Introducing the activity parameter for elementary cellular automata

Given an elementary cellular automaton (ECA) with local transition rule R, two different types of local transitions are identified: the ones in which a cell remains in its current state, called inactive transitions, and the ones in which the cell changes its current state, which are called active transitions. The number of active transitions of a rule is called its activity value. Based on latter identification, a rule R-1 is called a sub-rule of R-2 if the set of active transitions of R-1 is a subset of the active transitions of R-2. In this paper, the notion of sub-rule for elementary cellular automata is introduced and explored: first, we consider a lattice that illustrates relations of nonequivalent elementary cellular automata according to nearby sub-rules. Then, we introduce statistical measures that allow us to compare rules and sub-rules. Finally, we explore the possible similarities in the dynamics of a rule with respect to its sub-rules, obtaining both empirical and theoretical results.