Syntactic Complexity of Suffix-Free Languages

TL;DR

This paper establishes the maximum possible size of the syntactic semigroup for suffix-free languages with a given number of quotients, providing a tight bound and uniqueness results for minimal automata.

Contribution

It proves a tight upper bound on the syntactic semigroup size for suffix-free languages and shows the minimal alphabet size needed for witnesses, also establishing the uniqueness of the transition semigroup.

Findings

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

Witness languages require at least five letters in the alphabet.

Transition semigroup of minimal automata is unique for each $n$.

Abstract

We solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a suffix-free language with $n$ left quotients (that is, with state complexity $n$) is at most $(n-1)^{n-2}+n-2$ for $n\ge 6$. Since this bound is known to be reachable, this settles the problem. We also reduce the alphabet of the witness languages reaching this bound to five letters instead of $n+2$, and show that it cannot be any smaller. Finally, we prove that the transition semigroup of a minimal deterministic automaton accepting a witness language is unique for each $n$.