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Tree-shaped one-pass tableau systems for Linear Temporal Logic satisfiability checking Nicola Gigante University of Udine, Italy Joint work with Angelo Montanari, Mark Reynolds, Luca Geatti The need for formal verification Safety-critical

  1. Tree-shaped one-pass tableau systems for Linear Temporal Logic satisfiability checking Nicola Gigante University of Udine, Italy Joint work with Angelo Montanari, Mark Reynolds, Luca Geatti

  2. The need for formal verification Safety-critical systems need to avoid bugs at all costs. Formal Verification develops automatic techniques to provide mathematical proofs of software correctness. 2

  3. Formal verification and Logic In formal verification, an abstract model of the system is checked against a formal specification of the desired behavior. Systems are usually modeled as automata . Specifications are usually expressed as temporal logic formulas. This is the model checking problem. 3

  4. Linear Temporal Logic Linear Temporal Logic (LTL) is a propositional modal logic commonly used as specification language. X α α will be true at the next state. β will eventually be true, and α U β α always holds until then. F β ≡ ⊤ U β β will eventually be true. G β ≡ ¬ F ¬ β β will always be true. 4

  5. Linear Temporal Logic (2) If infinitely many requests are received, then infinitely many replies are sent. G F r − → G F q 5

  6. LTL satisfiability checking LTL satisfiability is the problem of checking whether there exists a model that satisfies a given LTL formula. Important to check consistency of specifications. Seemingly very hard problem: PSPACE -complete. Many algorithmic solutions to solve it: (Büchi) Automata-based Tableau methods Temporal resolution Reduction to model checking . . . 6

  7. Tableaux-based methods for LTL satisfiability Common tableaux-based methods for LTL are graph-shaped and require multiple passes to be built and checked for successful paths. ¬ p , q , X F p p , ¬ q , F p 7

  8. Tableaux-based methods for LTL satisfiability Common tableaux-based methods for LTL are graph-shaped and require multiple passes to be built and checked for successful paths. 7

  9. A One-Pass Tree-Shaped Tableau for LTL A one-pass tree-shaped tableau method for LTL satisfiability was recently proposed. GandALF 2016 M. Reynolds. “A New Rule for LTL Tableaux.” In: Proc. of the 7 th International Symposium on Games, Automata, Logics and Formal Verification . GandALF 2016 8

  10. A One-Pass Tree-Shaped Tableau for LTL A one-pass tree-shaped tableau method for LTL satisfiability was recently proposed, implemented. IJCAI 2016 M. Bertello, N. Gigante, A. Montanari, and M. Reynolds. “Leviathan: A New LTL Satisfiability Checking Tool Based on a One-Pass Tree-Shaped Tableau.” In: Proc. of the 25 th International Joint Conference on Artificial Intelligence . IJCAI 2016 http://www.github.com/corralx/leviathan 8

  11. A One-Pass Tree-Shaped Tableau for LTL A one-pass tree-shaped tableau method for LTL satisfiability was recently proposed, implemented, and parallelized. GandALF 2017 John Christopher McCabe-Dansted and Mark Reynolds. “A Parallel Linear Temporal Logic Tableau.” In: Proceedings 8th International Symposium on Games, Automata, Logics and Formal Verification . 2017, pp. 166–179 8

  12. A One-Pass Tree-Shaped Tableau for LTL A one-pass tree-shaped tableau method for LTL satisfiability was recently proposed Purely tree-shaped rule-based search procedure. A single pass is suffjcient to determine the acceptance of rejection of a given branch. Very simple set of rules, combining the simplicity of declarative tableaux with the effjciency of one-pass systems. Easy to extend . Easy to parallelize with huge speedups. 8

  13. Example { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  14. Example { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  15. Example { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  16. Example { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  17. Example p { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  18. Example p { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  19. Example p { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  20. Example p { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  21. Example p { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  22. Example p ¬ p { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  23. Example p ¬ p { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  24. Example p ¬ p { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } 9

  25. Example p ¬ p p ¬ p { G F ( p ∧ X ¬ p ) } { F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } {¬ p , G F ( p ∧ X ¬ p ) } {¬ p , F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } {¬ p , p , X ¬ p , . . . } {¬ p , X F ( p ∧ X ¬ p ) , X G F ( p ∧ X ¬ p ) } ✗ { F ( p ∧ X ¬ p ) , G F ( p ∧ X ¬ p ) } . . . { p , X ¬ p , X G F ( p ∧ X ¬ p ) } ✓ 9

  26. Example - unsatisfiable formula { G ¬ p ∧ q U p } { G ¬ p , q U p } {¬ p , X G ¬ p , p } {¬ p , X G ¬ p , q , X ( q U p ) } { G ¬ p , q U p } {¬ p , X G ¬ p , p } {¬ p , X G ¬ p , q , X ( q U p ) } ✗ { G ¬ p , q U p } {¬ p , X G ¬ p , p } {¬ p , X G ¬ p , q , X ( q U p ) } ✗ 10

  27. Example - unsatisfiable formula { G ¬ p ∧ q U p } { G ¬ p , q U p } {¬ p , X G ¬ p , p } {¬ p , X G ¬ p , q , X ( q U p ) } { G ¬ p , q U p } {¬ p , X G ¬ p , p } {¬ p , X G ¬ p , q , X ( q U p ) } ✗ { G ¬ p , q U p } {¬ p , X G ¬ p , p } {¬ p , X G ¬ p , q , X ( q U p ) } ✗ 10

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