eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-03-04
16:1
16:14
10.4230/LIPIcs.STACS.2020.16
article
Existential Length Universality
Gawrychowski, Paweł
1
Lange, Martin
2
Rampersad, Narad
3
Shallit, Jeffrey
4
Szykuła, Marek
1
Institute of Computer Science, University of Wrocław, Wrocław, Poland
School of Electr. Eng. and Comp. Sc., University of Kassel, Kassel, Germany
Department of Math/Stats, University of Winnipeg, Winnipeg, Canada
School of Computer Science, University of Waterloo, Waterloo, Canada
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol154-stacs2020/LIPIcs.STACS.2020.16/LIPIcs.STACS.2020.16.pdf
decision problem
deterministic automaton
nondeterministic automaton
pushdown automaton
regular expression
regular language
universality