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.

DOI: 10.1007/978-3-642-30870-3_12

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Cite this paper

@inproceedings{Celaya2012AnUN, title={An Undecidable Nested Recurrence Relation}, author={Marcel Celaya and Frank Ruskey}, booktitle={CiE}, year={2012} }