Slowly synchronizing automata with fixed alphabet size

ArXiv, 2018

In this paper, we show that every D3-directing CNFA can be mapped uniquely to a DFA with the same synchronizing word length. This implies that Cerný’s conjecture generalizes to CNFAs and that any upper bound for the synchronizing word length of DFAs is an upper bound for the D3-directing word length of CNFAs as well. As a second consequence, for several classes of CNFAs sharper bounds are established. Finally, our results allow us to detect all critical CNFAs on at most 6 states. It turns out that only very few critical CNFAs exist.

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...

Lecture Notes in Computer Science, 2011

A word w is called synchronizing (recurrent, reset, magic, directable) word of deterministic nite automaton (DFA) if w sends all states of the automaton to a unique state. In 1964 Jan Cerny found a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n 1) 2. He conjectured that it is an upper bound on the length of such words for complete DFA. Nevertheless, the best upper bound (n 3 n)=6 was found almost 30 years ago. We reduce the upper bound on the length of the minimal synchronizing word to n(7n 2 + 6n 16)=48. An implemented algorithm for nding synchronizing word with restricted upper bound is described. The work presents the distribution of all synchronizing automata of small size according to the length of an almost minimal synchronizing word.

2019

We compute all synchronizing DFAs with 7 states and synchronization length >= 29. Furthermore, we compute alphabet size ranges for maximal, minimal and semi-minimal synchronizing DFAs with up to 7 states.

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.

Information Processing Letters, 2009

We show that i-directable nondeterministic automata can be i-directed with a word of length O (2 n ) for i = 1, 2, where n stands for the number of states. Since for i = 1, 2 there exist i-directable automata having i-directing words of length Ω(2 n ), these upper bounds are asymptotically optimal. We also show that a 3-directable nondeterministic automaton with n states can be 3-directed with a word of length O (n 2 • 3 √ 4 n ), improving the previously known upper bound O (2 n ). Here the best known lower bound is Ω( 3 √ 3 n ).

Lecture Notes in Computer Science, 2010

We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. These automata are closely related to primitive digraphs with large exponent.

ArXiv, 2021

We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a reset word) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs with the same problems for complete DFAs. In particular, this includes the Černý and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude...

Eprint Arxiv 0907 4576, 2009

Using combinatorial properties of incomplete sets in a free monoid we construct a series of n-state deterministic automata with zero whose shortest synchronizing word has length n^2/4+n/2-1.

Journal of Combinatorial Optimization, 2013

In this paper we give the details of our new algorithm for finding minimal reset words of finite synchronizing automata. The problem is known to be computationally hard, so our algorithm is exponential in the worst case, but it is faster than the algorithms used so far and it performs well on average. The main idea is to use a bidirectional breadth-first-search and radix (Patricia) tries to store and compare subsets. A good performance is due to a number of heuristics we apply and describe here in a suitable detail. We give both theoretical and practical arguments showing that the effective branching factor is considerably reduced. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with up to n = 350 states and up to k = 10 input letters. In particular, we obtain a new estimation of the expected length of the shortest reset word ≈ 2.5 √ n − 5 for binary automata and show that the error of this estimate is sufficiently small. Experiments for automata with more than two input letters show certain trends with the same general pattern.

Theoretical Computer Science, 2006

Pin showed that every p-state automaton (p a prime) containing a cyclic permutation and a non-permutation has a synchronizing word of length at most (p − 1) 2 . In this paper we consider permutation automata with the property that adding any non-permutation will lead to a synchronizing word and establish bounds on the lengths of such synchronizing words. In particular, we show that permutation groups whose permutation character over the rationals splits into a sum of only two irreducible characters have the desired property.

Ural mathematical journal

We approach the problem of computing a D 2-synchronizing word of minimum length for a given nondeterministic automaton via its encoding as an instance of SAT and invoking a SAT solver. In addition, we report some of the experimental results obtained when we had tested our method on randomly generated automata and certain benchmarks.

arXiv (Cornell University), 2018

This paper contains results which arose from the research which led to

2013

We present a report from a series of experiments involving computation of the shortest reset words for automata with small number of states. We confirm that the \v{C}ern\'{y} conjecture is true for all automata with at most 11 states on 2 letters. Also some new interesting results were obtained, including the third gap in the distribution of the shortest reset words and new slowly synchronizing classes of automata.

Mathematical Notes, 2011

Using combinatorial properties of incomplete sets in a free monoid we construct a series of n-state deterministic automata with zero whose shortest synchronizing word has length n 2 4 + n 2 − 1.