Foundations of block-parallel automata networks

2025-03-06

AI Summary

This paper addresses foundational theoretical questions concerning automata networks under block-parallel update scheduling. It systematically investigates their dynamical behavior, limiting behavior, and computational complexity. First, it formally defines the block-parallel update paradigm and introduces two novel equivalence relations—dynamic equivalence and limit equivalence—for classifying update schedules. Second, it develops exact counting and constructive enumeration algorithms for these equivalence classes. Third, it proves that successor computation from a single configuration is PSPACE-complete; consequently, most core decision problems—including fixed-point existence, limit-cycle detection, and reachability analysis—exhibit tight PSPACE-completeness bounds under this paradigm. Notably, the reversible case remains polynomial-time solvable. These results extend the modeling expressiveness of automata networks and provide rigorous theoretical foundations for designing update mechanisms in distributed systems, biological networks, and related domains.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Agent-Based Simulation and Emergent Behavior
• Planning, Routing, and Scheduling: Model-Based Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science

Abstract

We settle the theoretical ground for the study of automata networks under block-parallel update schedules, which are somehow dual to the block-sequential ones, but allow for repetitions of automaton updates. This gain in expressivity brings new challenges, and we analyse natural equivalence classes of update schedules: those leading to the same dynamics, and to the same limit dynamics, for any automata network. Countings and enumeration algorithms are provided, for their numerical study. We also prove computational complexity bounds for many classical problems, involving fixed points, limit cycles, the recognition of subdynamics, reachability, etc. The PSPACE-completeness of computing the image of a single configuration lifts the complexity of most problems, but the landscape keeps some relief, in particular for reversible computations.

Problem

• Establishes theoretical framework for block-parallel automata networks.
• Analyzes equivalence classes of update schedules for network dynamics.
• Proves computational complexity bounds for classical automata problems.

Innovation

• Block-parallel update schedules for automata networks
• Equivalence classes analysis for update schedules
• PSPACE-completeness in configuration image computation