Analyzing PosSLP and Sum of Squares Complexity

Core Concepts

The author explores the complexity of PosSLP variants, connecting them to problems related to sums of squares. By delving into these variations, a deeper understanding of decision problems associated with SLPs is achieved.

Abstract

The content discusses the problem PosSLP and its connection to determining whether an integer can be represented as the sum of squares. It explores various extensions of this problem, introduces new intriguing problems like Div2SLP, and establishes relationships between different complexities. The paper also highlights the coNP-hardness of the polynomial variant of PosSLP and opens avenues for further research in this area. The study analyzes the connections between different problems related to straight-line programs (SLPs) and provides insights into their computational complexity. It delves into various variants of the original problem, introducing new challenges and establishing links between them. The content emphasizes the importance of understanding decision problems associated with SLPs and offers directions for future research.

Stats

Given a straight-line program representing N ∈ Z, decide whether N > 0. For every n ∈ N, at least one element in the set {n,n+2} is 3SoS. If a constructive variant of the radical conjecture is true and PosSLP ∈ BPP then NP ⊆ BPP. EquSLP reduces to both 3SoSSLP and 2SoSSLP. GTNC is polynomial-time Turing equivalent to PosSLP.

Quotes

"Given a straight-line program representing N ∈ Z, decide if ∃g ∈ Z[x] such that f = g^2." - Problem 4.2 (SqPolySLP) "We showed it is DegSLP hard." - Open Problem on Div2SLP complexity

Key Insights Distilled From

PosSLP and Sum of Squares

by Mark... at arxiv.org 03-04-2024

https://arxiv.org/pdf/2403.00115.pdf

Deeper Inquiries

What is the exact complexity of Div2SLP?

The exact complexity of Div2SLP is in the counting hierarchy CH. This means that it falls within a specific level of the counting hierarchy, indicating its computational complexity in relation to other problems.

Can we prove Theorem 1.5 without relying on Conjecture 3.1?

It is currently challenging to prove Theorem 1.5 without relying on Conjecture 3.1. The theorem's dependency on Conjecture 3.1 suggests that there may not be a straightforward way to establish this result unconditionally at present.

How does Div2SLP relate to PosSLP?

Div2SLP and PosSLP are related through their connections with decision problems associated with straight-line programs (SLPs). In particular, Div2SLP involves determining if an integer computed by an SLP can be expressed as divisible by a given power of two, while PosSLP focuses on deciding whether a given SLP computes a positive integer or not. These problems share underlying concepts related to arithmetic circuits and polynomial computations but address different aspects such as divisibility and positivity in their respective contexts within computational complexity theory.