Computer Aided Verification: 24th International Conference, by Wolfgang Thomas (auth.), P. Madhusudan, Sanjit A. Seshia

By Wolfgang Thomas (auth.), P. Madhusudan, Sanjit A. Seshia (eds.)

This publication constitutes the refereed lawsuits of the twenty fourth foreign convention on machine Aided Verification, CAV 2012, held in Berkeley, CA, united states in July 2012. The 38 common and 20 device papers awarded have been conscientiously reviewed and chosen from 185 submissions. The papers are equipped in topical sections on automata and synthesis, inductive inference and termination, abstraction, concurrency and software program verification, biology and probabilistic structures, embedded and regulate platforms, SAT/SMT fixing and SMT-based verification, timed and hybrid structures, verification, protection, verification and synthesis, and power demonstration.

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S | · + 1, . . , d! · |S | · + 1) ∈ Safe(s) ∩ Cover(s). An immediate consequence of Lemma 3 is that Safe(s) = ∅ iff Cover(s) = ∅. Our next lemma shows that the existence of a winning strategy in Streett games is polynomially reducible to the problem whether Safe(s) = ∅ in consumption games. Lemma 4. Let S = (V, →, (V , V ), A) be a Streett game where A = {(G1 , R1 ), . . , (Gm , Rm )}. Let CS = (V, →, (V , V ), L) be a consumption game of dimension m where L(u, v)(i) is either −1, ω, or 0, depending on whether (u, v) ∈ Gi , (u, v) ∈ Ri , or (u, v) Gi ∪ Ri , respectively.

For example, Z<0 is the set of all negative integers, and Zω<0 is the set Z<0 ∪ {ω}. We use Greek letters α, β, . . to denote vectors over Z 0 or Zω0 , and 0 to denote the vector of zeros. The i-th component of a given α is denoted by α(i). The standard component-wise ordering over vectors is denoted by ≤, and we also write α < β to indicate that α(i) < β(i) for every i. Let M be a finite or countably infinite alphabet. A word over M is a finite or infinite sequence of elements of M. The empty word is denoted by ε, and the set of all finite words over M is denoted by M ∗ .

Hence, the acceptance condition can now be expressed as a positive Boolean combination over Rabin pairs in a similar way as the standard Rabin condition is a disjunction of Rabin pairs. Example 13. Let us consider the (strong) fairness constraint ϕ = FGa ∨ GFb. e. of size 1 + 22 = 5. Furthermore, the syntactic tree of U(ϕ) = XFGa ∨ (XGa ∧ a) ∨ (XGFb ∧ (XFb ∨ b)) immediately determines possible sets I. These either contain Ga (possibly with also FGa or some other elements) or GFb, Fb. The first option generates the requirement to visit states with ¬a only finitely often, the second one to visit b infinitely often.

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