Large Aperiodic Semigroups

TL;DR

This paper investigates the maximum size of syntactic semigroups for star-free languages, introducing new classes of aperiodic semigroups and establishing bounds on language reversal complexity.

Contribution

It introduces unitary and semiconstant semigroups, providing new structures that surpass known sizes and resolving open problems in star-free language complexity.

Findings

Largest known aperiodic semigroups are semiconstant tree semigroups.

Proved an upper bound of 2^n-1 for reversal complexity of star-free languages.

Resolved an open problem on concatenation complexity for star-free languages.

Abstract

The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of the minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having $n$ left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with $n$ states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For $n \ge 4$ there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that $2^n-1$ is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.