On the decomposition of k-valued rational relations

TL;DR

This paper presents a new, more understandable proof for decomposing k-valued rational relations into unambiguous functional components, improving complexity and extending to relations with bounded length-degree.

Contribution

It introduces a simplified, more efficient construction for decomposition and generalizes the method to handle relations with bounded length-degree.

Findings

Reduces the complexity of decomposition by one exponential

Provides a constructive method based on lexicographic ordering

Extends decomposition to rational relations with bounded length-degree

Abstract

We give a new, and hopefully more easily understandable, structural proof of the decomposition of a $k$-valued transducer into $k$ unambiguous functional ones, a result established by A. Weber in 1996. Our construction is based on a lexicographic ordering of computations of automata and on two coverings that can be build by means of this ordering. The complexity of the construction, measured as the number of states of the transducers involved in the decomposition, improves the original one by one exponential. Moreover, this method allows further generalisation that solves the problem of decomposition of rational relations with bounded length-degree, which was left open in Weber's paper.