Abstract
We study the combinatorial structure of concurrent programs with non-deterministic choice and a fork-join style of coordination. As a first step we establish a link between these concurrent programs and a class of combinatorial structures. Based on this combinatorial interpretation, we develop and experiment algorithms aimed at the statistical exploration of the state-space of programs. The first algorithm is a uniform random sampler of bounded executions, providing a suitable default exploration strategy. The second algorithm is a random sampler of execution prefixes that allows to control the exploration with respect to the uniform distribution. The fundamental characteristic of these algorithms is that they work on the control graph of the programs and not directly on their state-space, thus providing a way to tackle the state explosion problem.
This research was partially supported by the ANR MetACOnc project ANR-15-CE40-0014.
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Notes
- 1.
A function f is in \(\sharp P\) if there is a polynomial-time non-deterministic Turing machine M such that for any instance x, f(x) is the number of executions of M that accept x as input [23].
- 2.
An implementation of our algorithms and all the scripts used for the experiments can be found on the companion repository at https://gitlab.com/ParComb/libnfj.
- 3.
Since we do not interpret the action symbols, no distinction is possible in the provided semantics between a choice being triggered internally or externally.
- 4.
In general, the notation \(P \overset{a}{\Rightarrow } P'\) is ambiguous since there may be several different proof-trees with the same conclusion \(P \overset{a}{\rightarrow } P'\). An example of this is given in Remark 1. For the sake of simplicity and in order to keep the notations light, in the examples given here, each step \(P \overset{a}{\Rightarrow } P'\) identifies only one possible proof-tree.
- 5.
The word “labelled” is not particularly relevant in our setup. It refers to the fact that another way to represent the interleaving of two executions is to put an integer label on each step of both execution carrying the position of each step in the interleaving.
- 6.
Grammar descriptions involving non-disjoint unions are referred to as “ambiguous” and lack most of the benefits, if not all, of the symbolic method, essentially because some objects may be counted multiple times when applying the method.
- 7.
This is the only “non-standard” computation rule we use in this example. All the rest is usual polynomial manipulations. General rules for computing \(A(z) \circledcirc B(z)\) are beyond the scope of this article.
- 8.
All the benchmarks were run on a standard laptop with an Intel Core i7-8665U and 32G of RAM running Ubuntu 19.10 with kernel version 5.3.0–46-generic. We used FLINT version 2.5.2 and GMP version 6.1.2.
- 9.
The interquartile range of a set of measures is the difference between the third and the first quartiles. Compared with the value of the median, it gives a rough estimate of the dispersion of the measures.
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Genitrini, A., Pépin, M., Peschanski, F. (2020). Statistical Analysis of Non-deterministic Fork-Join Processes. In: Pun, V.K.I., Stolz, V., Simao, A. (eds) Theoretical Aspects of Computing – ICTAC 2020. ICTAC 2020. Lecture Notes in Computer Science(), vol 12545. Springer, Cham. https://doi.org/10.1007/978-3-030-64276-1_5
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