Operator systems

One can represent the possible values of expressions like f[f[p, q], p] by rule numbers analogous to those used for cellular automata. Specifying an operator f (taken in general to have n arguments with k possible values) by giving the rule number u for f[p, q, …], the rule number for an expression with variables vars can be obtained from

With[{m = Length[vars]}, FromDigits[Block[{f = Reverse[IntegerDigits[u, k, kⁿ]]〚FromDigits[{##}, k] + 1〛 &}, Apply[Function[Evaluate[vars], expr], Reverse[Array[IntegerDigits[# - 1, k, m] &, k^m]], {1}]], k]]