Stochastic well-structured transition systems

TL;DR

This paper introduces stochastic well-structured transition systems, extending traditional models with probabilistic rules, and characterizes their computational power, showing they compute exactly the languages in BPP when augmented with order or equivalence relations.

Contribution

It defines a new class of stochastic systems incorporating probabilistic scheduling and analyzes their computational capabilities, including phase clock behavior and language recognition power.

Findings

Phase clock implementation either stops or ticks too fast after polynomial steps.

Terminating computations finish or fail in expected polynomial time.

Augmented systems compute exactly the languages in BPP, unaugmented compute symmetric languages in BPL.

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.