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Received Nov 15, 2017; Revised Jan 6, 2018; Accepted Jan 18, 2018
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1. Introduction
In the research field of theoretical computer science, finite automaton is one of the simplest models of computation. Finite automaton is a device whose states take values from a finite set. It receives a discrete sequence of inputs from the outside world and changes its state according to the inputs. The study of finite automata has received many scholars’ research interest in the last century [1–5] due to its wide applications in engineering, computer science, and so on.
As we all know, controllability and stabilizability analysis of finite automata are fundamental topics, which are important and necessary to the solvability of many related problems [1, 4, 6]. The concepts of controllability, reachability, and stabilizability of finite automata were defined in [2] by resorting to the classic control theory. The controllability of a deterministic Rabin automaton was studied in [7] by defining the “controllability subset.” Kobayashi et al. [8] investigated the state feedback stabilization of a deterministic finite automaton and presented some new results.
Recently, a new matrix product, namely, the semitensor product (STP) of matrices, has been proposed by Cheng et al. [9]. Up to now, STP has been successfully applied to many research fields related to finite-valued systems like Boolean networks [10–20], multivalued logical networks [21–23], game theory [24, 25], finite automata [5, 26], and so on [27–35]. The main feature of STP is to convert a finite-valued system into an equivalent algebraic form [22]. Thus, STP provides a convenient way for the construction and analysis of finite automata [5, 26]. Xu and Hong [5] provided a matrix-based algebraic approach for the reachability analysis of finite automata with the help of STP. Yan et al. [26] studied the controllability and stabilizability analysis of finite automata based on STP and presented some novel results. It should be pointed out that although the concepts of controllability, reachability, and stabilizability of finite automata come from classic control theory, there exist fewer results on the construction of controllability matrix for finite automata.
In this...