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Improving the upper bound on the length of the shortest reset words
[article]

2017
*
arXiv
*
pre-print

*on*

*the*

*length*

*of*

*the*

*shortest*

*reset*

*words*. ... We

*improve*

*the*best known

*upper*

*bound*

*on*

*the*

*length*

*of*

*the*

*shortest*

*reset*

*words*

*of*synchronizing automata.

*The*new

*bound*is slightly better than 114 n^3 / 685 + O(n^2). ... This

*bound*is too large to provide further

*improvement*(at least within

*the*cubic coefficient) for

*the*

*upper*

*bound*

*on*

*the*

*length*

*of*

*the*

*shortest*

*reset*

*words*. ...

##
###
Improving the Upper Bound on the Length of the Shortest Reset Words

unpublished

*on*

*the*

*length*

*of*

*the*

*shortest*

*reset*

*words*. ... We

*improve*

*the*best known

*upper*

*bound*

*on*

*the*

*length*

*of*

*the*

*shortest*

*reset*

*words*

*of*synchronizing automata.

*The*new

*bound*is slightly better than 114n 3 /685+O(n 2). ... This

*bound*is too large to provide a further

*improvement*(at least within

*the*cubic coefficient) for

*the*

*upper*

*bound*

*on*

*the*

*length*

*of*

*the*

*shortest*

*reset*

*words*. ...

##
###
An Extremal Series of Eulerian Synchronizing Automata
[chapter]

2016
*
Lecture Notes in Computer Science
*

This

doi:10.1007/978-3-662-53132-7_31
fatcat:ex4hjccwo5bf5mtootc5odlbhm
*improves**the*current lower*bound**on**the**length**of**shortest**reset**words*in Eulerian automata. ... We present an infinite series*of*n-state Eulerian automata whose*reset**words*have*length*at least (n^2-3)/2. ... For n ≥ 3, (n 2 − 3)/2 is an*upper**bound*for*the**reset*threshold*of*an n-state Eulerian synchronizing automaton. If |Σ| = 2, then*the**bound*can be*improved*to (n 2 − 5)/2. ...##
###
Experiments with Synchronizing Automata
[chapter]

2016
*
Lecture Notes in Computer Science
*

We have

doi:10.1007/978-3-319-40946-7_15
fatcat:w4tqaniirzeb3bwy3lt2uficwa
*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*. ... We thank Mikhail Volkov for suggesting Conjecture 4, and Mikhail Berlinkov for observing that*the**bound*for*one*-cluster automata can be*improved*for periodic subsets*on**the*cycle, which leaded to an*improvement*...##
###
Algebraic synchronization criterion and computing reset words
[article]

2015
*
arXiv
*
pre-print

We

arXiv:1412.8363v3
fatcat:gbqag5pxprbnjmbelrnv2pglny
*improve**the*best general*upper**bound*for*reset*thresholds*of*finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a*reset**word**of**length*at most O(n ^3 n). ... Moreover,*reset**words**of**lengths*within all*of*our*bounds*are computable in polynomial time. ...*One*can verify that its*shortest**reset**word*is ba 3 ba 3 b.*The**length**of**the**shortest**reset**word*is called*the**reset*threshold and is denoted by rt(A ). ...##
###
Algebraic synchronization criterion and computing reset words

2016
*
Information Sciences
*

We

doi:10.1016/j.ins.2016.07.049
fatcat:ihfimthllvghxju7hika46fpxu
*improve**the*best general*upper**bound*for*reset*thresholds*of*finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a*reset**word**of**length*at most O(n log 3 n). ... Moreover,*reset**words**of**lengths*within all*of*our*bounds*are computable in polynomial time. ... . 13-01-00852,*the*Ministry*of*Education and Science*of**the*Russian Federation, project no. 1.1999 ...##
###
Synchronizing Automata with Extremal Properties
[chapter]

2015
*
Lecture Notes in Computer Science
*

We also discuss possible relaxations

doi:10.1007/978-3-662-48057-1_26
fatcat:jjy6bgvfj5fafk7fb424poz6de
*of**the*conjecture, and propose*the*image-extension conjecture, which would lead to a quadratic*upper**bound**on**the**length**of**the**shortest**reset**words*. ...*the**shortest**reset**words*cannot be*improved*generally by means*of**the*extension method. ... us to prove a quadratic*bound*for*the**length**of**the**shortest**reset**words*). ...##
###
An Improved Algorithm for Finding the Shortest Synchronizing Words
[article]

