The Context-Freeness Problem Is coNP-Complete for Flat Counter Systems

  • Jérôme Leroux
  • Vincent Penelle
  • Grégoire Sutre
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8837)


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.


Regular Language Counter System Integral Cone Presburger Arithmetic Emptiness Problem 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jérôme Leroux
    • 1
  • Vincent Penelle
    • 2
  • Grégoire Sutre
    • 1
  1. 1.LaBRI, UMR 5800Univ. Bordeaux & CNRSTalenceFrance
  2. 2.LIGM, UMR 8049Univ. Paris Est & CNRSMarne-la-ValléeFrance

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