An undecidable nested recurrence relation

Abstract

Roughly speaking, a recurrence relation is nested if it contains a subexpression of the form ...A(...A(...)...). Many nested recurrence relations occur in the literature, and determining their behavior seems to be quite difficult and highly dependent on their initial conditions. A nested recurrence relation A(n) is said to be undecidable if the following problem is undecidable: given a finite set of initial conditions for A(n), is the recurrence relation calculable? Here calculable means that for every n ≥ 0, either A(n) is an initial condition or the calculation of A(n) involves only invocations of A on arguments in {0, 1, ..., n - }. We show that the recurrence relation A(n) = A (n - 4 - A(A(n - 4))) + 4A(A(n - 4)) + A(2A(n - 4 - A(n-2)) + A(n - 2)) is undecidable by showing how it can be used, together with carefully chosen initial conditions, to simulate Post 2-tag systems, a known Turing complete problem. © 2012 Springer-Verlag.