A Note on Minimal Boolean Formula Size of One-Dimensional Cellular Automata

In [4] Wolfram asserts to have found minimal Boolean formulas for (what he denotes) rules of one-dimensional, two-state, nearest neighbor cellular automata (CA) or simply elementary rules. These formulas are minimal in the sense that they “use the minimum possible number of operators” over basis Ω1 = {0,1,¬,∧,∨,⊕}. Provided that elementary rules can be interpreted as 3-input Boolean functions and visualized via their respective truth table representation, we would like to draw attention to result (a) of [2], which states that the maximal formula size for 3-input Boolean functions over basis Ω1 is 5. This result clearly disproves the minimalistic nature of Wolfram’s 8 Boolean formulas in [4] of size 6 and sets a new upper bound on formula size. We enumerate all 256 3-input Boolean functions via their respective truth table representation and their output column Boolean vector α̂ where αi ∈ {0,1}. Each of the 256 functions represents one permutation of eight binary bits in the output column Boolean vector