In this paper we develop the theory of how to count, in thin concurrent games, the configurations of a strategy witnessing that it reaches a certain configuration of the game. This plays a central role in many recent developments in concurrent games, whenever one aims to relate concurrent strategies with weighted relational models. The difficulty, of course, is symmetry: in the presence of symmetry many configurations of the strategy are, morally, different instances of the same, only differing on the inessential choice of copy indices. How do we know which ones to count? The purpose of the paper is to clarify that, uncovering many strange phenomena and fascinating pathological examples along the way. To illustrate the results, we show that a collapse operation to a simple weighted relational model simply counting witnesses is preserved under composition, provided the strategies involved do not deadlock.

中文翻译：

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学习数到对称

在本文中，我们开发了如何在细并发游戏中计算策略配置的理论，以证明它达到了游戏的特定配置。这在并发游戏的许多最新发展中起着核心作用，每当人们旨在将并发策略与加权关系模型相关联时。当然，困难在于对称性：在存在对称性的情况下，策略的许多配置在道德上是相同的不同实例，只是在复制索引的非本质选择上有所不同。我们怎么知道要计算哪些呢？这篇论文的目的是澄清这一点，在此过程中发现许多奇怪的现象和引人入胜的病理例子。为了说明结果，