Quasi-symmetric logic networks have Lyapunov functions

The authors consider that asynchronous behavior of logic networks which have a certain type of Lyapunov function associated with them. After reviewing the necessary and sufficient conditions for the existence of Lyapunov functions, they analyze the linear case to have a generalized quasi-symmetric condition for component Boolean functions with higher order threshold representation. They also derive explicit forms of Lyapunov functions under this condition. It is shown that generalized quasi-symmetric conditions are sufficient for a logic net to have Lyapunov functions