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Automata and Formal Lanugages Büchi Automata and Model Checking Ralf Möller based on slides by Chang-Beom Choi Provable Software Lab, KAIST Transition System to Mealy-Style Automaton Translation Example transition system p,q Corresponding automaton i 1 But: No accept states {p,q} 1 q 2 3 p {p,q} {q} {q} 2 Each state is labeled with the propositions that hold in that state 3 {p} Overview Büchi Automata • Büchi Automata – Automata which accept infinite words – named after Julius Richard Büchi, Swiss Logician • Usually used for modeling systems with infinite sequences of states, each of which satisfies certain atomic propositions • Büchi Automaton M accepts sequences of labels for program states: L(M) describes all potential sequences of state labels of the system (and therefore describes system behavior) Overview Büchi Automaton (deterministic version) • Definition – M = (Σ, S, s0, d, F) Σ : alphabet (set of “labels for program states”) S : set of automaton states s0 : initial state d : a transition function (S x Σ x S) F : a set of accepting states Overview Büchi Automaton – M = (Σ, S, s0, d, F) – The input of M is infinite w : a0, a1, … (∈ Σω) – A run is a sequence of states r: s0,s1, … (∈ Sω) • Initiation: s0 ∈ S0 • Consecution : si+1∈ d (si, ai) – Accepting run (r = s0,s1, … ) • There exists an infinite number of integers i ∈ N such that si ∈ F Overview Büchi Automata • P must eventually occur, and if it occurs P holds forever • • • • • Σ = {P, true} S = {q0, q1} s0 = {q0} d = {(q0,true, q0), (q0, P, q1), (q1,P, q1) F = {q1} run : q0, q1, q1, q1, … Overview Büchi Automata • P must eventually occur, and if it occurs P holds forever • • • • • Σ = {P, true} Relation to S = {q0, q1} Linear Temporal s0 = {q0} d = {(q0,true, q0), (q0, P, q1), (q1,P, q1) F = {q1} Logic LTL Properties Büchi automata true Gp p p true Fp G (F p) p p p p p p The size of the property automaton can be exponential in the size of the LTL formula Overview • Model checking – Specify requirement properties and build system model – Generate possible states from the model and then check whether given requirement properties are satisfied within the state space OK Target Program Model Check Requirement Properties (F W) or Error Trace Found Overview • A process of Model Checking – Modeling • Build a model of program or system – Specification • Describe requirement properties – Verification • Checking that a model of the program or system satisfies a given specification Overview • How can we model check of a program or system? – Modeling • Build a Büchi automaton for a given program or system – Specification • Describe requirement properties using Temporal Logic – Verification • Automatically (semi-automatic) Overview Process of Model Checking Requirement Properties Target Program (F Model Checker W)