Syntactic complexity of bifix-free languages

TL;DR

This paper determines the maximum size of the syntactic semigroup for bifix-free regular languages with a given state complexity, providing exact bounds and showing the uniqueness of the largest transition semigroups.

Contribution

It proves the exact maximum syntactic semigroup size for bifix-free languages and establishes the minimal alphabet size needed to achieve this bound, resolving an open problem.

Findings

Maximum syntactic semigroup size: $(n-1)^{n-3}+(n-2)^{n-3}+(n-3)2^{n-3}$ for $n \\ge 6$

Minimal alphabet size to meet the bound: $(n-2)^{n-3} + (n-3)2^{n-3} - 1$ for $n \\ge 6$

Largest transition semigroups are unique up to renaming states

Abstract

We study the properties of syntactic monoids of bifix-free regular languages. In particular, we solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a bifix-free language with state complexity $n$ is at most $(n-1)^{n-3}+(n-2)^{n-3}+(n-3)2^{n-3}$ for $n\ge 6$. The main proof uses a large construction with the method of injective function. Since this bound is known to be reachable, and the values for $n \le 5$ are known, this completely settles the problem. We also prove that $(n-2)^{n-3} + (n-3)2^{n-3} - 1$ is the minimal size of the alphabet required to meet the bound for $n \ge 6$. Finally, we show that the largest transition semigroups of minimal DFAs which recognize bifix-free languages are unique up to renaming the states.