On the base-dependence of sets of numbers recognizable by finite automata

Cobham, Alan

doi:10.1007/bf01746527

Cited by 231 publications

(158 citation statements)

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“…A finite automata approach to Presburger arithmetic was pioneered in [12,15], and continues to be an active area of research (see, for example, [10,16,34,55]). This approach is quite different from the present paper's, but it can attack similar questions: for example, see [39] for results on counting solutions to Presburger formulas (non-parametrically).…”

Section: Property 2 Gives a Nice Geometric Characterization Of Presbumentioning

confidence: 99%

Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

Woods

2013

Automata, Languages, and Programming

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Abstract. Presburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p = (p1, . . . , pn) are a subset of the free variables in a Presburger formula, we can define a counting function g(p) to be the number of solutions to the formula, for a given p. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

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Section: Property 2 Gives a Nice Geometric Characterization Of Presbumentioning

confidence: 99%

Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

Woods

2013

Automata, Languages, and Programming

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“…Recall that two integers k and l larger than 1 are multiplicatively independent if log(k)/ log(l) ∈ Q. Then Cobham's theorem says that only very well-behaved sets of integers can be automatic with respect to two multiplicatively independent numbers [9]. In 1969, Cobham proved the following result.…”

Section: Definition 12 a Compact Set X ⊆ [0 1]mentioning

confidence: 99%

An analogue of Cobham’s theorem for fractals

Adamczewski

Bell

2011

Trans. Amer. Math. Soc.

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Abstract. We introduce the notion of k-self-similarity for compact subsets of R n and show that it is a natural analogue of the notion of k-automatic subsets of integers. We show that various well-known fractals such as the triadic Cantor set, the Sierpiński carpet or the Menger sponge turn out to be k-selfsimilar for some integers k. We then prove an analogue of Cobham's theorem for compact sets of R that are self-similar with respect to two multiplicatively independent bases k and . Namely, we show that X is both a k-and an -self-similar compact subset of R if and only if it is a finite union of closed intervals with rational endpoints.

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“…This is a consequence of the following result in [Cob69] where it is stated in terms of automatic sequences. Here we rephrase it in terms of algebraicity over finite fields.…”

Section: Preserving Algebraicitymentioning

confidence: 76%