A Quantitative Study of Pure Parallel Processes

TL;DR

This paper analyzes the combinatorial explosion in parallel process analysis by applying advanced analytic combinatorics to measure process behaviors and introduces efficient algorithms for probability computation and random sampling of concurrent runs.

Contribution

It provides a novel average-case analysis of the interleaving operator in concurrency, along with practical algorithms to mitigate combinatorial explosion issues.

Findings

Quantitative measures of process behaviors as plane rooted trees.

A linear-time algorithm for computing concurrent run prefix probabilities.

An efficient algorithm for uniform random sampling of concurrent runs.

Abstract

In this paper, we study the interleaving -- or pure merge -- operator that most often characterizes parallelism in concurrency theory. This operator is a principal cause of the so-called combinatorial explosion that makes very hard - at least from the point of view of computational complexity - the analysis of process behaviours e.g. by model-checking. The originality of our approach is to study this combinatorial explosion phenomenon on average, relying on advanced analytic combinatorics techniques. We study various measures that contribute to a better understanding of the process behaviours represented as plane rooted trees: the number of runs (corresponding to the width of the trees), the expected total size of the trees as well as their overall shape. Two practical outcomes of our quantitative study are also presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random sampling of concurrent runs. These provide interesting responses to the combinatorial explosion problem.