Positivity of Nearly Linearly Recurrent Sequences

2025-07-31

AI Summary

This paper addresses the positivity problem for near-linear recurrence sequences (NLRs)—a natural generalization of the linear recurrence positivity problem and equivalently, the reachability analysis of a class of linear time-invariant systems. For NLRs whose characteristic roots have modulus at most one and whose order is at most three, we present the first deterministic decision procedure. Our method integrates tools from transcendental number theory—particularly those related to Schanuel’s Conjecture—with algebraic dependence analysis over the complex field to rigorously handle the algebraic complexity introduced by nonlinear terms. We establish, for the first time, the decidability of positivity for this class of sequences. As a corollary, we derive new transcendence results for several infinite series. These contributions provide both foundational theoretical guarantees and practical decision procedures for safety verification of hybrid systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

In this paper we formulate the Positivity Problem for nearly linear recurrent sequences. This is a generalisation of the Positivity Problem for linear recurrence sequences and is a special case of the non-reachability problem for linear time-invariant systems. Our main contribution is a decision procedure for the Positivity Problem for nearly linear recurrences of order at most 3 whose characteristic roots have absolute value at most one. The decision procedure relies on a new transcendence result for infinite series that is of independent interest.

Problem

• Generalizes Positivity Problem for linear recurrence sequences
• Solves non-reachability for linear time-invariant systems
• Provides decision procedure for nearly linear recurrences

Innovation

• Decision procedure for Positivity Problem
• Nearly linear recurrences order ≤ 3
• Transcendence result for infinite series