Lower Bounds on Avoiding Thresholds

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F62690094%3A18450%2F21%3A50019040" target="_blank" >RIV/62690094:18450/21:50019040 - isvavai.cz</a>

Výsledek na webu

<a href="https://drops.dagstuhl.de/opus/portals/lipics/index.php?semnr=16204" target="_blank" >https://drops.dagstuhl.de/opus/portals/lipics/index.php?semnr=16204</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.MFCS.2021.46" target="_blank" >10.4230/LIPIcs.MFCS.2021.46</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Lower Bounds on Avoiding Thresholds

Popis výsledku v původním jazyce

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(nk). This meets the known upper bound for k ≥ 3. For k = 1 and k = 2, the known upper bounds are respectively in O(n2) and in O(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 O(n)) also implies a tight upper bound in O(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.

Název v anglickém jazyce

Lower Bounds on Avoiding Thresholds

Popis výsledku anglicky

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(nk). This meets the known upper bound for k ≥ 3. For k = 1 and k = 2, the known upper bounds are respectively in O(n2) and in O(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 O(n)) also implies a tight upper bound in O(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.

Druh

D - Stať ve sborníku

CEP obor

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OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

ISBN

978-3-95977-201-3

ISSN

1868-8969

e-ISSN

—

Počet stran výsledku

14

Strana od-do

"Article number: 46"

Název nakladatele

LIPICS

Místo vydání

Dagstuhl

Místo konání akce

Tallinn

Datum konání akce

23. 8. 2021

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

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