Slowly synchronizing automata with zero and incomplete sets

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.

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 .

Information and Computation, 2011

A synchronizing word for a given synchronizing DFA is called minimal if none of its proper factors is synchronizing. We characterize the class of synchronizing automata having only finitely many minimal synchronizing words (the class of such automata is denoted by FG).

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.

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.

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.

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...

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.

Jcst, 2008

A word w is called synchronizing (recurrent, reset, directable) word of deterministic finite automata (DFA) if w brings all states of the automaton to a unique state. According to the famous conjecture ofČerný from 1964, every n-state synchronizing automaton possesses a synchronizing word of length at most (n − 1) 2. The problem is still open. It will be proved that theČerný conjecture holds good for synchronizing DFA with transition monoid having no involutions and for every n-state (n > 2) synchronizing DFA with transition monoid having only trivial subgroups the minimal length of synchronizing word is not greater than (n − 1) 2 /2. The last important class of DFA involved and studied by Schȗtzenberger is called aperiodic; its automata accept precisely star-free languages. Some properties of an arbitrary synchronizing DFA were established. See