Existential Length Universality

Authors Paweł Gawrychowski, Martin Lange, Narad Rampersad, Jeffrey Shallit, Marek Szykuła

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Author Details

Paweł Gawrychowski
  • Institute of Computer Science, University of Wrocław, Wrocław, Poland
Martin Lange
  • School of Electr. Eng. and Comp. Sc., University of Kassel, Kassel, Germany
Narad Rampersad
  • Department of Math/Stats, University of Winnipeg, Winnipeg, Canada
Jeffrey Shallit
  • School of Computer Science, University of Waterloo, Waterloo, Canada
Marek Szykuła
  • Institute of Computer Science, University of Wrocław, Wrocław, Poland

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Paweł Gawrychowski, Martin Lange, Narad Rampersad, Jeffrey Shallit, and Marek Szykuła. Existential Length Universality. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study the following natural variation on the classical universality problem: given a language L(M) represented by M (e.g., a DFA/RE/NFA/PDA), does there exist an integer ? ≥ 0 such that Σ^? ⊆ L(M)? In the case of an NFA, we show that this problem is NEXPTIME-complete, and the smallest such ? can be doubly exponential in the number of states. This particular case was formulated as an open problem in 2009, and our solution uses a novel and involved construction. In the case of a PDA, we show that it is recursively unsolvable, while the smallest such ? is not bounded by any computable function of the number of states. In the case of a DFA, we show that the problem is NP-complete, and e^{√{n log n} (1+o(1))} is an asymptotically tight upper bound for the smallest such ?, where n is the number of states. Finally, we prove that in all these cases, the problem becomes computationally easier when the length ? is also given in binary in the input: it is polynomially solvable for a DFA, PSPACE-complete for an NFA, and co-NEXPTIME-complete for a PDA.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Formal languages and automata theory
  • decision problem
  • deterministic automaton
  • nondeterministic automaton
  • pushdown automaton
  • regular expression
  • regular language
  • universality


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