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Liveness Properties Typical property: F p Bounded check: Is there a path of length k, that ends in a loop, such that never p ? ¬p ¬p ¬p ¬p ¬p . . . s0 s1 s2 sk-1 sk Counterexamples for liveness properties end in a loop Comp 3153 Ansgar Fehnker Liveness Properties Given a Kripke Stucture M=(S, S 0, R, L) The property p never holds in the states 1..k if ¬p ¬p ¬p ¬p ¬p . . . s0 s1 s2 sk-1 sk Comp 3153 Ansgar Fehnker Liveness Properties Formulation as SAT problem � The liveness property Fp is valid up to cycle k iff Ω(k) is unsatisfiable: Comp 3153 Ansgar Fehnker Liveness Properties Given a Kripke Stucture M=(S, S 0, R, L) The reachable states in k steps are captured by: ¬p ¬p ¬p ¬p ¬p . . . s0 s1 s2 sk-1 sk Comp 3153 Ansgar Fehnker Liveness Properties Given a Kripke Stucture M=(S, S 0, R, L) The path ends in a loop if ¬p ¬p ¬p ¬p ¬p . . . s0 s1 s2 sk-1 sk Comp 3153 Ansgar Fehnker Liveness Properties Example: another bit counter 00 11 Transition: R: l’ = (l ≠ r) ∧ 01 10 r’ = ¬ r Property: F (l ∧ r). The property holds within 2 steps if Ω(k) is unsatisfiable Comp 3153 Ansgar Fehnker Initial state: I 0= ¬ l ∧ ¬ r 4
LTL Checking an arbitrary formula φ LTL 1. Build Buechi Automaton for ¬ φ 2. Compose A x A ¬ φ 3. Check if L(A x A ¬ φ ) is empty Bounded check: Is there a path of length k, that ends in a loop with an accepting state? Comp 3153 Ansgar Fehnker Given Buechi Automaton A= 〈Σ, Q, Q 0,δ, F〉 LTL The reachable states in k steps are captured by: F . . . s0 s1 s2 sk-1 sk Comp 3153 Ansgar Fehnker Formulation as SAT problem � The LTL formula formula φ is valid up to cycle k iff Ω(k) for Buechi automaton A x A ¬ φ is not satisfiable: Comp 3153 Ansgar Fehnker LTL Given Buechi Automaton A= 〈Σ, Q, Q 0,δ, F〉 LTL Check fairness condition GF f L(A) is non-empty if a witness exists Witness ends in a loop with some state satisfying f F . . . s0 s1 s2 sk-1 sk Comp 3153 Ansgar Fehnker Given Buechi Automaton 〈Σ, Q, Q 0,δ, F〉 Path ends in a loop, with accepting state F . . . s0 s1 s2 sk-1 sk Comp 3153 Ansgar Fehnker Bounded Model Checking Safety � Check if counterexample of length k exists Liveness LTL � Check if counterexample of length k exists that ends in loop � Check if accepting run of length k exists in Buechi automaton of the negation Comp 3153 Ansgar Fehnker 5
Page 1 and 2: Overview Algorithmic Verification C
Page 3: Bounded Model Checking Reminder �
Page 7: Tuning SAT for BMC Replicating Clau

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