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

2025-05-15

AI Summary

This work addresses the logical characterization of languages accepted by higher-dimensional automata (HDAs), aiming to generalize the classical Büchi–Elgot–Trakhtenbrot theorem to concurrent models. The central challenge lies in the absence of a unified definability and recognizability theory for HDA languages. To resolve this, we adopt interface-equipped partially ordered multisets (ipomsets) as fundamental semantic objects, introduce their step-sequence encodings, and define a novel interface-aware HDA model. We establish that the class of ipomset languages recognizable by HDAs coincides precisely with those definable in monadic second-order (MSO) logic, subject to bounded width and closure under order refinement. This yields the first rigorous equivalence between automata recognizability and higher-order logical definability in concurrent systems, providing a unifying algebraic–logical foundation for verification and synthesis of concurrent programs.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Extending Büchi-Elgot-Trakhtenbrot theorem to higher-dimensional automata languages
• Characterizing languages of higher-dimensional automata via MSO logic
• Establishing equivalence between automaton model and logical definability

Innovation

• Step sequences represent ipomsets as word equivalence classes
• Automaton model recognizes bounded-width interval pomset languages
• Lifts Büchi-Elgot-Trakhtenbrot Theorem to higher-dimensional automata