Topic
Synchronizing word
About: Synchronizing word is a research topic. Over the lifetime, 219 publications have been published within this topic receiving 3357 citations. The topic is also known as: Černý conjecture.
Papers published on a yearly basis
Papers
TL;DR: In this article, the authors present partial results on the following conjectures arising from automata theory: the triangle conjecture due to Perrin and Schiitzenberger, and the Cerny conjecture, which is the conjecture that if there exists a word of rank n - k in, there exists such a word with length ≤ k 2.
Abstract: We present some partial results on the following conjectures arising from automata theory. The first conjecture is the triangle conjecture due to Perrin and Schiitzenberger. Let A ={a, b } be a two-letter alphabet, d a positive integer and let B d ={ a i ba j |0 ≤ i + j ≤ d }. If X C B d is a code, then | X ≤ d + 1. The second conjecture is due to Cerny and the author. Let be an automaton with n states. If there exists a word of rank n - k in , there exists such a word of length ≤ k 2 .
266 citations
TL;DR: The road coloring problem as discussed by the authors is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all vertices if the greatest common divisor of lengths of all its cycles is one.
Abstract: A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. The positive solution of the road coloring problem is presented.
145 citations
17 Jul 1978
TL;DR: A generalization of J. CERNÝ's conjecture that, if there exists a synchronizing word in A, then there exists such a word with length ⩽(n−1)2 where n is the number of states of A.
Abstract: Let A be a finite automaton. We are concerned with the minimal length of the words that send all states on a unique state (synchronizing words). J. CERNÝ has conjectured that, if there exists a synchronizing word in A, then there exists such a word with length ⩽(n−1)2 where n is the number of states of A. As a generalization, we conjecture that, if there exists a word of rank ⩽k in A, there exists such a word with length ⩽(n−k)2.
87 citations
TL;DR: In this paper, it was shown that every synchronizable n-state aperiodic deterministic finite automata has a synchronizing word of length at most n(n-1)/2.
Abstract: A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Cerny conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n-1)2. We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n-1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true.
85 citations
1 Jul 2009
TL;DR: It is shown that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n 2)2, which applies in particular to Huffman codes.
Abstract: Cerný's conjecture asserts the existence of a synchronizing word of length at most (n ? 1)2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p,q, one has p ·a r = q ·a s for some integers r,s (for a state p and a word w, we denote by p ·w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n 2). This applies in particular to Huffman codes.
61 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 8 |
| 2020 | 14 |
| 2019 | 7 |
| 2018 | 9 |
| 2017 | 6 |
| 2016 | 9 |