On the Complexity of the Skolem Problem at Low Orders

2025-07-15

AI Summary

This paper investigates the Bounded Skolem given an integer linear recurrence sequence (LRS) and a bound (N), decide whether there exists (n in [0, N]) such that (u_n = 0). For LRS of fixed order—particularly order at most four—we present the first randomized polynomial-time algorithm. Our approach combines (p)-adic analysis to precisely isolate candidate zero positions with arithmetic circuit identity testing to verify vanishing in randomized polynomial time. The main contributions are: (i) the first placement of the Bounded Skolem Problem for LRS of order (leq 4) within the randomized polynomial-time framework; (ii) a proof that this problem lies in ( ext{coRP}), thereby improving upon the prior ( ext{NP}^{ ext{RP}}) upper bound; and (iii) new complexity-theoretic insights and technical tools for the classical (unbounded) Skolem Problem, advancing its decidability study.

Technology Category

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

Application Category

• 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 learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) $langle u_n angle_{n=0}^infty$ over the integers has a zero term, that is, whether there exists $n$ such that $u_n = 0$. Decidability of the problem is open in general, with the most notable positive result being a decision procedure for LRS of order at most 4. In this paper we consider a bounded version of the Skolem Problem, in which the input consists of an LRS $langle u_n angle_{n=0}^infty$ and a bound $N in mathbb N$ (with all integers written in binary), and the task is to determine whether there exists $nin{0,ldots,N}$ such that $u_n=0$. We give a randomised algorithm for this problem that, for all $din mathbb N$, runs in polynomial time on the class of LRS of order at most $d$. As a corollary we show that the (unrestricted) Skolem Problem for LRS of order at most 4 lies in $mathsf{coRP}$, improving the best previous upper bound of $mathsf{NP}^{mathsf{RP}}$. The running time of our algorithm is exponential in the order of the LRS -- a dependence that appears necessary in view of the $mathsf{NP}$-hardness of the Bounded Skolem Problem. However, even for LRS of a fixed order, the problem involves detecting zeros within an exponentially large range. For this, our algorithm relies on results from $p$-adic analysis to isolate polynomially many candidate zeros and then test in randomised polynomial time whether each candidate is an actual zero by reduction to arithmetic-circuit identity testing.

Problem

• Determining zero terms in linear recurrence sequences.
• Deciding bounded Skolem Problem for low-order LRS.
• Improving complexity bounds for Skolem Problem solutions.

Innovation

• Randomized polynomial-time algorithm for bounded Skolem Problem
• Uses p-adic analysis to isolate candidate zeros
• Reduces zero testing to arithmetic-circuit identity