[Paper Review] An Improvement to a Recent Upper Bound for Synchronizing Words of Finite Automata

Yaroslav Shitov|arXiv (Cornell University)|Jan 1, 2019

semigroups and automata theory19 citations

TL;DR

This paper improves the upper bound for the reset threshold of synchronizing finite automata from α ≈ 0.1664 to α ≤ 0.1654 by refining Szyku{ł}a's 2017 method through a novel counting argument based on corank progression and optimized word length gaps. The approach combines a modified application of Theorem 1 with a refined analysis of word rank transitions, leading to a tighter cubic bound on the length of the shortest reset word in n-state automata.

ABSTRACT

It has been known since the 60's that any complete discrete $n$-state automaton admits a reset word of length not exceeding $αn^3+o(n^3)$ for some absolute constant $α$. J.-E. Pin and P. Frankl proved this statement with $α=1/6=0.1666...$ in 1982, and this bound remained best known until 2017, when M. Szykuła decreased its value to $α\approx0.1664$. In this note, we present a modification to the latest approach and develop a different counting argument which leads to a more substantial improvement of $α\leqslant 0.1654$.

Motivation & Objective

• To improve the long-standing upper bound on the reset threshold of synchronizing finite automata.
• To refine Szyku{ł}a's 2017 method that reduced the bound to α ≈ 0.1664.
• To develop a new counting argument based on corank progression and word length gaps to achieve a more substantial improvement.
• To provide a tighter cubic upper bound of the form αn³ + o(n³) for the shortest reset word length.
• To resolve the optimization problem underlying the bound by analyzing the distribution of rank transitions via constrained summation.

Proposed method

• Introduces a modified version of Theorem 1 (from Szyku{ł}a's work) to construct words that increase corank more efficiently.
• Defines λi as the minimal length of a word with corank at least i, and δj = λj+1 − λj to track length increments between corank levels.
• Applies Theorem 5 to bound the length needed to increase corank from r to r+1 using two alternatives: Pin–Frankl’s bound or a new counting-based bound involving sr (number of δj values in {2r−1, 2r}).
• Uses Corollary 6 to express the total reset threshold as a sum involving min{r²/4, 1s₁ + ⋯ + rsr} terms, which are then optimized.
• Applies optimization techniques under constraints s₁ + ⋯ + sk ≤ ρ and 1s₁ + ⋯ + rsr ≤ r²/4 for r ≥ ρ, proving the maximum of the sum is bounded by 15625n³/1597536.
• Employs calculus on a compact feasible region to maximize the objective function, confirming the bound is achieved at (25n/129, 125n/258) + o(n).

Experimental results

Research questions

• RQ1 Can the upper bound for the reset threshold of n-state synchronizing automata be improved beyond α ≈ 0.1664?
• RQ2 What is the optimal way to bound the length increment required to increase the corank of a word by one, given constraints on rank transitions?
• RQ3 How can the counting argument in Szyku{ł}a's method be refined to yield a more substantial improvement in the cubic coefficient?
• RQ4 What is the maximum value of the sum ∑_{r=ρ}^k min{r²/4, 1s₁ + ⋯ + rsr} under the constraints s₁ + ⋯ + sk ≤ ρ and 1s₁ + ⋯ + rsr ≤ r²/4 for all r ≥ ρ?
• RQ5 Can the optimization of the sum be completed analytically to yield a closed-form upper bound on the reset threshold?

Key findings

• The paper establishes a new upper bound for the reset threshold of any synchronizing n-state automaton: rt(A) ≤ 0.1654n³ + O(n²).
• The bound is achieved by refining Szyku{ł}a's method through a new counting argument that improves the coefficient α from 0.1664 to 0.1654.
• The optimization of the sum ∑_{r=ρ}^k min{r²/4, 1s₁ + ⋯ + rsr} is shown to be bounded by 15625n³/1597536 + o(n³), which is the key analytical component of the improvement.
• The maximum of the objective function is attained at the point (25n/129, 125n/258) + o(n), confirming the tightness of the derived bound.
• The o(n³) error term is shown to be O(n² log n), and a more careful analysis could reduce it to O(n²) with explicit, small coefficients.
• The result provides a significant step toward resolving the Černý conjecture, which posits a (n−1)² bound, by narrowing the gap in the cubic coefficient.