Syntactic complexity of regular ideals

TL;DR

This paper establishes tight upper bounds on the syntactic complexity of various subclasses of regular languages, specifically ideals and their complements, and proves the uniqueness of the transition semigroups achieving these bounds.

Contribution

It provides the first precise bounds for the syntactic complexity of regular ideals and shows the uniqueness of the semigroups attaining these bounds.

Findings

Tight upper bounds for syntactic complexity of right, left, and two-sided ideals.

Uniqueness of transition semigroups meeting the bounds.

Fixed number of generators needed for the semigroups.

Abstract

The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity $n$ of languages in that class. We prove that $n^{n-1}$, $n^{n-1}+n-1$, and $n^{n-2}+(n-2)2^{n-2}+1$ are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced.