Arnaud Carayol ; Philippe Duchon ; Florent Koechlin ; Cyril Nicaud - Random Deterministic Automata With One Added Transition

lmcs:13044 - Logical Methods in Computer Science, January 30, 2025, Volume 21, Issue 1 - https://doi.org/10.46298/lmcs-21(1:11)2025
Random Deterministic Automata With One Added TransitionArticle

Authors: Arnaud Carayol ; Philippe Duchon ; Florent Koechlin ; Cyril Nicaud

    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 $2^n$ 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 $d\geq 1$, 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 $\Omega(\frac1{\sqrt{n}})$ to be precise, therefore allowing models with a sublinear number of final states in expectation.


    Volume: Volume 21, Issue 1
    Published on: January 30, 2025
    Accepted on: November 1, 2024
    Submitted on: February 12, 2024
    Keywords: Computer Science - Formal Languages and Automata Theory,Computer Science - Logic in Computer Science

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