Synchronizing Boolean Networks Asynchronously

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

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.

Information and Computation, 2008

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 ).

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 .

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.

Theoretical Computer Science, 2007

Reversible cellular automata (RCA) are models of massively parallel computation that preserve information. We generalize these systems by introducing the class of ωω bijective finite automata. It consists of those finite automata where for any bi-infinite word there exists a unique path labelled by that word. These systems are strictly included in the class of local automata. Although the synchronization delay of an n-state local automaton is known to be Θ(n 2 ) in the worst case, we prove that in the case of ωω bijective finite automata the synchronization delay is at most n − 1. Based on this we prove that for a one-dimensional n-state RCA where the neighborhood consists of m consecutive cells, the neighbourhood of the inverse automaton consists of at most n m−1 − (m − 1) cells. Similar bounds are obtained also in [E. Czeizler, J. Kari, A tight linear bound on the neighborhood of inverse cellular automata, in: Proceedings of ICALP 2005, in: LNCS, vol. 3580, 2005, pp. 410-420] but here the result comes as a direct consequence of the more general result. We also construct examples of RCA with large inverse neighbourhoods proving that the upper bounds provided here are the best possible in the case m = 2.

Computing Research Repository, 2011

We show that synchronism can significantly impact on network behaviours, in particular by filtering unstable attractors induced by a constraint of asynchronism. We investigate and classify the different possible impacts that an addition of synchronism may have on the behaviour of a Boolean automata network. We show how these relate to some strong specific structural properties, thus supporting the idea that for most networks, synchronism only shortcuts asynchronous trajectories. We end with a discussion on the close relation that apparently exists between sensitivity to synchronism and non-monotony.

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.

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.