This paper investigates the transition function and the reachability conditions of finite automata by using a semitensor product of matrices, which is a new powerful matrix analysis tool. The states and input symbols are first expressed in vector forms, then the transition function is described in an algebraic form. Using this algebraic representation, a sufficient and necessary condition of the reachability of any two states is proposed, based on which an algorithm is developed for discovering all the paths from one state to another. Furthermore, a mechanism is established to recognize the language acceptable by a finite automaton. Finally, illustrative examples show that the results/algorithms presented in this paper are suitable for both deterministic finite automata （DFA） and nondeterministic finite automata （NFA）.
The problem of solving type-2 fuzzy relation equations is investigated. In order to apply semi-tensor product of matrices, a new matrix analysis method and tool, to solve type-2 fuzzy relation equations, a type-2 fuzzy relation is decomposed into two parts as principal sub-matrices and secondary sub-matrices; an r-ary symmetrical-valued type-2 fuzzy relation model and its corresponding symmetrical-valued type-2 fuzzy relation equation model are established. Then, two algorithms are developed for solving type-2 fuzzy relation equations, one of which gives a theoretical description for general type-2 fuzzy relation equations; the other one can find all the solutions to the symmetrical-valued ones. The results can improve designing type-2 fuzzy controllers, because it provides knowledge to search the optimal solutions or to find the reason if there is no solution. Finally some numerical examples verify the correctness of the results/algorithms.
The semi-tensor product （STP） of matrices was used in the article, as a new matrix analysis tool, to investigate the problem of verification of self-verifying automata （SVA）. SVA is a special variant of finite automata which is essential to nondeterministic communication with a limited number of advice bits. The status, input and output symbols are expressed in vector forms, the dynamic behaviour of SVA is modelled as matrix product is STP. By such algebraic formulation, three an algebraic equation of the states and inputs, in which the necessary and sufficient conditions are presented for the verification problem, by which three algorithms are established to find out all the strings which are accepted, rejected, or unrecognized by a SVA. Testing examples show the correctness of the results.
