Synchronous orders on the set of integers

TL;DR

This paper studies synchronous automata recognizing strict orders over integers, providing methods to analyze their chains, antichains, linearity, and equivalence, thus advancing understanding of automata-recognized order structures.

Contribution

It introduces algorithms for analyzing the structure of orders recognized by synchronous automata over integers, including chain and antichain detection and order equivalence.

Findings

Decidable detection of infinite chains and antichains in automata-recognized orders

Characterization of linear orders recognized by synchronous automata

Method to determine equivalence of two linear synchronous orders

Abstract

A binary relation over a free monoid is synchronous if it can be recognized by a synchronous automaton that reads its two tapes simultaneously. We consider the case where the free monoid is generated by a single element (which makes it isomorphic to the additive monoid of integers) and where the binary relation recognized is a strict order. Our main results are: given such an automaton it is possible to determine whether or not is has infinite chains or antichains; we characterize the orders that are linear; given two linear synchronous orders we show how to determine whether or not they are equivalent.