithm NP-Algorithm for PLTL[C] over CPS guess a compatible unfolding and intervals for each term on each node and construct P = p 1 l 1 + p 2l l + k 1 p klk ω guess y [1, 2td(φ) + 5] k 1 guess y [1, 2 p (size(s)+size(c 0)+size(φ)) ] k 1 check that p 1 l y[1] 1 p 2 l y[2] 2... l y[k 1] k 1 p k lk ω = symb φ build the constraint system E over the variables y 1,..., y k 1 for P with initial counter values v 0. for i = 1 k 1 do if y[i] = 2td(φ) + 5 then ψ i y i 2td(φ) + 5 else ψ i y i = y[i] end if end for check that y = E ψ 1 ψ k 1 Checking Guesses Ensures the validity of run with respect to updates and guards Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

67 PLTL[C] over CPS, CFS NP-Algorithm for PLTL[C] over CPS PLTL[C] over CPS and CFS Theorem MC(PLTL[C], CFS) is NP-complete. Proof. Guess a path schema in the flat counter system. Apply the algorithm for model checking of CPS. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

68 Table of Contents PLTL[C] over CPS, CFS Fixing the number of loops 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

69 PLTL[C] over CPS(n) PLTL[C] over CPS, CFS Fixing the number of loops Lemma MC(PLTL[C], CPS(n)) is NP-complete for n 2. Proof. x 1 2 n, , q 0 q 1, 2 n 2 n 1 ω. 2 1 The formula ψ is defined defined from φ by replacing each occurrence of p i by F(q 1 x i 2 n 2 n i 2 n i+1 x i 2 n 1 2 n i+1 ). (p i is true iff i th bit of the counter value is 1). Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

70 PLTL[C] over CPS, CFS PLTL[C] over CPS(n) (contd.) Fixing the number of loops p p p Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

71 PLTL[C] over CPS, CFS PLTL[C] over CPS(n) (contd.) Fixing the number of loops p p p Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

72 PLTL[C] over CPS, CFS PLTL[C] over CPS(n) (contd.) Fixing the number of loops p p p Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

73 PLTL[C] over CPS, CFS Fixing the number of loops PLTL[C] over CPS(1) Lemma MC(PLTL[C], CPS(1)) is in PTime. Proof. Unfold the single loop deterministically to reflect the intervals of terms on the unique run. check if the unfolded path schema satisfies the formula. Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

74 Table of Contents Conclusion 1 Introduction Models Logic Related Works Problem 2 Path Schemas 3 PLTL[ ] over KPS,KFS Stuttering Theorem 4 PLTL[C] over CPS, CFS Characterizing runs by equations Elimination of Disjunction and Constraints NP-Algorithm for PLTL[C] over CPS Fixing the number of loops 5 Conclusion Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

75 Conclusion Conclusion On Going Works: 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,... Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52

76 Conclusion Thank You! Amit Kumar Dhar (LIAFA) Taming Past LTL and Flat Counter Systems April 2, / 52