In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language - representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication - it is possible to decide precisely whether a program has certain growth-rate properties, in particular whether a computed value, or the program's running time, has a polynomial growth rate. A natural and intriguing problem was to move from answering the decision problem to giving a quantitative result, namely, a tight polynomial upper bound. This paper shows how to obtain asymptotically-tight, multivariate, disjunctive polynomial bounds for this class of programs. This is a complete solution: whenever a polynomial bound exists it will be found. A pleasant surprise is that the algorithm is quite simple; but it relies on some subtle reasoning. An important ingredient in the proof is the forest factorization theorem, a strong structural result on homomorphisms into a finite monoid.

中文翻译：

---

循环程序的紧多项式最坏情况界限

2008 年，Ben-Amram、Jones 和 Kristiansen 表明，对于一种简单的编程语言——用有界循环表示非确定性命令式程序，以及仅限于加法和乘法的算术——可以精确地确定程序是否具有一定的增长率属性，特别是计算值或程序的运行时间是否具有多项式增长率。一个自然而有趣的问题是从回答决策问题转向给出定量结果，即紧多项式上限。本文展示了如何获得此类程序的渐近紧、多元、析取多项式边界。这是一个完整的解决方案：只要存在多项式边界，就会找到它。令人惊喜的是，算法非常简单；但它依赖于一些微妙的推理。证明中的一个重要组成部分是森林分解定理，这是将同态转化为有限幺半群的强结构结果。