2022
*
arXiv
*
pre-print

Given a modest time limit, we compute

arXiv:2207.05495v1
fatcat:5egqgjvw7jd4vojzyitovg5iau
*the**lengths**of**the**shortest*synchronizing*words*for random binary automata up to 570 states, significantly beating*the*previous record. ... Because*the*problem*of*finding a*shortest*synchronizing*word*is computationally hard, among exact algorithms only exponential*ones*are known. ... Conclusions We have*improved**the*best-known algorithm for computing*the*(*length**of**the*)*shortest**reset**words*. ...##
###
Modifying the Upper Bound on the Length of Minimal Synchronizing Word
[chapter]

2011
*
Lecture Notes in Computer Science
*

He conjectured that it is an

doi:10.1007/978-3-642-22953-4_15
fatcat:3uehwvpvivdsfm5bon6i2kqjzy
*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. ... 4 Acknowledgments I would like to express my gratitude to*the*referees for many helpful and useful remarks and for*improving**the*calculation*of**the*result. ...##
###
Modifying the upper bound on the length of minimal synchronizing word
[article]

2014
*
arXiv
*
pre-print

He conjectured that it is an

arXiv:1104.2409v7
fatcat:ixgnklkkqbfktm4ne6ggly5xmi
*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 finding synchronizing*word*with restricted*upper**bound*is described. ... Acknowledgments I would like to express my gratitude to*the*referees for many helpful and useful remarks and for*improving**the*calculation*of**the*result. ...##
###
Generating Synchronizing Automata with Large Reset Lengths
[article]

2018
*
arXiv
*
pre-print

First, we refine

arXiv:1404.3311v4
fatcat:6sch33vh5jfclldradfa2x6jea
*the*Frankl-Pin result*on**the**length**of**the**shortest**words**of*rank m, and*the*Béal, Berlinkov, Perrin, and Steinberg results*on**the**length**of**the**shortest**reset**words*in*one*-cluster automata ... We study synchronizing automata with*the**shortest**reset**words**of*relatively large*length*. ...*The*computations were performed*on*a grid that belongs to Institute*of*Computer Science*of*Jagiellonian University. ...##
###
Synchronizing automata with a letter of deficiency 2

2007
*
Theoretical Computer Science
*

We present two infinite series

doi:10.1016/j.tcs.2007.01.010
fatcat:rhggja2qp5bcnaoy6nfi2engfm
*of*synchronizing automata with a letter*of*deficiency 2 whose*shortest**reset**words*are longer than those for synchronizing automata obtained by a straightforward modification ... Acknowledgement This work was supported by*the*Russian Foundation for Basic Research, grant 05-01-00540. ...*The*problem is known to be NP-complete (see [4] or [12] ), but*on**the*other hand, there are some*upper**bounds**on**the*minimum*length**of**reset**words*for synchronizing automata with a given number*of*states ...##
###
Synchronizing Automata with a Letter of Deficiency 2
[chapter]

2006
*
Lecture Notes in Computer Science
*

We present two infinite series

doi:10.1007/11779148_39
fatcat:ax3h7rytobbvfe4rgn3qoaf3ym
*of*synchronizing automata with a letter*of*deficiency 2 whose*shortest**reset**words*are longer than those for synchronizing automata obtained by a straightforward modification ... Acknowledgement This work was supported by*the*Russian Foundation for Basic Research, grant 05-01-00540. ...*The*problem is known to be NP-complete (see [4] or [12] ), but*on**the*other hand, there are some*upper**bounds**on**the*minimum*length**of**reset**words*for synchronizing automata with a given number*of*states ...##
###
A linear bound on the k-rendezvous time for primitive sets of NZ matrices
[article]

2021
*
arXiv
*
pre-print

We provide two

arXiv:1903.10421v3
fatcat:urybw4m5cbhuplfqtkfhebrdnq
*upper**bounds**on**the*k-RT:*the*second is an*improvement**of**the*first*one*, although*the*latter can be written in closed form. ... We then report numerical results comparing our*upper**bounds**on**the*k-RT with heuristic approximation methods. ... We are usually interested in*the**length**of**the**shortest**reset**word**of*a synchronizing automaton A, called its*reset*threshold and denoted by rt(A). ...##
###
An efficient algorithm finds noticeable trends and examples concerning the Černy conjecture
[article]

2007
*
arXiv
*
pre-print

He had conjectured that it is an

arXiv:0709.1197v1
fatcat:ro6r52gncvcw5hcv2tuvhrjtte
*upper**bound*for*the**length**of**the**shortest*synchronizing*word*for any n-state complete DFA. ... A*word*w is called synchronizing (recurrent,*reset*, directed)*word**of*a deterministic finite automaton (DFA) if w sends all states*of**the*automaton*on*a unique state. ... He had conjectured that it is an*upper**bound*for*the**length**of**the**shortest*synchronizing*word*for any n-state complete DFA. ...
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