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Bounded underapproximations

Abstract : We show a new and constructive proof of the following language-theoretic result: for every context-free language L, there is a bounded context-free language L'⊆L which has the same Parikh (commutative) image as L. Bounded languages, introduced by Ginsburg and Spanier, are subsets of regular languages of the form w1*w2*... wm* for some w1,...,wm∈ Σ*. In particular bounded context-free languages have nice structural and decidability properties. Our proof proceeds in two parts. First, we give a new construction that shows that each context free language L has a subset LN that has the same Parikh image as L and that can be represented as a sequence of substitutions on a linear language. Second, we inductively construct a Parikh-equivalent bounded context-free subset of LN. We show two applications of this result in model checking: to underapproximate the reachable state space of multithreaded procedural programs and to underapproximate the reachable state space of recursive counter programs. The bounded language constructed above provides a decidable underapproximation for the original problems. By iterating the construction, we get a semi-algorithm for the original problems that constructs a sequence of underapproximations such that no two underapproximations of the sequence can be compared. This provides a progress guarantee: every word w∈L is in some underapproximation of the sequence, and hence, a program bug is guaranteed to be found. In particular, we show that verification with bounded languages generalizes context-bounded reachability for multithreaded programs.
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Contributor : Benedikt Bollig <>
Submitted on : Wednesday, January 16, 2013 - 11:16:05 AM
Last modification on : Monday, February 15, 2021 - 10:49:18 AM

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Pierre Ganty, Rupak Majumdar, Benjamin Monmege. Bounded underapproximations. Formal Methods in System Design, Springer Verlag, 2012, 40 (2), pp.206-231. ⟨10.1007/s10703-011-0136-y⟩. ⟨hal-00776791⟩



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