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Computing Least and Greatest Fixed Points in Absorptive Semirings
arXiv - CS - Computational Complexity Pub Date : 2021-06-01 , DOI: arxiv-2106.00399
Matthias Naaf

We present two methods to algorithmically compute both least and greatest solutions of polynomial equation systems over absorptive semirings (with certain completeness and continuity assumptions), such as the tropical semiring. Both methods require a polynomial number of semiring operations, including semiring addition, multiplication and an infinitary power operation. Our main result is a closed-form solution for least and greatest fixed points based on the fixed-point iteration. The proof builds on the notion of (possibly infinite) derivation trees; a careful analysis of the shape of these trees allows us to collapse the fixed-point iteration to a linear number of steps. The second method is an iterative symbolic computation in the semiring of absorptive polynomials, largely based on results on Kleene algebras.

中文翻译:

计算吸收半环中的最小和最大不动点

我们提出了两种方法来算法计算多项式方程组在吸收半环(具有一定的完整性和连续性假设)上的最小解和最大解,例如热带半环。这两种方法都需要多项式的半环运算,包括半环加法、乘法和无穷次幂运算。我们的主要结果是基于不动点迭代的最小和最大不动点的封闭形式解决方案。证明建立在(可能是无限的)派生树的概念之上;仔细分析这些树的形状使我们能够将定点迭代折叠为线性步数。第二种方法是吸收多项式半环中的迭代符号计算,主要基于 Kleene 代数的结果。
更新日期:2021-06-02
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