The Context-Freeness Problem Is coNP-Complete for Flat Counter Systems
Bounded languages have recently proved to be an important class of languages for the analysis of Turing-powerful models. For instance, bounded context-free languages are used to under-approximate the behaviors of recursive programs. Ginsburg and Spanier have shown in 1966 that a bounded language \(L \subseteq a_1^* \cdots a_d^*\) is context-free if, and only if, its Parikh image is a stratifiable semilinear set. However, the question whether a semilinear set is stratifiable, hereafter called the stratifiability problem, was left open, and remains so. In this paper, we give a partial answer to this problem. We focus on semilinear sets that are given as finite systems of linear inequalities, and we show that stratifiability is coNP-complete in this case. Then, we apply our techniques to the context-freeness problem for flat counter systems, that asks whether the trace language of a counter system intersected with a bounded regular language is context-free. As main result of the paper, we show that this problem is coNP-complete.
KeywordsRegular Language Counter System Integral Cone Presburger Arithmetic Emptiness Problem
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