Characterizing regular languages with polynomial densities

  • Andrew Szilard
  • Sheng Yu
  • Kaizhong Zhang
  • Jeffrey Shallit
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 629)


A language L is said to have a polynomial density if the function pL.(n)=¦L∩∑n¦ of L is bounded by a polynomial. We show that the function p R(n) of a regular language R is O(n k ), for some k≥0, if and only if R can be represented as a finite union of the regular expressions of the form xy 1 * z1 ...y t * zt with a nonnegative integer tk+1, where x,y 1,z 1,..., yt, zt are all strings in ∑*.

We prove a characterization for the (restricted) starheight-one languages. We show that a regular language is starheight one if and only if it is the image of a regular language of polynomial density under a finite substitution. We also show that the set of starheight-one languages includes all the regular languages with polynomial densities and their complements.


Regular languages population functions of languages density functions polynomial densities starheight-one languages 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Andrew Szilard
    • 1
  • Sheng Yu
    • 1
  • Kaizhong Zhang
    • 1
  • Jeffrey Shallit
    • 2
  1. 1.Department of Computer ScienceThe University of Western OntarioLondonCanada
  2. 2.Department of Computer ScienceUniversity of WaterlooWaterlooCanada

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