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.
The notion of resource plays a central role in concurrent systems. In its purest form a resource is simply a unique identity one can create, use and ultimately destruct. In this paper we propose a simple yet effective characterization of resource usages and develop for it a complete analysis framework. We address qualitative issues such as the classification of resources and whether two systems exhibit similar patterns of resource usages-namely equivalent resource profiles. From the quantitative point of view, we develop the omniscient garbage collector (OGC), which decides precisely when a resource can be reclaimed or reused. This allows to bound precisely the number of resources consumed by a given system. To illustrate the approach, we study experimentally the resource consumption of pi-calculus processes using a prototype analysis tool. We propose two different resource abstractions for pi-processes: one based on the labelled transitions for open systems, and another one for closed systems. The latter notably provides a refined view of behaviors, less opaque than reductions. Beyond this experiment, the proposed framework is quite generic and can apply to many different formalisms and situations.
In this paper, we study two operators for composing combinatorial classes: the ordered product and its dual, the colored product. These operators have a natural interpretation in terms of Analytic Combinatorics, in relation with combinations of Borel and Laplace transforms. Based on these new constructions, we exhibit a set of transfer theorems and closure properties. We also illustrate the use of these operators to specify increasingly labeled structures tightly related to Series-Parallel constructions and concurrent processes. In particular, we provide a quantitative analysis of Fork/Join (FJ) parallel processes, a particularly expressive example of such a class.
The Pi-calculus is a formalism to model and reason about highly concurrent and dynamic systems. Most of the expressive power of the language comes from the ability to pass communication channels among concurrent processes, as any other value. We present in this paper the CubeVM, an interpreter architecture for an applied variant of the Pi-calculus, focusing on its operational semantics. The main characteristic of the CubeVM comes from its stackless architecture. We show, in a formal way, that the resource management model inside the VM may be greatly simplified without the need for nested stack frames. This is particularly true for the garbage collection of processes and channels. The proposed GC, based on a reference counting scheme, is highly concurrent and, most interestingly, does automatically detect and reclaim cycles of disabled processes. We also address the main performance issues raised by the fine-grained concurrency model of the Pi-calculus. We introduce the reactive variant of the semantics that allows, when applicable, to increase the performance drastically by bypassing the scheduler. We define the language subset of processes in so called chain-reaction forms for which the sequential semantics can be proved statically. We illustrate the expressive power and performance gains of such chain-reactions with examples of functional, dataflow and object-oriented systems. Encodings for the pure Picalculus are also demonstrated.
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