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The compress-with-another threshold of Szykuła's Figure 3 family

  • Qichao Wang
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Abstract

We give a self-contained pair-automaton proof of the exact compress-with-another threshold of the corrected Figure 3 family from a recent survey of open problems in synchronizing automata. For every p ≥ 3, the automaton has n = 3p states and μ(q0) = 4p = 4n/3. The word (ba)p(ab)p attains this value. Two entrance potentials and an excluded region yield the lower bound, with the endpoint exception at p = 3 treated explicitly. A separate reset construction proves rt(Ap) ≤ 3p2 + 4p − 1 for every parameter. Reproducible computations verify the transition and entrance identities; the all-parameter results follow from the explicit proofs.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026compresswithanother,
  title = {{The compress-with-another threshold of Szykuła's Figure 3 family}},
  author = {Qichao Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07044},
  eprint = {2610.07044},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07044},
}
RIS
TY  - UNPB
TI  - The compress-with-another threshold of Szykuła's Figure 3 family
AU  - Qichao Wang
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07044
UR  - https://arxiv.org/abs/2610.07044
AB  - We give a self-contained pair-automaton proof of the exact compress-with-another threshold of the corrected Figure 3 family from a recent survey of open problems in synchronizing automata. For every p ≥ 3, the automaton has n = 3p states and μ(q_0) = 4p = 4n/3. The word (ba)^p(ab)^p attains this value. Two entrance potentials and an excluded region yield the lower bound, with the endpoint exception at p = 3 treated explicitly. A separate reset construction proves rt(A_p) ≤ 3p^2 + 4p − 1 for every parameter. Reproducible computations verify the transition and entrance identities; the all-parameter results follow from the explicit proofs.
ER  -