Extremal Binary PFAs with Small Number of States

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 $n\geq 6$. 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 \v{C}ern\'y automata; for $n\leq 10$ it contains a binary PFA with maximal possible reset threshold; for all $n\geq 6$ 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. Asymptotically the \v{C}ern\'y family gives reset thresholds of polynomial order. We prove that PFAs in the family are not extremal for $n\geq 41$. For that purpose, we present an improvement of Martyugin's prime number construction of binary PFAs.