### Abstract

For PFAs (partial automata) on ≤7≤7 states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding (n−1)2(n−1)2 for n≥4n≥4. 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 ≤10≤10 states and two symbols we investigate all occurring synchronizing word lengths.

We give series of PFAs on two and three symbols, reaching the maximal possible length for some small values of nn. For n=6,7,8,9n=6,7,8,9, the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than (n−1)2(n−1)2, although still quadratic.

Based on string rewriting, for arbitrary size we 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. Both PFAs are transitive.

Finally, we show that exponential lengths are even possible with just one single undefined transition, again with transitive constructions.

Original language | English |
---|---|

Pages (from-to) | 29-60 |

Number of pages | 32 |

Journal | International Journal of Foundations of Computer Science |

Volume | 30 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2019 |

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### Keywords

- careful synchronization
- conjecture
- DFA
- erný
- PFA

### Cite this

*International Journal of Foundations of Computer Science*,

*30*(1), 29-60. https://doi.org/10.1142/S0129054119400021

}

*International Journal of Foundations of Computer Science*, vol. 30, no. 1, pp. 29-60. https://doi.org/10.1142/S0129054119400021

**Lower bounds for synchronizing word lengths in partial automata.** / de Bondt, M.; Don, H.M.; Zantema, Hans.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Lower bounds for synchronizing word lengths in partial automata

AU - de Bondt, M.

AU - Don, H.M.

AU - Zantema, Hans

PY - 2019

Y1 - 2019

N2 - It was conjectured by Černý in 1964, that a synchronizing DFA on nn states always has a synchronizing word of length at most (n−1)2(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≤5n≤5, and with bounds on the number of symbols for n≤12n≤12. Here we give the full analysis for n≤7n≤7, without bounds on the number of symbols.For PFAs (partial automata) on ≤7≤7 states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding (n−1)2(n−1)2 for n≥4n≥4. 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 ≤10≤10 states and two symbols we investigate all occurring synchronizing word lengths.We give series of PFAs on two and three symbols, reaching the maximal possible length for some small values of nn. For n=6,7,8,9n=6,7,8,9, the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than (n−1)2(n−1)2, although still quadratic.Based on string rewriting, for arbitrary size we 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. Both PFAs are transitive.Finally, we show that exponential lengths are even possible with just one single undefined transition, again with transitive constructions.

AB - It was conjectured by Černý in 1964, that a synchronizing DFA on nn states always has a synchronizing word of length at most (n−1)2(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≤5n≤5, and with bounds on the number of symbols for n≤12n≤12. Here we give the full analysis for n≤7n≤7, without bounds on the number of symbols.For PFAs (partial automata) on ≤7≤7 states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding (n−1)2(n−1)2 for n≥4n≥4. 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 ≤10≤10 states and two symbols we investigate all occurring synchronizing word lengths.We give series of PFAs on two and three symbols, reaching the maximal possible length for some small values of nn. For n=6,7,8,9n=6,7,8,9, the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than (n−1)2(n−1)2, although still quadratic.Based on string rewriting, for arbitrary size we 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. Both PFAs are transitive.Finally, we show that exponential lengths are even possible with just one single undefined transition, again with transitive constructions.

KW - careful synchronization

KW - conjecture

KW - DFA

KW - erný

KW - PFA

UR - http://www.scopus.com/inward/record.url?scp=85062488392&partnerID=8YFLogxK

U2 - 10.1142/S0129054119400021

DO - 10.1142/S0129054119400021

M3 - Article

AN - SCOPUS:85062488392

VL - 30

SP - 29

EP - 60

JO - International Journal of Foundations of Computer Science

JF - International Journal of Foundations of Computer Science

SN - 0129-0541

IS - 1

ER -