On Varieties of Ordered Automata

TL;DR

This paper explores a generalized correspondence between classes of languages and classes of ordered automata, providing new insights and discussing complexity aspects of membership problems.

Contribution

It extends the Eilenberg correspondence to positive -varieties of languages and ordered automata, offering new perspectives and specific instances.

Findings

Revealed new instances of the language-automata correspondence.

Provided insights into complexity of membership problems.

Connected known results with novel observations.

Abstract

The Eilenberg correspondence relates varieties of regular languages to pseudovarieties of finite monoids. Various modifications of this correspondence have been found with more general classes of regular languages on one hand and classes of more complex algebraic structures on the other hand. It is also possible to consider classes of automata instead of algebraic structures as a natural counterpart of classes of languages. Here we deal with the correspondence relating positive $\mathcal C$-varieties of languages to positive $\mathcal C$-varieties of ordered automata and we present various specific instances of this correspondence. These bring certain well-known results from a new perspective and also some new observations. Moreover, complexity aspects of the membership problem are discussed both in the particular examples and in a general setting.