Abstract
Automata, Logic and Semantics First, we close the multi-parameter analysis of a canonical problem concerning short reset words (SYN) initiated by Fernau et al. (2013). Namely, we prove that the problem, parameterized by the number of states, does not admit a polynomial kernel unless the polynomial hierarchy collapses. Second, we consider a related canonical problem concerning synchronizing road colorings (SRCP). Here we give a similar complete multi-parameter analysis. Namely, we show that the problem, parameterized by the number of states, admits a polynomial kernel and we close the previous research of restrictions to particular values of both the alphabet size and the maximum length of a reset word.
Highlights
Questions about synchronization of finite automata have been studied since the early times of automata theory
The Cernyconjecture, a longstanding open problem in automata theory, claims that each synchronizing automaton has a reset word of length at most (|Q| − 1)2
If the time-bounding polynomials for different values are all of the same degree, we get into the class FPT: A parameterized problem is fixed-parameter tractable (FPT) if there is an algorithm that solves it in time f (P ) · r(|x|), where x is the input string, P ∈ N is the parameter of x, r is an appropriate polynomial, and f is any computable function
Summary
Questions about synchronization of finite automata have been studied since the early times of automata theory. Tab. 1: Results of the complete multi-parameter analysis of SYN and SRCP. Diamonds mark the results of the present paper proving that the edges of any strongly connected aperiodic directed multigraph with constant out-degree can be colored such that a synchronizing automaton arises. Motivation for this problem comes from symbolic dynamics [1]. The need to compute a synchronizing labeling for a suitable graph, possibly with a request for a short reset word, may arise as well It turns out, as we describe below, that such computational problems are typically NP-hard, even under heavy restrictions.
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