Minimum Equivalence in Random Boolean Networks, Elementary Cellular Automata, and Beyond

Abstract

Random Boolean networks (RBN) and Cellular Automata (CA) operate in a very similar way. They update their state with simple deterministic functions called Boolean function or Transition Table (TT), both being essentially the same mechanism under different names. This paper applies a concept most known from CA called Minimum Equivalence (ME). ME is applied to RBN and shows how to calculate the number of unique computations for a given number of neighbours. Crucially, it is shown how RBN rules are even more equivalent than in CA, how the set can be reduced into even fewer unique rules, and how the concept becomes more relevant with larger neighbourhoods. For example, switching transformation alone reduces the number of unique rules in RBN with 4 neighbours from 65 536 to only 3 984 (6.1%) rules. Additionally, this paper examines the ME and transformations in substrates beyond Elementary CA (ECA), such as CA with additional spatial dimensions and number of states.