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PosSLP and Sum of Squares

Authors Markus Bläser , Julian Dörfler , Gorav Jindal

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Author Details

Markus Bläser
  • Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Julian Dörfler
  • Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Gorav Jindal
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany

Acknowledgements

We would like to thank Robert Andrews for providing a simpler proof of Lemma 2.6. We had a proof of it that was a bit longer. Robert Andrews simplified the proof after reviewing our proof in a personal communication.

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Markus Bläser, Julian Dörfler, and Gorav Jindal. PosSLP and Sum of Squares. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.13

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Problems, reductions and completeness
Keywords
  • PosSLP
  • Straight-line program
  • Polynomial identity testing
  • Sum of squares

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