The largest known reset thresholds for DFAs are equal to (n-1)^2, where n is the number of states. This is conjectured to be the maximum possible. PFAs (with partial transition function) can have exponentially large reset thresholds. This is still true if we restrict to binary PFAs. However, asymptotics do not give conclusions for fixed n. We prove that the maximal reset threshold for binary PFAs is strictly greater than (n-1)^2 if and only if n > 5. These results are mostly based on the analysis of synchronizing word lengths for a certain family of binary PFAs. This family has the following properties: it contains the well-known Cerny automata; for n < 11 it contains a binary PFA with maximal possible reset threshold; for all n > 5 it contains a PFA with reset threshold larger than the maximum known for DFAs. Analysis of this family reveals remarkable patterns involving the Fibonacci numbers and related sequences such as the Padovan sequence. We derive explicit formulas for the reset thresholds in terms of these recurrent sequences. Furthermore, we prove that the family asymptotically still gives reset thresholds of polynomial order.

中文翻译：

---

Cerny 家族中的极值二元 PFA

DFA 的最大已知重置阈值等于 (n-1)^2，其中 n 是状态数。推测这是最大可能的。PFA（具有部分转换功能）可以具有指数级大的复位阈值。如果我们限制为二进制 PFA，这仍然是正确的。然而，渐近法不会给出固定 n 的结论。我们证明了当且仅当 n > 5 时，二进制 PFA 的最大复位阈值严格大于 (n-1)^2。这些结果主要基于对某个二进制 PFA 系列的同步字长的分析。该家族具有以下特性：它包含著名的 Cerny 自动机；对于 n < 11，它包含一个具有最大可能复位阈值的二进制 PFA；对于所有 n > 5，它包含一个 PFA，其重置阈值大于 DFA 已知的最大值。对该族的分析揭示了涉及斐波那契数列和相关序列（例如 Padovan 序列）的非凡模式。我们根据这些循环序列推导出重置阈值的明确公式。此外，我们证明该族渐近地仍然给出多项式阶的重置阈值。