Variants of Higher-Dimensional Automata

2026-01-24

AI Summary

Higher-dimensional automata (HDA) are often too rigid in their formalism, limiting their direct applicability in modeling and reasoning. This work systematically integrates several weakened variants—such as HDA with interfaces, partial HDA, ST-automata, and relational HDA—and demonstrates, through formal language theory, automata transformations, and algebraic analysis, that these variants fall into only two distinct classes at the language level: those closed under inclusion and those that are not. The paper’s core contributions include the first proof that partial HDA satisfy a Kleene theorem and admit determinization, alongside the establishment of a unified framework that clarifies the expressive power boundaries among all considered variants. These results lay the foundational groundwork for regular expression characterizations and determinization procedures for partial HDA.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The theory of higher-dimensional automata (HDAs) has seen rapid progress in recent years, and first applications, notably to Petri net analysis, are starting to show. It has, however, emerged that HDAs themselves often are too strict a formalism to use and reason about. In order to solve specific problems, weaker variants of HDAs have been introduced, such as HDAs with interfaces, partial HDAs, ST-automata or even relational HDAs. In this paper we collect definitions of these and a few other variants into a coherent whole and explore their properties and translations between them. We show that with regard to languages, the spectrum of variants collapses into two classes, languages closed under subsumption and those that are not. We also show that partial HDAs admit a Kleene theorem and that, contrary to HDAs, they are determinizable.

Problem

• higher-dimensional automata
• variants
• formalism
• weakness
• unified framework

Innovation

• higher-dimensional automata
• partial HDAs
• Kleene theorem
• determinizability
• subsumption-closed languages