Lower Bounds for Synchronizing Word Lengths in Partial Automata

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.

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...

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.

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.

Theoretical Computer Science, 2015

Recently, Dassow et al. connected partial words and regular languages. Partial words are sequences in which some positions may be undefined, represented with a "hole" symbol . If we restrict what the symbol can represent, we can use partial words to compress the representation of regular languages. Doing so allows the creation of so-called -DFAs, smaller than the DFAs recognizing the original language L, which recognize the compressed language. However, the -DFAs may be larger than the NFAs recognizing L. In this paper, we investigate a question of Dassow et al. as to how these sizes are related.

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).

arXiv (Cornell University), 2018

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

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.

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 ).