Bounded Underapproximations

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 w∗ 1w ∗ 2 · · ·w∗ k for some w1, . . . , wk ∈ Σ∗. In particular bounded subsets of context-free languages have nice structural and decidability properties. Our proof proceeds in two parts. First, using Newton’s iterations on the language semiring, we construct a context-free subset LN of L that can be represented as a sequence of substitutions on a linear language and has the same Parikh image as L. Second, we inductively construct a Parikh-equivalent bounded context-free subset of LN . As an application of this result in model checking we show how to underapproximate the reachable state space of multithreaded procedural programs. The bounded language constructed above provides a decidable underapproximation for the original problem. 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. Incidentally, we show that our approach subsumes context-bounded reachability for multithreaded programs.