Bounded Model Checking with Parametric Data Structures

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1 Bounded Model Checking with Marc Herbstritt (joint work with Erika Ábrahám, Bernd Becker, Martin Steffen) August th International Workshop on Bounded Model Checking

2 Context Automated analysis of complex systems is mandatory Especially safety-critical ones Real-world scenarios embed discrete control in continuous environments Modeling relies on hybrid automata Bounded Model Checking for correctness analysis Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

3 Idea Symmetry is inherent to Bounded Model Checking! Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

4 Idea Symmetry is inherent to Bounded Model Checking! Exploiting symmetry algorithmically. Strichmann (FMSD 2004): Constraints replication Constraints sharing Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

5 Idea Symmetry is inherent to Bounded Model Checking! Exploiting symmetry algorithmically. Strichmann (FMSD 2004): Constraints replication Constraints sharing Exploiting symmetry by data structure. Our proposal: Parametric variables Parametric clauses Parametric watch literals Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

6 Idea Symmetry is inherent to Bounded Model Checking! Exploiting symmetry algorithmically. Strichmann (FMSD 2004): Constraints replication Constraints sharing Exploiting symmetry by data structure. Our proposal: Parametric variables Parametric clauses Parametric watch literals Reduces CPU time. Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

7 Idea Symmetry is inherent to Bounded Model Checking! Exploiting symmetry algorithmically. Strichmann (FMSD 2004): Constraints replication Constraints sharing Exploiting symmetry by data structure. Our proposal: Parametric variables Parametric clauses Parametric watch literals Reduces CPU time. Reduces memory. Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

8 Outline Bounded Model Checking for Linear Hybrid Automata Leads to high memory requirements Memory-aware storage Parametric data structures Experimental results Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

9 Bounded Model Checking for Linear Hybrid Automata Linear Hybrid Automata Bounded Model Checking Hybrid automaton (Thermostat controller) x=x max off ẋ 0 x x min x=x min x=x max on ẋ 0 x x max Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

10 Bounded Model Checking for Linear Hybrid Automata Linear Hybrid Automata Bounded Model Checking Hybrid automaton (Fischer s mutual exclusion protocol) used for asynchronous distributed systems H 1: k=0 idle 1 k=0 x1:=0 x 1 B k 1 test x ẋ1 1 1 A k,x 1:=1,0 wait ẋ1 1 x 1 B k=1 crit 1 k:=0 H 2: k=0 idle 2 k=0 x2:=0 x 2 B k 2 test 2 1 ẋ x 2 A k,x 2:=2,0 wait 2 1 ẋ x 2 B k=2 crit 2 k:=0 Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

11 BMC of Linear Hybrid Automata Linear Hybrid Automata Bounded Model Checking Counterexamples of length k described by first-order logic formulas over (R, +, <, 0, 1): ϕ k (s 0,..., s k ) : Init(s 0 ) Trans(s 0, s 1 )... Trans(s k 1, s k ) Bad(s k ) ϕ k is satisfiable exists run of length k leading to an unsafe state Check ϕ k incrementally for k = 0, 1,... using a suitable solver [BMC for discrete systems: Biere et al. (TACAS 1999)] Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

12 Linear Hybrid Automata Bounded Model Checking SAT-LP-Solver: Lazy SMT approach φ Boolean abstraction SAT solver unsat UNSAT sat (In)equation set Explanation unsat LP solver sat SAT [Ábrahám et al. (VMCAI 2005)] Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

13 Learning Learning Boolean Conflicts Learning Real-Valued Conflicts Two kinds of learning: B-conflicts in boolean domain R-conflicts in real-valued domain sat (In)equation set φ Boolean abstraction SAT solver unsat unsat Explanation UNSAT LP solver sat SAT [Boolean constraints sharing: Strichmann (CAV 2000)] [Boolean constraints replication: Strichmann (CHARME 2001)] Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

14 Learning: B-Conflicts Learning Boolean Conflicts Learning Real-Valued Conflicts Iteration k: I 0 T 0,1 T 1,2... T k 2,k 1 T k 1,k S k boolean conflict a(0) & b(0) boolean conflict a(2) & b(2) boolean conflict a(k) & b(k) Iteration k+1: I 0 T 0,1 T 1,2... Tk 2,k 1 T k 1,k T k,k+1 S k+1 boolean conflict a(0) & b(0) boolean conflict a(k+1) & b(k+1) boolean conflict a(1) & b(1)... boolean conflict a(k) & b(k) Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

