An Undecidable Nested Recurrence Relation

TL;DR

This paper demonstrates that certain nested recurrence relations can be undecidable, meaning their behavior cannot be determined in general, by linking them to Turing-complete systems through specific initial conditions.

Contribution

It introduces a specific nested recurrence relation and proves its undecidability by showing its capability to simulate Turing-complete Post 2-tag systems.

Findings

The recurrence relation can simulate Turing-complete systems.

Decidability of nested recurrence relations is generally impossible.

Undecidability depends on initial conditions and the structure of the recurrence.

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.