Stochastic well-structured transition systems

2025-12-23

AI Summary

Well-structured transition systems (WSTS) lack probabilistic scheduling semantics, limiting their applicability to stochastic population protocols, chemical reaction networks, and epidemic spreading models. Method: We introduce randomized well-structured transition systems (RSWSTS), a unifying framework incorporating population protocols, CRNs, and propagation models; we enhance expressiveness via inter-agent total orders or equivalence relations. Contributions: First, we establish a rigorous semantic and analytical foundation for RSWSTS. Second, we prove an inherent time bottleneck in phase-clock implementations: any terminating computation requires expected polynomial time, and every phase clock must halt or overshoot within polynomially many steps. Third, we precisely characterize computational power: enhanced RSWSTS exactly capture the class BPP, while non-enhanced variants exactly compute symmetric BPL—thereby delineating their fundamental computational boundaries.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Social Networks and Social Media: Computational social science
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Extending well-structured transition systems to incorporate a probabilistic scheduling rule, we define a new class of stochastic well-structured transition systems that includes population protocols, chemical reaction networks, and many common gossip models; as well as augmentations of these systems by an oracle that exposes a total order on agents as in population protocols in the comparison model or an equivalence relation as in population protocols with unordered data. We show that any implementation of a phase clock in these systems either stops or ticks too fast after polynomially many expected steps, and that any terminating computation in these systems finishes or fails in expected polynomial time. This latter property allows an exact characterization of the computational power of many stochastic well-structured transition systems augmented with a total order or equivalence relation on agents, showing that these compute exactly the languages in BPP, while the corresponding unaugmented systems compute just the symmetric languages in BPL.

Problem

• Extending transition systems with probabilistic scheduling rules
• Analyzing phase clock behavior and termination in stochastic systems
• Characterizing computational power of augmented stochastic transition systems

Innovation

• Extends well-structured systems with probabilistic scheduling
• Introduces stochastic well-structured transition systems class
• Characterizes computational power as BPP with augmentations