Approximating Heterotypic Bisimulations for Weighted Finite Automata over the Field of Real Numbers

This paper examines the existence and approximation of bisimulations between weighted finite automata (WFAs) over the real numbers. It shows that forward–backward bisimulation (fbb) and backward–forward bisimulation (bfb) between two WFAs are equivalent to solving specific homogeneous Sylvester equations and two vector equations. The transition matrices of the automata serve as the coefficient matrices in these equations. This approach reformulates the WFA’s problem as a linear algebra task involving real coefficient matrices. Obtained systems of vector and matrix equations frequently lack consistency. Multi-criteria optimization can address these inconsistencies. We apply continuous-time zeroing neural network (ZNN) dynamics to find approximate solutions for these inconsistent vector-matrix systems. Since ZNN dynamics are globally convergent, they generate approximate solutions that evolve over time. Simulation experiments are conducted on random transition matrices, using various initial state matrices and two activation functions.