The Application Of The Simi-tensor Product Of Matrices In Logical Networks And Multilinear Mapping

Using matrices to express logic equivalently, we know under the framework ofsemi-tensor product of matrices, both Boolean network and multi-valued logical net-work can be represented as discrete-time dynamic system, then they can be convertedinto algebraic form x(t+1)=Lx(t). Meanwhile Boolean control network and multi-valued logical control network can be represented as the form x(t+1)=Lu(t)x(t); Inaddition, use the method of semi-tensor product of matrices, we can also structure astructure matrix of a multilinear mapping. About Boolean network, combined semi-tensor product of matrices and digital transformation we discussed its stability andstabilization. Then the fixed points and stability of k-valued logical net-works withimpulsive effects were discussed too. Besides, to semi-tensor product of matrices it-self, we obtained some new properties. The main content in this paper are:1. A one-to-one correspondence between the structure matrix of algebraic formand a digital transformation were set up. Using the method of digital transformation,we obtained a necessary and sufficient condition for the stability and stabilization ofBoolean networks.2.The k-valued logical networks with impulsive effects were discussed. And weobtained a necessary and sufficient condition for the stability and stabilization of thesystem.3. Some new properties of semi-tensor product of matrices were obtained. Thenwe put the results into the application of multilinear mappings, eventually some newmethods to improve the problems of multilinear mappings were obtained.