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-1}. 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.
Cite
@article{arxiv.1203.0586,
title = {An Undecidable Nested Recurrence Relation},
author = {Marcel Celaya and Frank Ruskey},
journal= {arXiv preprint arXiv:1203.0586},
year = {2012}
}