The v{C}ern '{y} conjecture for small automata: experimental report

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, 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

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 .

Lecture Notes in Computer Science, 2013

In this paper we present a new fast algorithm for finding minimal reset words for finite synchronizing automata, which is a problem appearing in many practical applications. 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 BFS and radix (Patricia) tries to store and compare subsets. Also a number of heuristics are applied. 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 n ≤ 300 states and 2 input letters. In particular, we obtain a new estimation of the expected length of the shortest reset word ≈ 2.5 √ n − 5.

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...

Lecture Notes in Computer Science, 2017

It was conjectured byČerný in 1964, that a synchronizing DFA on n states always has a synchronizing word of length at most (n−1) 2 , 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 n ≤ 4, and with bounds on the number of symbols for n ≤ 10. Here we give the full analysis for n ≤ 6, without bounds on the number of symbols. For PFAs on n ≤ 6 states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding (n − 1) 2 for n = 4, 5, 6. For arbitrary n we use rewrite systems to construct a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation.

Developments in Language Theory, 2021

The largest known reset thresholds for DFAs are equal to (n − 1) 2 , where n is the number of states. This is conjectured to be the maximum possible. PFAs (with partial transition function) can have exponentially large reset thresholds. This is still true if we restrict to binary PFAs. However, asymptotics do not give conclusions for fixed n. We prove that the maximal reset threshold for binary PFAs is strictly greater than (n − 1) 2 if and only if n ≥ 6. These results are mostly based on the analysis of synchronizing word lengths for a certain family of binary PFAs. This family has the following properties: it contains the well-knownČerný automata; for n ≤ 10 it contains a binary PFA with maximal possible reset threshold; for all n ≥ 6 it contains a PFA with reset threshold larger than the maximum known for DFAs. Analysis of this family reveals remarkable patterns involving the Fibonacci numbers and related sequences such as the Padovan sequence. We derive explicit formulas for the reset thresholds in terms of these recurrent sequences. Furthermore, we prove that the family asymptotically still gives reset thresholds of polynomial order.

arXiv (Cornell University), 2021

The largest known reset thresholds for DFAs are equal to (n − 1) 2 , where n is the number of states. This is conjectured to be the maximum possible. PFAs (with partial transition function) can have exponentially large reset thresholds. This is still true if we restrict to binary PFAs. However, asymptotics do not give conclusions for fixed n. We prove that the maximal reset threshold for binary PFAs is strictly greater than (n − 1) 2 if and only if n ≥ 6. These results are mostly based on the analysis of synchronizing word lengths for a certain family of binary PFAs. This family has the following properties: it contains the well-knownČerný automata; for n ≤ 10 it contains a binary PFA with maximal possible reset threshold; for all n ≥ 6 it contains a PFA with reset threshold larger than the maximum known for DFAs. Analysis of this family reveals remarkable patterns involving the Fibonacci numbers and related sequences such as the Padovan sequence. We derive explicit formulas for the reset thresholds in terms of these recurrent sequences. Asymptotically theČerný family gives reset thresholds of polynomial order. We prove that PFAs in the family are not extremal for n ≥ 41. For that purpose, we present an improvement of Martyugin's prime number construction of binary PFAs.

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.

Language and Automata Theory and Applications

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 2006 Trahtman conjectured that apart fromČerný'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 this latter condition, and we drop limits on alphabet size. For n ≤ 4 we do the full analysis yielding 19 new DFAs with smallest synchronizing word length (n − 1) 2 , refuting Trahtman's conjecture. Several of these new DFAs admit more than one synchronizing word of length (n − 1) 2 , and even the synchronizing state is not unique. All these new DFAs are extensions of DFAs that were known before. For n ≥ 5 we prove that none of the DFAs in Trahtman's analysis can be extended similarly. In particular, as a main result we prove that theČerný examples Cn do not admit non-trivial extensions keeping the same smallest synchronizing word length (n − 1) 2 .