Language acceptability of finite automata based on theory of semi-tensor product of matrices

Using the theories of many-valued logic and semi-tensor product of matrices (STP), this paper investigates how to mathematically determine whether or not a regular language is recognized by finite automata (FA). To this end, the dynamic behaviour of FA is first formulated as bilinear dynamic equations, which provides a uniform model for deterministic and non-deterministic FA. Based on the bilinear model, the recognition power of FA understanding of regular languages is investigated and several algebraic criteria are obtained. With the algebraic criteria, to judge whether a regular sentence is accepted by a FA or not, one only needs to calculate an STP of some vectors, rather than making the sentence run over the machine as traditional manners. Further, the inverse problem of recognition is considered, an algorithm is developed that can mathematically construct all the accepted sentences for a given FA. The algebraic approach of this paper may be a new angle and means to understand and analyse the dynamics of FA.