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

Marek Szykuła
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.  ... 
arXiv:1702.05455v4 fatcat:53wlidklzbcfflq6dcfqhe3p7a

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

Marek Szykuła, Mikhail Berlinkov, Costanza Catalano, Vladimir Gusev, Jakub Kośmider, Jakub Kowalski
unpublished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.  ... 
fatcat:osjgenlpcjgnbjtw6rgswojm7a

An Extremal Series of Eulerian Synchronizing Automata [chapter]

Marek Szykuła, Vojtěch Vorel
2016 Lecture Notes in Computer Science  
This 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.  ... 
doi:10.1007/978-3-662-53132-7_31 fatcat:ex4hjccwo5bf5mtootc5odlbhm

Experiments with Synchronizing Automata [chapter]

Andrzej Kisielewicz, Jakub Kowalski, Marek Szykuła
2016 Lecture Notes in Computer Science  
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.  ...  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  ... 
doi:10.1007/978-3-319-40946-7_15 fatcat:w4tqaniirzeb3bwy3lt2uficwa

Algebraic synchronization criterion and computing reset words [article]

Mikhail Berlinkov, Marek Szykuła
2015 arXiv   pre-print
We 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 ).  ... 
arXiv:1412.8363v3 fatcat:gbqag5pxprbnjmbelrnv2pglny

Algebraic synchronization criterion and computing reset words

Mikhail V. Berlinkov, Marek Szykuła
2016 Information Sciences  
We 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  ... 
doi:10.1016/j.ins.2016.07.049 fatcat:ihfimthllvghxju7hika46fpxu

Synchronizing Automata with Extremal Properties [chapter]

Andrzej Kisielewicz, Marek Szykuła
2015 Lecture Notes in Computer Science  
We also discuss possible relaxations 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).  ... 
doi:10.1007/978-3-662-48057-1_26 fatcat:jjy6bgvfj5fafk7fb424poz6de

An Improved Algorithm for Finding the Shortest Synchronizing Words [article]

Marek Szykuła, Adam Zyzik
2022 arXiv   pre-print
Given a modest time limit, we compute 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.  ... 
arXiv:2207.05495v1 fatcat:5egqgjvw7jd4vojzyitovg5iau

Modifying the Upper Bound on the Length of Minimal Synchronizing Word [chapter]

A. N. Trahtman
2011 Lecture Notes in Computer Science  
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.  ...  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.  ... 
doi:10.1007/978-3-642-22953-4_15 fatcat:3uehwvpvivdsfm5bon6i2kqjzy

Modifying the upper bound on the length of minimal synchronizing word [article]

A. N. Trahtman
2014 arXiv   pre-print
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 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.  ... 
arXiv:1104.2409v7 fatcat:ixgnklkkqbfktm4ne6ggly5xmi

Generating Synchronizing Automata with Large Reset Lengths [article]

Andrzej Kisielewicz, Marek Szykuła
2018 arXiv   pre-print
First, we refine 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.  ... 
arXiv:1404.3311v4 fatcat:6sch33vh5jfclldradfa2x6jea

Synchronizing automata with a letter of deficiency 2

D.S. Ananichev, M.V. Volkov, Yu.I. Zaks
2007 Theoretical Computer Science  
We present two infinite series 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  ... 
doi:10.1016/j.tcs.2007.01.010 fatcat:rhggja2qp5bcnaoy6nfi2engfm

Synchronizing Automata with a Letter of Deficiency 2 [chapter]

D. S. Ananichev, M. V. Volkov, Yu. I. Zaks
2006 Lecture Notes in Computer Science  
We present two infinite series 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  ... 
doi:10.1007/11779148_39 fatcat:ax3h7rytobbvfe4rgn3qoaf3ym

A linear bound on the k-rendezvous time for primitive sets of NZ matrices [article]

Costanza Catalano, Umer Azfar, Ludovic Charlier, Raphaël Jungers
2021 arXiv   pre-print
We provide two 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).  ... 
arXiv:1903.10421v3 fatcat:urybw4m5cbhuplfqtkfhebrdnq

An efficient algorithm finds noticeable trends and examples concerning the Černy conjecture [article]

A.N. Trahtman
2007 arXiv   pre-print
He had conjectured that it is an 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.  ... 
arXiv:0709.1197v1 fatcat:ro6r52gncvcw5hcv2tuvhrjtte
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