Single-Step and Asymptotic Mutual Information in Bipartite Boolean Nets

Abstract

In this paper we contrast two fundamentally different ways to approach the analysis of transition system behaviours. Both methods refer to the (finite) global state transition graph; but while method A, familiar to software system designers and process algebraists, deals with execution paths of virtually unbounded length, typically starting from a precise initial state, method B, associated with counterfactual reasoning, looks at single-step evolutions starting from all conceivable system states.

Among various possible state transition models we pick boolean nets – a generalisation of cellular automata in which all nodes fire synchronously. Our nets shall be composed of parts P and Q that interact by shared variables. At first we adopt approach B and a simple information-theoretic measure – mutual information $M(y_P,y_Q)$ – for detecting the degree of coupling between the two components after one transition step from the uniform distribution of all global states. Then we consider an asymptotic version $M(y_P^*,y_Q^*)$ of mutual information, somehow mixing methods A and B, and illustrate a technique for obtaining accurate approximations of $M(y_P^*,y_Q^*)$ based on the attractors of the global graph.