Decompositions and complexity of linear automata

TL;DR

This paper introduces a complexity measure for linear automata based on their decompositions into indecomposable components using triangular and wreath product operations, extending Krohn-Rhodes theory.

Contribution

It defines the complexity of linear automata through new decomposition operations and establishes a parallel between wreath and triangular products in automata theory.

Findings

Defined the complexity of linear automata via minimal decomposition operations.

Established the triangular product as a terminal object in cascade connections of linear automata.

Extended Krohn-Rhodes complexity theory to linear automata.

Abstract

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.