Quantitative and Algorithmic aspects of Barrier Synchronization in Concurrency

Abstract

International audience In this paper we address the problem of understanding Concurrency Theory from a combinatorial point of view. We are interested in quantitative results and algorithmic tools to refine our understanding of the classical combinatorial explosion phenomenon arising in concurrency. This paper is essentially focusing on the the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space.

A Combinatorial Study of Async/Await Processes

“…In [4] we introduce the class of BIT-decomposable processes together with a compact representation of the counting problem as a multivariate integral formula. While this does not directly yield a counting algorithm, computer algebra systems can be used for the numerical resolution.…”

“…[14] (for sparse posets, with interesting applications in artificial intelligence) and [19] (based on a fast enumeration of linear extensions, not suitable for actual counting). In [4] we describe a linear extension sampler for BIT-decomposable processes, which we experiment on async/await processes in the present paper. An alternative approach is proposed in [13] which is based on a coupling from the past (MCMC) procedure.…”

“…In this section we present a very simple process calculus whose purpose is to capture the fundamental ingredients of async/await concurrency. Listing 1.1 shows a somewhat minimal Javascript example 4 . Putting aside the classical language features (function calls, assignments, etc.…”

“…At the other end of the spectrum, the lack of any obvious control structure makes counting executions akin to counting linear extensions of arbitrary posets, a ♯P -complete problem (cf. [4]).…”

“…As a first contribution, we show that the corresponding structure falls into the subclass of what we call the BIT-decomposable posets. Based on our previous work [4], we obtain a way to formulate the counting problem as a compact multivariate integral which can be solved by a computer algebra system. This is discussed in Section 3.…”