Finding DFAs with Maximal Shortest Synchronizing Word Length

  title={Finding DFAs with Maximal Shortest Synchronizing Word Length},
  author={Henk Don and Hans Zantema},
It was conjectured by Cerný 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 2006 Trahtman conjectured that apart from Cerný’s sequence only 8 DFAs exist attaining the bound. He gave an investigation of all DFAs up to certain size for which the bound is reached, and which do not contain other synchronizing DFAs. Here we extend this analysis in two ways: we drop… Expand
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It is proved that the Cern\'y automaton on states does not admit non-trivial extensions with the same smallest synchronizing word length $(n-1)^2$. Expand
Synchronizing non-deterministic finite automata
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Matrix approach to synchronizing automata
A special class of matrices of mapping induced by words in the alphabet of letters on edges of underlying graph are considered and it is claimed that $(n-1)^2$ is a precise upper bound on the length of such a word for every complete $n$-state DFA. Expand
Counting symbol switches in synchronizing automata
It is proved that switch count has the same complexity as synchronizing word length, and all binary automata on at most 9 states and the maximal possible switch count are investigated. Expand
On the synchronization of finite state automata
It is proved that recognizing the planar games that can be won by the synchronizer is a co-NP hard problem and some additional results indicating that pla- nar games are as hard as nonplanar games amount to show that planar automata are representative of the intricacies of automata synchronization. Expand
Adversarial Models for Deterministic Finite Automata
This work investigates a finer-grained understanding of the characteristics of particular deterministic finite automata (DFA) and proposes an adversarial model that reveals the sensitive transitions embedded in a DFA. Expand
The algebra of row monomial matrices
An algebra with non-standard operations on the class of row monomial matrices (having one unit and rest of zeros in every row) plays an important role in the study of DFA, especially for synchronizing automata. Expand
Cerny-Starke conjecture from the sixties of XX century.
A special classes of matrices induced by words in the alphabet of labels on edges of the underlying graph of DFA are used to prove the Cerny conjecture. Expand


DFAs and PFAs with Long Shortest Synchronizing Word Length
A full analysis of all DFAs reaching this bound was only given for n states, with bounds on the number of symbols for \(n 4\), but here it is given without bounds. Expand
An efficient algorithm finds noticeable trends and examples concerning the Černy conjecture
By help of a program based on some effective algorithms, a wide class of automata of size less than 11 was checked and some new examples of n-state DFA with minimal synchronizing word of length (n–1)2 were discovered. Expand
A Quadratic Upper Bound on the Size of a Synchronizing Word in One-Cluster Automata
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. Expand
The Černý Conjecture for Aperiodic Automata
  • A. Trakhtman
  • Computer Science
  • Discret. Math. Theor. Comput. Sci.
  • 2007
It is shown that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n − 1)/2, which means that the Černý conjecture holds true as well as for automata accepting only star-free languages. Expand
The Černý Conjecture and 1-Contracting Automata
  • H. Don
  • Mathematics, Computer Science
  • Electron. J. Comb.
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The notion of aperiodically $1-$contracting automata is introduced and it is proved that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Expand
Sur un Cas Particulier de la Conjecture de Cerny
  • J. Pin
  • Mathematics, Computer Science
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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. Expand
Sur Les Automates Circulaires et la Conjecture de Cerný
  • L. Dubuc
  • Computer Science, Mathematics
  • RAIRO Theor. Informatics Appl.
  • 1998
This paper generalizes Cerný's earliest result (the proof of the conjecture about biaised circular automata) to all circular Automata. Expand
A Note on Černý Conjecture for Automata over 3-Letter Alphabet
  • A. Roman
  • Mathematics, Computer Science
  • J. Autom. Lang. Comb.
  • 2008
It is shown that the alphabet size can play an essential role in the issue of automata synchronization, and an example of 5-state automaton not isomorphic to Cerny's one is given. Expand
Reset Sequences for Monotonic Automata
  • D. Eppstein
  • Mathematics, Computer Science
  • SIAM J. Comput.
  • 1990
A new algorithm based on breadth-first search is presented that runs in faster asymptotic time than Natarajan’s algorithms, and in addition finds the shortest possible reset sequence if such a sequence exists. Expand
On two Combinatorial Problems Arising from Automata Theory
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 aExpand