Factorization in Formal Languages

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". X Paul Bell, Daniel Reidenbach, and Jeffrey Shallit Proof. We just use the example of Proposition 5.⊓ ⊔ Theorem 20. If L is regular then ufs(L) need not be a CFL.Proof. We use a variation of the construction in the proof of Theorem 16. Let L = (ab) + (ac) + aa + (ba) + (ca) + + aa + aaa. Then (using the notation in the proof of Theorem 16), if w := aa(ab) r (ac) s aa(ba) t (ca) q aaa ∈ ufs(L) ∩ R with r, s, t, q ≥ 1 then there are two different factorizations of w:aa which are subset-invariant if and only if r = t and s = q. So ufs(L) ∩ R = {aa(ab) r (ac) s aa(ba) r (ca) s aaa : r, s ≥ 1}, which is not a CFL. ⊓ ⊔ AcknowledgmentThe idea of considering semi-unique factorization was inspired by a talk of Nasir Sohail at the University of Waterloo in April 2014.

Unique decipherability in formal languages

“…A preliminary version of this paper appeared in [1]. In the present version, we improve the upper bound on the maximal length of a minimal word having subset-invariant factorization, if any such word exists, from O(m 2 n 2 ) to O(m 2 n) and study various complexity issues in Section 7.…”