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arXiv · 2505.10461

B\"uchi-Elgot-Trakhtenbrot Theorem for Higher-Dimensional Automata

Amazigh Amrane·Hugo Bazille·Emily Clement·Uli Fahrenberg·Marie Fortin·Krzysztof Ziemiański
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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 B\"uchi-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.

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Amazigh Amrane, Hugo Bazille, Emily Clement, Uli Fahrenberg, Marie Fortin, Krzysztof Ziemiański. 2025-05-15. B\"uchi-Elgot-Trakhtenbrot Theorem for Higher-Dimensional Automata. https://arxiv.org/abs/2505.10461

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