Numeration systems, linear recurrences, and regular sets (1994)
| Venue: | Inform. and Comput |
| Citations: | 27 - 1 self |
BibTeX
@ARTICLE{Shallit94numerationsystems,,
author = {Jeffrey Shallit},
title = {Numeration systems, linear recurrences, and regular sets},
journal = {Inform. and Comput},
year = {1994},
volume = {113},
pages = {331--347}
}
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Abstract
A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a non-negative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is order-preserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an order-preserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1
