Computing Least and Greatest Fixed Points in Absorptive Semirings

TL;DR

This paper introduces efficient algorithms for computing least and greatest fixed points of polynomial systems over absorptive semirings, with applications to structures like the tropical semiring, using fixed-point iteration and symbolic methods.

Contribution

It provides the first closed-form solutions and two efficient algorithms for fixed points in absorptive semirings, expanding computational tools in algebraic structures.

Findings

Fixed-point iteration collapses to linear steps.

Algorithms require polynomially many semiring operations.

Applicable to tropical and other absorptive semirings.

Abstract

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.