We develop new algebraic tools to reason about concurrent behaviours modelled as languages of Mazurkiewicz traces and asynchronous automata. These tools reflect the distributed nature of traces and the underlying causality and concurrency between events, and can be said to support true concurrency. They generalize the tools that have been so efficient in understanding, classifying and reasoning about word languages. In particular, we introduce an asynchronous version of the wreath product operation and we describe the trace languages recognized by such products (the so-called asynchronous wreath product principle). We then propose a decomposition result for recognizable trace languages, analogous to the Krohn-Rhodes theorem, and we prove this decomposition result in the special case of acyclic architectures. Finally, we introduce and analyze two distributed automata-theoretic operations. One, the local cascade product, is a direct implementation of the asynchronous wreath product operation. The other, global cascade sequences, although conceptually and operationally similar to the local cascade product, translates to a more complex asynchronous implementation which uses the gossip automaton of Mukund and Sohoni. This leads to interesting applications to the characterization of trace languages definable in first-order logic: they are accepted by a restricted local cascade product of the gossip automaton and 2-state asynchronous reset automata, and also by a global cascade sequence of 2-state asynchronous reset automata. Over distributed alphabets for which the asynchronous Krohn-Rhodes theorem holds, a local cascade product of such automata is sufficient and this, in turn, leads to the identification of a simple temporal logic which is expressively complete for such alphabets.

中文翻译：

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并发行为的异步花圈乘积和级联分解

我们开发了新的代数工具来推理并发行为，这些并发行为被建模为Mazurkiewicz迹线和异步自动机的语言。这些工具反映了跟踪的分布式性质以及事件之间潜在的因果关系和并发性，可以说它们支持真正的并发性。他们概括了在理解，分类和推理单词语言方面非常有效的工具。特别是，我们介绍了花圈产品操作的异步版本，并描述了此类产品可识别的跟踪语言（所谓的异步花圈产品原理）。然后，我们针对可识别的跟踪语言提出分解结果，类似于Krohn-Rhodes定理，并且我们在非循环体系结构的特殊情况下证明了这种分解结果。最后，我们介绍并分析了两种分布式自动机理论操作。一种是本地级联产品，是异步花圈产品操作的直接实现。其他全局级联序列，尽管在概念上和操作上与本地级联产品相似，但转换为使用Mukund和Sohoni的八卦自动机的更复杂的异步实现。这将导致有趣的应用程序表征一阶逻辑中定义的跟踪语言：八卦自动机和2状态异步复位自动机的受限局部级联乘积，以及2状态的全局级联序列接受它们异步重置自动机。在异步Krohn-Rhodes定理所适用的分布式字母表中，