The Continuous Skolem-Pisot Problem: On the Complexity of Reachability for Linear Ordinary Differential Equations

TL;DR

This paper investigates the complexity and decidability of the continuous Skolem-Pisot problem, focusing on zeros and nonnegativity of solutions to linear differential equations, revealing NP-hardness and decidability in specific cases.

Contribution

It introduces the continuous Skolem-Pisot problem, analyzes its complexity, and identifies cases where zero presence is decidable, advancing understanding of reachability in linear ODEs.

Findings

Nonnegativity problem is NP-hard in general.

Zero presence is decidable for depth-two or less cases.

Decidability remains open for the general case.

Abstract

We study decidability and complexity questions related to a continuous analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of a linear recurrent sequence. In particular, we show that the continuous version of the nonnegativity problem is NP-hard in general and we show that the presence of a zero is decidable for several subcases, including instances of depth two or less, although the decidability in general is left open. The problems may also be stated as reachability problems related to real zeros of exponential polynomials or solutions to initial value problems of linear differential equations, which are interesting problems in their own right.