Büchi-Elgot-Trakhtenbrot Theorem for Higher-Dimensional Automata

TL;DR

It is proved that a set of interval ipomsets is the language of an HDA if and only if it is simultaneously MSO-definable, of bounded width, and closed under order refinement.

Abstract

In this paper we explore languages of higher-dimensional automata (HDAs) from an algebraic and logical point of view. Such languages are sets of finite width-bounded interval pomsets with interfaces (ipomsets) closed under order extension. We show that ipomsets can be represented as equivalence classes of words over a particular alphabet, called step sequences. We introduce an automaton model that recognize such languages. Doing so allows us to lift the classical Büchi-Elgot-Trakhtenbrot Theorem to languages of HDAs: we prove that a set of interval ipomsets is the language of an HDA if and only if it is simultaneously MSO-definable, of bounded width, and closed under order refinement.