A functional proof pearl: inverting the Ackermann hierarchy

Abstract

We implement in Gallina a hierarchy of functions that calculate the upper inverses to the hyperoperation/Ackermann hierarchy. Our functions run in Θ(b) for inputs expressed in unary, and in O(b2) for inputs expressed in binary (where b = bitlength). We use our inverses to define linear-time functions—Θ(b) for both unary-represented and binary-represented inputs—that compute the upper inverse of the diagonal Ackermann function A(n). We show that these functions are consistent with the usual definition of the inverse Ackermann function α(n).