Simulations and Bisimulations between Weighted Finite Automata Based on Time-Varying Models over Real Numbers

Abstract

The zeroing neural network (ZNN) is an important kind of continuous-time recurrent neural network (RNN). Meanwhile, the existence of forward and backward simulations and bisimulations for weighted finite automata (WFA) over the field of real numbers has been widely investigated. Two types of quantitative simulations and two types of bisimulations between WFA are determined as solutions to particular systems of matrix and vector inequations over the field of real numbers R. The approach used in this research is unique and based on the application of a ZNN dynamical evolution in solving underlying matrix and vector inequations. This research is aimed at the development and analysis of four novel ZNN dynamical systems for addressing the systems of matrix and/or vector inequalities involved in simulations and bisimulations between WFA. The problem considered in this paper requires solving a system of two vector inequations and a couple of matrix inequations. Using positive slack matrices, required matrix and vector inequations are transformed into corresponding equations and then the derived system of matrix and vector equations is transformed into a system of linear equations utilizing vectorization and the Kronecker product. The solution to the ZNN dynamics is defined using the pseudoinverse solution of the generated linear system. A detailed convergence analysis of the proposed ZNN dynamics is presented. Numerical examples are performed under different initial state matrices. A comparison between the ZNN and linear programming (LP) approach is presented.