15 Learning: B-Conflicts Learning Boolean Conflicts Learning Real-Valued Conflicts Constraints harvesting Combining constraints sharing and constraints replication: Constraint is added in next BMC iteration, shifted by 1 time frame only. Iteration k: I 0 T 0,1 T 1,2... T k 2,k 1 T k 1,k S k boolean conflict a(0) & b(0) boolean conflict a(2) & b(2) boolean conflict a(k) & b(k) Iteration k+1: I 0 T 0,1 T 1,2... Tk 2,k 1 T k 1,k T k,k+1 S k+1 boolean conflict a(0) & b(0) boolean conflict a(2) & b(2) boolean conflict a(k+1) & b(k+1) boolean conflict a(3) & b(3) Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

16 Learning: R-Conflicts Learning Boolean Conflicts Learning Real-Valued Conflicts Iteration k: I 0 T 0,1 T 1,2... Iteration k+1: T k 2,k 1 real conflict x(1)>3 & x(1)<0 real conflict x(2)>3 & x(2)<0... real conflict x(k 1)>3 & x(k 1)<0 T k 1,k S k I 0 T 0,1 T 1,2... Tk 2,k 1 T k 1,k T k,k+1 S k+1 real conflict x(1)>3 & x(1)<0 real conflict x(2)>3 & x(2)<0... real conflict x(k 1)>3 & x(k 1)<0 real conflict x(k)>3 & x(k)<0 Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

17 Memory requirements Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Is memory really a limiting factor? Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

18 Memory requirements Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Our experience: Yes! Is memory really a limiting factor? Learning decreases CPU time, but increases memory consumption. heap peak [MB] Fischer s protocol for 4 processes non-parametric iteration Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

19 Memory requirements Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Our experience: Yes! Is memory really a limiting factor? Learning decreases CPU time, but increases memory consumption. Other approaches for reducing memory: heap peak [MB] Fischer s protocol for 4 processes non-parametric iteration Ganai et al. (CHARME 2003): Distributed SAT, proprietary benchmarks Dershowitz et al. (SAT 2005): QBF formulation of BMC, proprietary benchmarks Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

20 Parametric solver structure Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Basic idea Two-Watch-Literal scheme (Moskewicz et al., DAC 2001): Need to watch only two literals instead of whole clause Overhead of learning: Non-Watch-Literals are also stored Solution: Parameterize variables and clauses Store only parameterized watch literals for (learned) clauses Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

21 Parametric solver structure: Variables Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Variables are represented by pairs (a, i) abstract id a identifies variable s name instance id i identifies variable s time frame Example: if variable x has abstract id 2, then x 0 is identified by (2, 0), and x 5 is identified by (2, 5). Clauses are also represented by pairs (a, i) index i is used as offset for instance id s of the variables Example: If the 7th (abstract) clause has literals {(5, 0), (8, 1)}, then (7, 0) identifies the clause {(5, 0), (8, 1)}, and (7, 2) identifies the clause {(5, 2), (8, 3)}. Advantage: Easy conflict shifting Advantage: Reduced memory requirements Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

22 Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Parametric solver structure: Two-Watch-Literal Scheme SAT-solver exploits Two-Watch-Literal scheme (Chaff) Memory reduction by compact clause representation Non parametric clauses: Parametric clauses: T1(1) T2(1) T1 T2 watches: watches: 1: watches: T1(2) T2(2) 2: k: watches: T1(k) T2(k) Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

23 Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Experimental Results: Tcp (discrete system) heap peak [MB] Benchmark TCP parametric non-parametric depth k Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

24 Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Experimental Results: Fischer s protocol for 3 processes Memory requirements heap peak [MB] parametric non-parametric iteration Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

25 Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Experimental Results: Fischer s protocol for 3 processes CPU times cpu time [secs] parametric non-parametric iteration Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

26 Memory Requirements Motivation for Parametric Data Structure Variable and Clause Parametrization Experimental Results Experimental Results: Fischer4, Railroad Fischer s protocol for 4 processes Railroad Crossing heap peak [MB] parametric non-parametric heap peak [MB] parametric non-parametric iteration iteration Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

27 Conclusion Symmetry-based learning sets high memory requirements Parametric data structures parameterize Two-Watch-Literal scheme strongly reduce memory consumption without slowing down the computation Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

28 Questions and Answers Thank you for your attention! Questions? Answers! Ábrahám,Herbstritt,Becker,Steffen BMC with [BMC 06]

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