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 $\left\{ 0,1,\ldots,n-1\right\}$. We show that the recurrence relation

$$\begin{aligned} A\left(n\right) & =A\left(n-4-A\left(A\left(n-4\right)\right)\right)+4A\left(A\left(n-4\right)\right)\\ & +A\left(2A\left(n-4-A\left(n-2\right)\right)+A\left(n-2\right)\right) \end{aligned}$$

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.