Unambiguous Forest Factorization

TL;DR

This paper introduces an unambiguous automaton construction for Simon's forest factorization theorem, simplifying proofs and enabling applications in automata theory and transducer synthesis.

Contribution

It constructs a universal unambiguous automaton for a given morphism, providing a simpler proof of the forest factorization theorem and applications in regular transducer expressions.

Findings

Automaton A is universal and unambiguous for the morphism 

Provides a Ramsey split with height bounded by automaton states

Enables construction of regular transducer expressions from automata

Abstract

In this paper, we look at an unambiguous version of Simon's forest factorization theorem, a very deep result which has wide connections in algebra, logic and automata. Given a morphism $\varphi$ from $\Sigma^+$ to a finite semigroup $S$, we construct a universal, unambiguous automaton A which is "good" for $\varphi$. The goodness of $\Aa$ gives a very easy proof for the forest factorization theorem, providing a Ramsey split for any word in $\Sigma^{\infty}$ such that the height of the Ramsey split is bounded by the number of states of A. An important application of synthesizing good automata from the morphim $\varphi$ is in the construction of regular transducer expressions (RTE) corresponding to deterministic two way transducers.