Decision problems for linear recurrences involving arbitrary real numbers

TL;DR

This paper investigates the decidability of key problems for linear recurrences with real coefficients and initial values, showing partial algorithms that halt in many cases and highlighting differences from algebraic coefficient cases.

Contribution

It introduces partial algorithms for the Skolem, Positivity, and Ultimate Positivity problems that halt in locally constant instances, advancing understanding of their decidability in the real number setting.

Findings

Partial algorithms halt for locally constant instances.

Algorithms for Positivity and Ultimate Positivity halt on almost all instances.

Decidability is limited compared to algebraic coefficient cases.

Abstract

We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show that for each problem there exists a correct partial algorithm which halts for all problem instances for which the answer is locally constant, thus establishing that all three problems are as close to decidable as one can expect them to be in this setting. We further show that the algorithms for the Positivity Problem and the Ultimate Positivity Problem halt on almost every instance with respect to the usual Lebesgue measure on Euclidean space. In comparison, the analogous problems for exact rational or real algebraic coefficients are known to be decidable only for linear recurrences of fairly low order.