Post on 22-Dec-2015
Automata and Formal Lanugages
Büchi Automata and Model Checking
Ralf Möller
based on slidesby Chang-Beom Choi
Provable Software Lab, KAIST
Transition System to Mealy-Style Automaton Translation
2
1
3
Each state is labeled with the propositions that hold in that state
Example transition system Corresponding automaton
{p,q}
{p}
{q}
{p,q}
{q}
i
1
2 3
p,q
q p
But: No accept states
OverviewBüchi 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 (andtherefore describes system behavior)
OverviewBüchi Automaton (deterministic version)
• Definition– M = (Σ, S, s0, , F)
Σ : alphabet (set of “labels for program states”)S : set of automaton statess0 : initial state : a transition function (S x Σ x S) F : a set of accepting states
– M = (Σ, S, s0, , 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∈ (si, ai)
– Accepting run (r = s0,s1, … )• There exists an infinite number of integers
i ∈ N such that si ∈ F
OverviewBüchi Automaton
OverviewBüchi Automata
• P must eventually occur, and if it occurs P holds forever
• Σ = {P, true}
• S = {q0, q1}
• s0 = {q0}
• = {(q0,true, q0), (q0, P, q1), (q1,P, q1)
• F = {q1}
run : q0, q1, q1, q1, …
OverviewBüchi Automata
• P must eventually occur, and if it occurs P holds forever
• Σ = {P, true}
• S = {q0, q1}
• s0 = {q0}
• = {(q0,true, q0), (q0, P, q1), (q1,P, q1)
• F = {q1}
Relation to Linear Temporal Logic
LTL Properties Büchi automata
G p p ptrue
F p pptrue
G (F p) p
The size of the property automaton can be exponential in the size of the LTL formula
p
p
p
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
Error TraceFound
or
TargetProgram
RequirementProperties
Model Check
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)
Model Checker
OverviewProcess of Model Checking
Target Program Requirement
Properties