License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2021.46
URN: urn:nbn:de:0030-drops-144869
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14486/
Ferens, Robert ;
Szykuła, Marek ;
Vorel, Vojtěch
Lower Bounds on Avoiding Thresholds
Abstract
For a DFA, a word avoids a subset of states, if after reading that word the automaton cannot be in any state from the subset regardless of its initial state. A subset that admits an avoiding word is avoidable. The k-avoiding threshold of a DFA is the smallest number such that every avoidable subset of size k can be avoided with a word no longer than that number. We study the problem of determining the maximum possible k-avoiding thresholds. For every fixed k ≥ 1, we show a general construction of strongly connected DFAs with n states and the k-avoiding threshold in Θ(n^k). This meets the known upper bound for k ≥ 3. For k = 1 and k = 2, the known upper bounds are respectively in ?(n²) and in ?(n³). For k = 1, we show that 2n-3 is attainable for every number of states n in the class of strongly connected synchronizing binary DFAs, which is supposed to be the best possible in the class of all DFAs for n ≥ 8. For k = 2, we show that the conjectured solution for k = 1 (an upper bound in ?(n)) also implies a tight upper bound in ?(n²) on 2-avoiding threshold. Finally, we discuss the possibility of using k-avoiding thresholds of synchronizing automata to improve upper bounds on the length of the shortest reset words.
BibTeX - Entry
@InProceedings{ferens_et_al:LIPIcs.MFCS.2021.46,
author = {Ferens, Robert and Szyku{\l}a, Marek and Vorel, Vojt\v{e}ch},
title = {{Lower Bounds on Avoiding Thresholds}},
booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
pages = {46:1--46:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-201-3},
ISSN = {1868-8969},
year = {2021},
volume = {202},
editor = {Bonchi, Filippo and Puglisi, Simon J.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14486},
URN = {urn:nbn:de:0030-drops-144869},
doi = {10.4230/LIPIcs.MFCS.2021.46},
annote = {Keywords: avoiding word, \v{C}ern\'{y} conjecture, rank conjecture, reset threshold, reset word, synchronizing automaton, synchronizing word}
}
Keywords: |
|
avoiding word, Černý conjecture, rank conjecture, reset threshold, reset word, synchronizing automaton, synchronizing word |
Collection: |
|
46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021) |
Issue Date: |
|
2021 |
Date of publication: |
|
18.08.2021 |