Existential Length Universality
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.
decision problem
deterministic automaton
nondeterministic automaton
pushdown automaton
regular expression
regular language
universality
Theory of computation~Problems, reductions and completeness
Theory of computation~Formal languages and automata theory
16:1-16:14
Regular Paper
A full version of the paper is available at https://arxiv.org/abs/1702.03961.
Paweł
Gawrychowski
Paweł Gawrychowski
Institute of Computer Science, University of Wrocław, Wrocław, Poland
Martin
Lange
Martin Lange
School of Electr. Eng. and Comp. Sc., University of Kassel, Kassel, Germany
Narad
Rampersad
Narad Rampersad
Department of Math/Stats, University of Winnipeg, Winnipeg, Canada
Supported in part by a grant from NSERC.
Jeffrey
Shallit
Jeffrey Shallit
School of Computer Science, University of Waterloo, Waterloo, Canada
Supported in part by a grant from NSERC.
Marek
Szykuła
Marek Szykuła
Institute of Computer Science, University of Wrocław, Wrocław, Poland
Supported in part by the National Science Centre, Poland under project number 2017/25/B/ST6/01920.
10.4230/LIPIcs.STACS.2020.16
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Paweł Gawrychowski, Martin Lange, Narad Rampersad, Jeffrey Shallit, and Marek Szykuła
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