Equivalential calculus

Expressions with variables vars are equivalent if they give the same results for

Mod[Map[Count[expr, #, {-1}] &, vars], 2]

With n variables, there are thus 2^{n} equivalence classes of expressions (compared to 2^{2n} for ordinary logic). The operator can be either Xor or Equal (6 or 9). With k=3 there are no operators that yield the same results; with k=4 {458142180, 1310450865, 2984516430, 3836825115} work (see below). The shortest axiom system that works up to k=2 is {(a ∘ b) ∘ a == b}. With *modus ponens* as the rule of inference, the shortest single-axiom system that works is known to be {(a ∘ b) ∘ ((c ∘ b) ∘ (a ∘ c))}. Note that equivalential calculus describes the smallest non-trivial group, and can be viewed as an extremely minimal model of algebra.