On Computational Complexity of Set Automata

  • Alexander A. RubtsovEmail author
  • Mikhail N. Vyalyi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)


We consider a computational model which is known as set automata. The set automata are one-way finite automata with an additional storage—the set. There are two kinds of set automata—the deterministic and the nondeterministic ones. We denote them as DSA and NSA respectively. The model was introduced by M. Kutrib, A. Malcher, M. Wendlandt in 2014 in [3, 4]. It was shown that DSA-languages look similar to DCFL due to their closure properties and NSA-languages look similar to CFL due to their undecidability properties.

In this paper we show that this similarity is natural: we prove that languages recognizable by NSA form a rational cone, so as CFL. The main topic of this paper is computational complexity: we prove that languages recognizable by DSA belong to \({\mathbf {P}}\), and the word membership problem is \({\mathbf {P}}\)-complete for DSA without \(\varepsilon \)-loops; languages recognizable by NSA are in \({\mathbf {NP}}\), and there are \({\mathbf {NP}}\)-complete languages among them. Also we prove that the emptiness problem is \({\mathbf {PSPACE}}\)-hard for DSA.


Set automata Automata theory Formal languages Rational cone Computational complexity Membership problem 



We thank Dmitry Chistikov for the feedback and discussion of the text’s results and suggestion for improvements and anonymous referees for helpful comments.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  3. 3.Dorodnicyn Computing Centre, FRC CSC RASMoscowRussia

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