Several Applications Of The Semi-tensor Product Of Matrices In Logic Switching Networks And Reversible Logic

This paper mainly investigates condition for the stability, controllability andobservability of multi-valued switched logic control networks and mixed-valued switchedlogic control networks by using the semi-tensor product. And the application of thesemi-tensor product in the synthesis of Toffoli gates.1. Promote the sufficiency and necessary conditions for the controllability and stabilityof switched Boolean networks to the multi-valued switched logic networks andmixed-valued switched logic networks. i.e. a kind of switching-input-state transfer graph isgiven, based on which we propose the switching-input-state incidence matrix that containsthe complete information of the switching-input-state transfer-graph. And by using theswitching-input-state incidence matrix, the sufficient and necessary conditions areobtained for the controllability and stability of multi-valued and mixed-valued switchedlogic control networks.2. Promote the sufficient condition for the observability and the sufficient andnecessary conditions for the observability under the assumption of controllability ofswitched Boolean networks to the multi-valued switched logic networks and mixed-valuedswitched logic networks. i.e. based on the switching-input-state incidence matrix, thesufficient condition for the observability of the multi-valued switched logic networks andmixed-valued switched logic networks is obtained. And by using the observabilitymatrix,the sufficient and necessary conditions for the for the observability of multi-valuedand mixed-valued switched logic networks under the assumption of controllability isobtained too.3. The matrix expression of reversible logic function is obtained by using thesemi-tensor product. Based on which we proposed the structure matrix that contains thecomplete information of the output states. Then the structure matrix of reversible gatescasecade is gained. A new Toffoli gate casecade synthesis algorithm is presented.