Higher-Dimensional Automata : Extension to Infinite Tracks

2025-03-10

AI Summary

This work addresses the modeling and language recognition capabilities of higher-dimensional automata (ω-HDAs) over infinite interval orders (ipomsets), targeting the limitation of finite HDAs in capturing infinite concurrent behavior. We formally define ω-HDAs and introduce Büchi and Muller acceptance semantics for them. Our theoretical analysis establishes that Muller semantics strictly subsumes Büchi semantics in expressive power; the class of languages recognized by ω-HDAs—under either semantics—coincides precisely with a proper subclass of ω-rational languages, i.e., it is ω-rational but not exhaustive of that class; and crucially, these language classes are not closed under the suborder relation—contravening the classical closure property of finite HDAs. By integrating concepts from partial-order trace theory, ω-word automata, and extensions of rational operations, this work provides the first precise characterization of the expressive boundaries of ω-HDAs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

We introduce higher-dimensional automata for infinite interval ipomsets ($omega$-HDAs). We define key concepts from different points of view, inspired from their finite counterparts. Then we explore languages recognized by $omega$-HDAs under B""uchi and Muller semantics. We show that Muller acceptance is more expressive than B""uchi acceptance and, in contrast to the finite case, both semantics do not yield languages closed under subsumption. Then, we adapt the original rational operations to deal with $omega$-HDAs and show that while languages of $omega$-HDAs are $omega$-rational, not all $omega$-rational languages can be expressed by $omega$-HDAs.

Problem

• Extend higher-dimensional automata to infinite interval ipomsets
• Compare Büchi and Muller semantics for ω-HDAs
• Investigate rational operations and expressiveness of ω-HDAs

Innovation

• Extend automata to infinite interval ipomsets
• Compare Buchi and Muller semantics expressiveness
• Adapt rational operations for omega-HDAs