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

翻译：PosSLP问题旨在判断给定直线程序（SLP）是否计算出一个正整数。该问题由Allender等人引入，用于研究数值分析的复杂性（Allender等，2009）。基于拉格朗日于1770年提出的著名定理（该定理表明每个自然数均可表示为四个非负整数的平方和），PosSLP亦可重新表述为：判断给定SLP所计算的整数能否表示为四个整数的平方和。本文通过探究给定SLP计算的整数能否表示为两个或三个整数的平方和，研究了该问题的若干自然延伸。我们深入分析了这些变体的复杂性，并论证了原始PosSLP问题的复杂性与这些相关问题复杂性之间的关联。此外，我们引入了一个名为Div2SLP的新颖问题，阐释了Div2SLP与DegSLP以及"SLP是否计算出一个可表示为三个平方和的整数"这一问题之间的联系。通过理解这些问题间的关联，我们的研究结果深化了对SLP相关判定问题的认识，并为未来进一步研究开辟了新途径。