Abstract
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 t≤k+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.
This research is supported in part by the Natural Sciences and Engineering Research Council of Canada grants OGP0041630 and OGP0046373
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© 1992 Springer-Verlag Berlin Heidelberg
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Szilard, A., Yu, S., Zhang, K., Shallit, J. (1992). Characterizing regular languages with polynomial densities. In: Havel, I.M., Koubek, V. (eds) Mathematical Foundations of Computer Science 1992. MFCS 1992. Lecture Notes in Computer Science, vol 629. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-55808-X_48
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DOI: https://doi.org/10.1007/3-540-55808-X_48
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