Factorization in Formal Languages

TL;DR

This paper explores unique and semi-unique factorization properties in formal languages, establishing regularity results, bounds, and demonstrating limitations for context-free languages, with various novel factorization notions.

Contribution

It introduces new concepts of semi-unique and permutation-based factorizations, providing bounds and counterexamples for regular and context-free languages.

Findings

uf(L) is regular if L is regular

Bounds on shortest words outside uf(L)

uf(L) need not be context-free for context-free L

Abstract

We consider several novel aspects of unique factorization in formal languages. We reprove the familiar fact that the set uf(L) of words having unique factorization into elements of L is regular if L is regular, and from this deduce an quadratic upper and lower bound on the length of the shortest word not in uf(L). We observe that uf(L) need not be context-free if L is context-free. Next, we consider variations on unique factorization. We define a notion of "semi-unique" factorization, where every factorization has the same number of terms, and show that, if L is regular or even finite, the set of words having such a factorization need not be context-free. Finally, we consider additional variations, such as unique factorization "up to permutation" and "up to subset".