Mechanizing the Minimization of Deterministic Generalized B¨ uchi Automata Souheib Baarir 1 , 2 Alexandre Duret-Lutz 3 1 Universit´ e Paris Ouest Nanterre la D´ efense, Nanterre, France 2 Sorbonne Universit´ es, UPMC Univ. Paris 6, UMR 7606, LIP6, Paris, France souheib.baarir@lip6.fr 3 LRDE, EPITA, Le Kremlin-Bicˆ etre, France adl@lrde.epita.fr FORTE’14, 3–5 June 2014 1 / 14
Context Model checking prop. LTL → BA y/n sys. ◮ B¨ uchi Automata are used in many formal methods, but with different requirements. Prob. model checking prop. LTL → DBA prob. sys. Synthesis prop. LTL → DBA ctrl. sys. 2 / 14
Context Model checking prop. LTL → BA y/n sys. ◮ B¨ uchi Automata are used in many formal methods, but with different requirements. Prob. model checking ◮ Small [D]BA helps prop. LTL → DBA prob. ◮ Minimization (NP-comp.), sys. ◮ Simulation-based algorithms, ◮ generalized acceptance , ◮ transition-based Synthesis acceptance . prop. LTL → DBA ctrl. sys. 2 / 14
Transion-based Generalized Acceptance Minimal B¨ uchi automaton for GF a ∧ GF b : ¯ ab s 1 s 2 a ¯ ab a ¯ ab b ¯ ab s 0 ¯ b BA 3 / 14
Transion-based Generalized Acceptance Minimal automata for GF a ∧ GF b : ¯ ab s 1 s 2 s 1 a ¯ ab ¯ a a ¯ a ab b b ¯ ab s 0 s 0 ¯ ¯ b b BA TBA Using Transition-based and Generalized acceptance allows more compact automata. 3 / 14
Transion-based Generalized Acceptance Minimal automata for GF a ∧ GF b : a ¯ b ¯ ab s 1 s 2 s 1 a ¯ ab a ¯ a s 0 ab ab ¯ ¯ a ab b b ¯ ab s 0 s 0 ¯ ¯ b b a ¯ ¯ b BA TBA TGBA with F = { , } Using Transition-based and Generalized acceptance allows more compact automata. 3 / 14
Objective Model checking prop. LTL → BA ◮ Small [D]BA helps y/n sys. ◮ Minimization (NP-comp.), ◮ Simulation-based algorithms, ◮ generalized acceptance, Prob. model checking ◮ transition-based prop. LTL → DBA acceptance. prob. sys. ◮ Our objective: building minimal DTGBA Synthesis prop. LTL → DBA ctrl. sys. 4 / 14
Objective Model checking prop. LTL → BA ◮ Small [D]BA helps y/n sys. ◮ Minimization (NP-comp.), ◮ Simulation-based algorithms, ◮ generalized acceptance, Prob. model checking ◮ transition-based prop. LTL → DBA acceptance. prob. sys. ◮ Our objective: building minimal DTGBA Synthesis LTL → mDTGBA prop. LTL → DBA ctrl. sys. 4 / 14
Objective Model checking prop. LTL → BA ◮ Small [D]BA helps y/n sys. ◮ Minimization (NP-comp.), ◮ Simulation-based algorithms, ◮ generalized acceptance, Prob. model checking ◮ transition-based prop. LTL → DBA acceptance. prob. sys. ◮ Our objective: building minimal DTGBA Synthesis LTL → mDTGBA prop. LTL → DBA ◮ We tackle NP-completeness ctrl. sys. via SAT solving 4 / 14
General Framework 1 Introduction 2 General Framework LTL Hierarchy: Determinization & Minimization Our Proposed Framework 3 SAT-based Minimization Equivalence Check of Two DTGBA SAT-Based Synthesis of Equivalent DTGBA Minimization by Iterative Synthesis 4 Conclusion 5 / 14
LTL Hierarchy: Determinization & Minimization BA Reactivity � GF p i ∨ FG q i Recurrence Persistence GF p FG p Obligation � G p i ∨ F q i Safety Guarantee G p F p Z. Manna and A. Pnueli. A hierarchy of temporal properties. PODC’90 6 / 14
LTL Hierarchy: Determinization & Minimization BA Reactivity ◮ Recurrence properties are � GF p i ∨ FG q i DBA DBA-realizable. (E.g. via Rabin) Recurrence Persistence GF p FG p Obligation � G p i ∨ F q i Safety Guarantee G p F p Z. Manna and A. Pnueli. A hierarchy of temporal properties. PODC’90 6 / 14
LTL Hierarchy: Determinization & Minimization BA Reactivity ◮ Recurrence properties are � GF p i ∨ FG q i DBA DBA-realizable. (E.g. via Rabin) Recurrence Persistence Weak BA GF p FG p Obligation � G p i ∨ F q i Safety Guarantee G p F p Z. Manna and A. Pnueli. A hierarchy of temporal properties. PODC’90 6 / 14
LTL Hierarchy: Determinization & Minimization BA Reactivity ◮ Recurrence properties are � GF p i ∨ FG q i DBA DBA-realizable. (E.g. via Rabin) ◮ WDBA can be minimized in Recurrence Persistence polynomial time. GF p FG p Obligation Weak � G p i ∨ F q i DBA Safety Guarantee G p F p C. Dax, J. Eisinger, and F. Klaedtke. Mechanizing the powerset construction for restricted classes of ω -automata. ATVA’07 6 / 14
