Munchausen Iteration

TL;DR

The paper introduces Munchausen iteration, a novel method for solving polynomial equations over idempotent omega-continuous semirings by iteratively refining solutions through substitution, avoiding direct computation.

Contribution

It presents a new iterative approach that uses function substitution and linear completions, connecting it to Newton iteration and automata theory techniques.

Findings

Evaluation of the n-th approximant matches the 2^n-th Newton approximant

Linear completions can be computed with automata theory methods

The method improves solution precision through substitution rather than direct calculation

Abstract

We present a method for solving polynomial equations over idempotent omega-continuous semirings. The idea is to iterate over the semiring of functions rather than the semiring of interest, and only evaluate when needed. The key operation is substitution. In the initial step, we compute a linear completion of the system of equations that exhaustively inserts the equations into one another. With functions as approximants, the following steps insert the current approximant into itself. Since the iteration improves its precision by substitution rather than computation we named it Munchausen, after the fictional baron that pulled himself out of a swamp by his own hair. The first result shows that an evaluation of the n-th Munchausen approximant coincides with the 2^n-th Newton approximant. Second, we show how to compute linear completions with standard techniques from automata theory. In particular, we are not bound to (but can use) the notion of differentials prominent in Newton iteration.