Alternating Nominal Automata with Name Allocation

2024-08-07

arXiv.org

AI Summary

For computational models over infinite alphabets equipped with unbounded registers, ensuring efficient decidability of language-theoretic problems—particularly nonemptiness and inclusion—is fundamentally challenging, especially when supporting alternation and selection mechanisms, without restricting machine power or memory capacity. Method: We introduce Regular Alternating Nominal Automata (RANA), a novel automaton model that integrates nominal set theory, orbit-finiteness analysis, and a new name-allocation mechanism to tame alternation in the presence of unbounded registers. Contribution/Results: RANA is the first alternating nominal automaton model supporting nonemptiness and inclusion checking under unbounded registers while retaining elementary-time complexity. Our construction achieves near-complete alternation elimination—leaving only a single universal deadlock state. Furthermore, we reduce the model-checking complexity of Bar-μTL over finite data words by one exponential level, advancing the state of the art in verification for infinite-alphabet systems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Agreement, Argumentation & Negotiation

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Formal languages over infinite alphabets serve as abstractions of structures and processes carrying data. Automata models over infinite alphabets, such as classical register automata or, equivalently, nominal orbit-finite automata, tend to have computationally hard or even undecidable reasoning problems unless stringent restrictions are imposed on either the power of control or the number of registers. This has been shown to be ameliorated in automata models with name allocation such as regular nondeterministic nominal automata, which allow for deciding language inclusion in elementary complexity even with unboundedly many registers while retaining a reasonable level of expressiveness. In the present work, we demonstrate that elementary complexity survives under extending the power of control to alternation: We introduce regular alternating nominal automata (RANAs), and show that their non-emptiness and inclusion problems have elementary complexity even when the number of registers is unbounded. Moreover, we show that RANAs allow for nearly complete de-alternation, specifically de-alternation up to a single deadlocked universal state. As a corollary to our results, we improve the complexity of model checking for a flavour of Bar-$mu$TL, a fixed-point logic with name allocation over finite data words, by one exponential level.

Problem

• Infinite Symbol Set
• Machine Model Efficiency
• Complex Operation Handling

Innovation

• Complexity Reduction
• Nominal Automata
• Bar-μTL Logic