LTL Hierarchy: Determinization & Minimization BA Reactivity ◮ Recurrence properties are � GF p i ∨ FG q i DBA DBA-realizable. (E.g. via Rabin) ◮ WDBA can be minimized in Recurrence Persistence polynomial time. GF p FG p TCONG ◮ Some recurrences (the TCONG class) can always be Obligation Weak � G p i ∨ F q i determininized to DTBA by DBA powerset construction. Safety Guarantee G p F p C. Dax, J. Eisinger, and F. Klaedtke. Mechanizing the powerset construction for restricted classes of ω -automata. ATVA’07 6 / 14
LTL Hierarchy: Determinization & Minimization BA Reactivity ◮ Recurrence properties are � GF p i ∨ FG q i DBA DBA-realizable. (E.g. via Rabin) ◮ WDBA can be minimized in Recurrence Persistence polynomial time. GF p FG p ◮ Some recurrences (the TCONG class) can always be Obligation Weak � G p i ∨ F q i determininized to DTBA by DBA powerset construction. ◮ So far, no technique for: Safety Guarantee ◮ Determinization of TGBA, G p F p ◮ Minimization of DTGBA. C. Dax, J. Eisinger, and F. Klaedtke. Mechanizing the powerset construction for restricted classes of ω -automata. ATVA’07 6 / 14
From LTL to Minimal D[T][G]BA Output: DBA. (Ehlers’ setup.) minimal DBA SAT DBA minimization LTL not a formula recurrence fail attempt ltl2dstar simplify conversion (DRA) success DBA to DBA R. Ehlers. Minimising DBA precisely using SAT solving. SAT’10 S. C. Krishnan et al. Deterministic ω -automata vis-a-vis DBA. ISAAC’94 7 / 14
From LTL to Minimal D[T][G]BA Output: DBA. minimal minimal DBA SAT WDBA DBA minimization LTL not a formula recurrence success fail attempt attempt ltl2dstar simplify conversion WDBA (DRA) success fail DBA to DBA minim. 8 / 14
From LTL to Minimal D[T][G]BA Output: DTBA. minimal minimal DTBA SAT WDBA DTBA minimization LTL not a formula recurrence success fail attempt attempt ltl2dstar simplify conversion WDBA (DRA) success fail DBA to DBA minim. 8 / 14
From LTL to Minimal D[T][G]BA Output: DTBA. |F | > 1 degen to TBA translate simplify to TGBA TGBA else minimal minimal DTBA SAT WDBA DTBA minimization LTL not a formula recurrence success fail attempt attempt ltl2dstar simplify conversion WDBA (DRA) success fail DBA to DBA minim. 8 / 14
From LTL to Minimal D[T][G]BA Output: DTBA. |F | > 1 degen to TBA attempt fail translate simplify WDBA to TGBA TGBA else minim. success minimal minimal DTBA SAT WDBA DTBA minimization LTL not a formula recurrence success fail attempt attempt ltl2dstar simplify conversion WDBA (DRA) success fail DBA to DBA minim. 8 / 14
From LTL to Minimal D[T][G]BA Output: DTBA. attempt not in |F | > 1 nondet. fail degen powerset TCONG to TBA attempt to DTBA success fail translate simplify WDBA to TGBA TGBA else det. minim. success minimal minimal DTBA SAT WDBA DTBA minimization LTL not a formula recurrence success fail attempt attempt ltl2dstar simplify conversion WDBA (DRA) success fail DBA to DBA minim. 8 / 14
From LTL to Minimal D[T][G]BA Output: DTGBA ( m > 1) or DTBA ( m = 1). nondet. or attempt not in |F | > m = 1 nondet. fail degen powerset TCONG to TBA attempt to DTBA success fail translate simplify WDBA to TGBA TGBA else det. minim. success m = 1 minimal minimal DTBA SAT WDBA DTBA minimization LTL not a m > 1 formula recurrence minimal DTGBA SAT success DTGBA minimization fail attempt attempt ltl2dstar simplify conversion WDBA (DRA) success fail DBA to DBA minim. 8 / 14
From LTL to Minimal D[T][G]BA Output: DTGBA ( m > 1) or DTBA ( m = 1). Our setup. nondet. or ltl2tgba attempt not in |F | > m = 1 nondet. fail degen powerset TCONG to TBA attempt to DTBA success fail translate simplify WDBA to TGBA TGBA else det. minim. success m = 1 minimal minimal DTBA SAT WDBA DTBA minimization LTL not a m > 1 formula recurrence minimal DTGBA SAT success DTGBA minimization fail attempt attempt ltl2dstar simplify conversion WDBA (DRA) success fail DBA to DBA minim. dstar2tgba 8 / 14
SAT-based Minimization 1 Introduction 2 General Framework LTL Hierarchy: Determinization & Minimization Our Proposed Framework 3 SAT-based Minimization Equivalence Check of Two DTGBA SAT-Based Synthesis of Equivalent DTGBA Minimization by Iterative Synthesis 4 Conclusion 9 / 14
Equivalence Check of Two DTGBA a ¯ b ¯ b a b ab ab ¯ a ¯ a ¯ ¯ b A B 10 / 14
Equivalence Check of Two DTGBA Two complete DTGBA A and B are equivalent iff: for each elementary cycle c of A ⊗ B , c | A is accepting ⇐⇒ c | B is accepting. a ¯ a ¯ a ¯ b ¯ b b ¯ b a a ¯ b ab ab ¯ ab b ab ¯ ab a ¯ a ¯ ¯ b a ¯ ab ¯ ¯ b A B A ⊗ B (acceptance marks omitted) 10 / 14
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