Regular Ideal Languages and Their Boolean Combinations

TL;DR

This paper characterizes regular ideals and their Boolean combinations using automaton models, providing algebraic and logical insights, and extending classical topological results to finite words.

Contribution

It introduces automaton models complete for regular Boolean combinations of ideals and connects these to logical definability and algebraic properties.

Findings

Automaton models like unions of flip automata are complete for regular Boolean ideals.

Regular Boolean combinations of right ideals correspond to languages recognizable by Staiger-Wagner automata.

The results extend classical topological theorems to finite words and relate to two-variable first-order logic.

Abstract

We consider ideals and Boolean combinations of ideals. For the regular languages within these classes we give expressively complete automaton models. In addition, we consider general properties of regular ideals and their Boolean combinations. These properties include effective algebraic characterizations and lattice identities. In the main part of this paper we consider the following deterministic one-way automaton models: unions of flip automata, weak automata, and Staiger-Wagner automata. We show that each of these models is expressively complete for regular Boolean combination of right ideals. Right ideals over finite words resemble the open sets in the Cantor topology over infinite words. An omega-regular language is a Boolean combination of open sets if and only if it is recognizable by a deterministic Staiger-Wagner automaton; and our result can be seen as a finitary version of this classical theorem. In addition, we also consider the canonical automaton models for right ideals, prefix-closed languages, and factorial languages. In the last section, we consider a two-way automaton model which is known to be expressively complete for two-variable first-order logic. We show that the above concepts can be adapted to these two-way automata such that the resulting languages are the right ideals (resp. prefix-closed languages, resp. Boolean combinations of right ideals) definable in two-variable first-order logic.