吸收半环中最小和最大不动点的计算
Computing Least and Greatest Fixed Points in Absorptive Semirings
摘要：
我们提出了两种方法来算法计算最小和最大的解决方案的多项式方程系统的吸收半环（具有一定的完整性和连续性的假设），如热带半环。这两种方法都需要多项式数量的半环运算，包括半环加法、乘法和无穷幂运算。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解。证明建立在（可能是无限的）派生树的概念上;仔细分析这些树的形状，我们可以将不动点迭代压缩为线性步数。第二种方法是广义吸收多项式半环上的迭代符号计算，主要基于Kleene代数上的结果。
英文摘要：
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 generalized absorptive polynomials, largely based on results on Kleene algebras.
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