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PosSLP and Sum of Squares

Markus Bläser, Julian Dörfler, Gorav Jindal

TL;DR

The paper investigates decision problems for constant-free, division-free straight-line programs (SLPs), focusing on PosSLP and its connections to representations of integers and polynomials as sums of squares. It introduces natural extensions (3SoSSLP, 2SoSSLP) and a new Div2SLP problem, establishing a network of reductions that relate PosSLP to these variants and to DegSLP/OrdSLP. The authors prove both unconditional and conditional hardness results (e.g., EquSLP ≤ 3SoSSLP and EquSLP ≤ 2SoSSLP) and place 3SoSSLP in P with access to Div2SLP and PosSLP oracles; they also analyze PosPolySLP (coNP-hard) and SqPolySLP (in coRP), linking square- and positivity-related questions for polynomials to SLP complexity. These results deepen understanding of SLP-based decision problems, connect number-theoretic square-sum representations to computational hardness, and outline important open questions about Div2SLP and unconditional hardness of PosSLP.

Abstract

The problem PosSLP is the problem of determining whether a given straight-line program (SLP) computes a positive integer. PosSLP was introduced by Allender et al. to study the complexity of numerical analysis (Allender et al., 2009). PosSLP can also be reformulated as the problem of deciding whether the integer computed by a given SLP can be expressed as the sum of squares of four integers, based on the well-known result by Lagrange in 1770, which demonstrated that every natural number can be represented as the sum of four non-negative integer squares. In this paper, we explore several natural extensions of this problem by investigating whether the positive integer computed by a given SLP can be written as the sum of squares of two or three integers. We delve into the complexity of these variations and demonstrate relations between the complexity of the original PosSLP problem and the complexity of these related problems. Additionally, we introduce a new intriguing problem called Div2SLP and illustrate how Div2SLP is connected to DegSLP and the problem of whether an SLP computes an integer expressible as the sum of three squares. By comprehending the connections between these problems, our results offer a deeper understanding of decision problems associated with SLPs and open avenues for further exciting research

PosSLP and Sum of Squares

TL;DR

The paper investigates decision problems for constant-free, division-free straight-line programs (SLPs), focusing on PosSLP and its connections to representations of integers and polynomials as sums of squares. It introduces natural extensions (3SoSSLP, 2SoSSLP) and a new Div2SLP problem, establishing a network of reductions that relate PosSLP to these variants and to DegSLP/OrdSLP. The authors prove both unconditional and conditional hardness results (e.g., EquSLP ≤ 3SoSSLP and EquSLP ≤ 2SoSSLP) and place 3SoSSLP in P with access to Div2SLP and PosSLP oracles; they also analyze PosPolySLP (coNP-hard) and SqPolySLP (in coRP), linking square- and positivity-related questions for polynomials to SLP complexity. These results deepen understanding of SLP-based decision problems, connect number-theoretic square-sum representations to computational hardness, and outline important open questions about Div2SLP and unconditional hardness of PosSLP.

Abstract

The problem PosSLP is the problem of determining whether a given straight-line program (SLP) computes a positive integer. PosSLP was introduced by Allender et al. to study the complexity of numerical analysis (Allender et al., 2009). PosSLP can also be reformulated as the problem of deciding whether the integer computed by a given SLP can be expressed as the sum of squares of four integers, based on the well-known result by Lagrange in 1770, which demonstrated that every natural number can be represented as the sum of four non-negative integer squares. In this paper, we explore several natural extensions of this problem by investigating whether the positive integer computed by a given SLP can be written as the sum of squares of two or three integers. We delve into the complexity of these variations and demonstrate relations between the complexity of the original PosSLP problem and the complexity of these related problems. Additionally, we introduce a new intriguing problem called Div2SLP and illustrate how Div2SLP is connected to DegSLP and the problem of whether an SLP computes an integer expressible as the sum of three squares. By comprehending the connections between these problems, our results offer a deeper understanding of decision problems associated with SLPs and open avenues for further exciting research
Paper Structure (18 sections, 30 theorems, 25 equations, 1 figure)

This paper contains 18 sections, 30 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Straight Line Programs and PosSLP
  3. How Hard is $\operatorname{PosSLP}$?
  4. Our Results
  5. SLPs as Sums of Three Squares
  6. Lower Bound for $\operatorname{3SoSSLP}$
  7. Upper Bound for $\operatorname{3SoSSLP}$
  8. Complexity of $\operatorname{Div2SLP}$
  9. SLPs as Sum of Two and Fewer Squares
  10. Lower Bounds for $\operatorname{2SoSSLP}$
  11. Polynomials as Sum of Squares
  12. Positivity of Polynomials
  13. Checking if a Polynomial is a Square
  14. Conclusion and Open Problems
  15. Acknowledgments:
  16. ...and 3 more sections

Key Result

Theorem 1.1

We have $\P^{\operatorname{PosSLP}} = \operatorname{BP}(\P_{\mathbb{R}}^{0})$.

Figures (1)

  • Figure 1: A visualization of the relations between the problems studied in this work. An arrow means that there is a Turing reduction. A thicker arrow indicates a polynomial time many one reduction. The reduction from $\operatorname{PosSLP}$ to $\operatorname{2SoSSLP}$ is nondeterministic and depends on \ref{['conj:gencramconj']}.

Theorems & Definitions (51)

  • Theorem 1.1: Proposition 1.1 in Allender06onthe
  • Theorem 1.2: Proposition 1.2 in Allender06onthe
  • Theorem 1.3: Theorem 1.2 in burgisser2023hardness
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 2.1: Legendre1797gauss1801disquisitionesAnkeny1957SumsOTmordell1958representation
  • Lemma 2.1
  • ...and 41 more