线性自动机的分解与复杂性
Decompositions and complexity of linear automata
摘要：
纯（无线性）自动机的Krohn-Rhodes复杂性理论是众所周知的。该理论使用圈积运算作为分解工具。本文的主要目的是引入线性自动机复杂性的概念。这个概念最终与线性自动机的分解有关。本文的第二个目的是研究这些分解。为了定义线性自动机的复杂性，必须使用三种运算，即线性自动机的三角积、纯自动机的圈积和线性自动机与返回线性自动机的纯自动机的圈积。我们将线性自动机的复杂性定义为将自动机分解为不可分解的组件（原子）的最小操作数。这一理论依赖于环积和三角积之间的以下平行性：它们都是自动机级联连接范畴中的终对象。圈积是纯自动机Krohn-Rhodes理论的终对象，而三角积则为线性自动机的级联提供了终对象。
英文摘要：
The Krohn-Rhodes complexity theory for pure (without linearity) automata is well-known. This theory uses an operation of wreath product as a decomposition tool. The main goal of the paper is to introduce the notion of complexity of linear automata. This notion is ultimately related with decompositions of linear automata. The study of these decompositions is the second objective of the paper. In order to define complexity for linear automata, we have to use three operations, namely, triangular product of linear automata, wreath product of pure automata and wreath product of a linear automaton with a pure one which returns a linear automaton. We define the complexity of a linear automaton as the minimal number of operations in the decompositions of the automaton into indecomposable components (atoms). This theory relies on the following parallelism between wreath and triangular products: both of them are terminal objects in the categories of cascade connections of automata. The wreath product is the terminal object in the Krohn-Rhodes theory for pure automata, while the triangular product provides the terminal object for the cascade connections of linear automata.
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