Confluence and Convergence in Probabilistically Terminating Reduction Systems

TL;DR

This paper extends the concept of convergence in abstract reduction systems to probabilistic systems, introducing almost-sure convergence and providing properties to prove it, thereby generalizing classical ARS results.

Contribution

It introduces a probabilistic generalization of ARS convergence, defining almost-sure convergence and establishing properties for its proof, expanding the theoretical framework.

Findings

Defines almost-sure convergence for probabilistic ARS

Provides properties for proving almost-sure convergence

Generalizes classical ARS convergence results

Abstract

Convergence of an abstract reduction system (ARS) is the property that any derivation from an initial state will end in the same final state, a.k.a. normal form. We generalize this for probabilistic ARS as almost-sure convergence, meaning that the normal form is reached with probability one, even if diverging derivations may exist. We show and exemplify properties that can be used for proving almost-sure convergence of probabilistic ARS, generalizing known results from ARS.