Factorization in Formal Languages

  • Paul C. Bell
  • Daniel Reidenbach
  • Jeffrey ShallitEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9168)


We consider several language-theoretic 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 some 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”. Although all these variations have been considered before, it appears that the languages of words having these properties have not been positioned in the Chomsky hierarchy up to now. We also consider the length of the shortest word not having the desired property.


Characteristic Vector Formal Language Regular Language Unique Factorization Distinct Factorization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Paul C. Bell
    • 1
  • Daniel Reidenbach
    • 1
  • Jeffrey Shallit
    • 2
    Email author
  1. 1.Department of Computer ScienceLoughborough UniversityLoughboroughUK
  2. 2. School of Computer ScienceUniversity of WaterlooWaterlooCanada

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