Extending word problems in deterministic finite automata

2018

Given a subset of states S of a deterministic finite automaton and a word w, the preimage is the subset of all states that are mapped to a state from S by the action of w. We study the computational complexity of three problems related to the existence of words yielding certain preimages, which are especially motivated by the theory of synchronizing automata. The first problem is whether, for a given subset, there exists a word extending the subset (giving a larger preimage). The second problem is whether there exists a word totally extending the subset (giving the whole set of states) - it is equivalent to the problem whether there exists an avoiding word for the complementary subset. The third problem is whether there exists a word resizing the subset (giving a preimage of a different size). We also consider the variants of the problem where an upper bound on the length of the word is given in the input. Because in most cases our problems are computationally hard, we additionally ...

2020

Given a subset of states S of a deterministic finite automaton and a word w, the preimage is the subset of all states mapped to a state in S by the action of w. We study three natural problems concerning words giving certain preimages. The first problem is whether, for a given subset, there exists a word extending the subset (giving a larger preimage). The second problem is whether there exists a totally extending word (giving the whole set of states as a preimage)—equivalently, whether there exists an avoiding word for the complementary subset. The third problem is whether there exists a resizing word. We also consider variants where the length of the word is upper bounded, where the size of the given subset is restricted, and where the automaton is strongly connected, synchronizing, or binary. We conclude with a summary of the complexities in all combinations of the cases.

2022

A deterministic finite automaton (DFA) with a set of states Q is completely reachable if every non-empty subset of Q is the image of the action of some word applied to Q. The concept was first introduced by Bondar and Volkov (2016), who also raised the question of the complexity of deciding if an automaton is completely reachable. We develop a polynomial-time algorithm for this problem, which is based on a complement-intersecting technique for finding an extending word for a subset of states. Additionally, we prove a weak Don's conjecture for this class of automata: a subset of size k is reachable with a word of length at most 2(|Q| − k)|Q|. This implies a quadratic upper bound in |Q| on the length of the shortest synchronizing words (reset threshold) and generalizes earlier upper bounds derived for subclasses of completely reachable automata. 2012 ACM Subject Classification Theory of computation → Formal languages and automata theory; Mathematics of computing → Discrete mathematics

Lecture Notes in Computer Science, 2002

In spite of its simple formulation, the problem about the synchronization of a finite deterministic automaton is not yet properly understood. The present paper investigates this and related problems within the general framework of a composition theory for functions over a finite domain N with n elements. The notion of depth introduced in this connection is a good indication of the complexity of a given function, namely, the complexity with respect to the length of composition sequences in terms of functions belonging to a basic set. Our results show that the depth may vary considerably with the target function. We also establish criteria about the reachability of some target functions, notably constants. Properties of n such as primality or being a power of 2 turn out to be important, independently of the semantic interpretation. Most of the questions about depth, as well as about the comparison of different notions of depth, remain open. Our results show that the study of functions of several variables may shed light also to the case where all functions considered are unary.

International Journal of Foundations of Computer Science

It was conjectured by Černý in 1964, that a synchronizing DFA on [Formula: see text] states always has a synchronizing word of length at most [Formula: see text], and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for [Formula: see text], and with bounds on the number of symbols for [Formula: see text]. Here we give the full analysis for [Formula: see text], without bounds on the number of symbols. For PFAs (partial automata) on [Formula: see text] states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding [Formula: see text] for [Formula: see text]. Where DFAs with long synchronization typically have very few symbols, for PFAs we observe that more symbols may increase the synchronizing word length. For PFAs on [Formula: see text] states and two symbols we investigate all occurring synchronizing word lengths. We give series of PFAs on two and thr...

2014

A deterministic finite automaton A is said to be synchronizing if it has a reset word, i.e. a word that brings all states of the automaton A to a particular one. We prove that it is a PSPACEcomplete problem to check whether the language of reset words for a given automaton coincides with the language of reset words for some particular automaton.

Journal of Computer and System Sciences, 2015

We initiate a multi-parameter analysis of two well-known NP-hard problems on deterministic finite automata (DFAs): the problem of finding a short synchronizing word, and that of finding a DFA on few states consistent with a given sample of the intended language and its complement. For both problems, we study natural parameterizations and classify them with the tools provided by Parameterized Complexity. Somewhat surprisingly, in both cases, rather simple FPT algorithms can be shown to be optimal, mostly assuming the (Strong) Exponential Time Hypothesis.

Lecture Notes in Computer Science, 2016

We have improved an algorithm generating synchronizing automata with a large length of the shortest reset words. This has been done by refining some known results concerning bounds on the reset length. Our improvements make possible to consider a number of conjectures and open questions concerning synchronizing automata, checking them for automata with a small number of states and discussing the results. In particular, we have verified the Černý conjecture for all binary automata with at most 12 states, and all ternary automata with at most 8 states.

2020

We approach the task of computing a carefully synchronizing word of optimum length for a given partial deterministic automaton, encoding the problem as an instance of SAT and invoking a SAT solver. Our experiments demonstrate that this approach gives satisfactory results for automata with up to 100 states even if very modest computational resources are used. We compare our results with the ones obtained by the first author for exact synchronization, which is another version of synchronization studied in the literature, and draw some theoretical conclusions.

Information and Computation

It was conjectured byČerný in 1964 that a synchronizing DFA on n states always has a shortest synchronizing word of length at most (n − 1) 2 , and he gave a sequence of DFAs for which this bound is reached. In this paper, we investigate the role of the alphabet size. For each possible alphabet size, we count DFAs on n ≤ 6 states which synchronize in (n − 1) 2 − e steps, for all e < 2 n/2. Furthermore, we give constructions of automata with any number of states, and 3, 4, or 5 symbols, which synchronize slowly, namely in n 2 − 3n + O(1) steps. In addition, our results proveČerný's conjecture for n ≤ 6. Our computation has led to 27 DFAs on 3, 4, 5 or 6 states, which synchronize in (n − 1) 2 steps, but do not belong toČerný's sequence. Of these 27 DFA's, 19 are new, and the remaining 8 which were already known are exactly the minimal ones: they will not synchronize any more after removing a symbol. So the 19 new DFAs are extensions of automata which were already known, including theČerný automaton on 3 states. But for n > 3, we prove that thě Cerný automaton on n states does not admit non-trivial extensions with the same smallest synchronizing word length (n − 1) 2 .