eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-03-03
19:1
19:14
10.4230/LIPIcs.STACS.2023.19
article
One Drop of Non-Determinism in a Random Deterministic Automaton
Carayol, Arnaud
1
Duchon, Philippe
2
Koechlin, Florent
3
Nicaud, Cyril
1
Univ Gustave Eiffel, CNRS, LIGM, F-77454 Marne-la-Vallée, France
Univ. Bordeaux, CNRS UMR 5800, LaBRI, F-33400 Talence, France
Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Every language recognized by a non-deterministic finite automaton can be recognized by a deterministic automaton, at the cost of a potential increase of the number of states, which in the worst case can go from n states to 2ⁿ states. In this article, we investigate this classical result in a probabilistic setting where we take a deterministic automaton with n states uniformly at random and add just one random transition. These automata are almost deterministic in the sense that only one state has a non-deterministic choice when reading an input letter. In our model each state has a fixed probability to be final. We prove that for any d ≥ 1, with non-negligible probability the minimal (deterministic) automaton of the language recognized by such an automaton has more than n^d states; as a byproduct, the expected size of its minimal automaton grows faster than any polynomial. Our result also holds when each state is final with some probability that depends on n, as long as it is not too close to 0 and 1, at distance at least Ω(1/√n) to be precise, therefore allowing models with a sublinear number of final states in expectation.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol254-stacs2023/LIPIcs.STACS.2023.19/LIPIcs.STACS.2023.19.pdf
non-deterministic automaton
powerset construction
probabilistic analysis