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2018 Ostrowski Numeration Systems, Addition, and Finite Automata
Philipp Hieronymi, Alonza Terry Jr.
Notre Dame J. Formal Logic 59(2): 215-232 (2018). DOI: 10.1215/00294527-2017-0027


We present an elementary three-pass algorithm for computing addition in Ostrowski numeration systems. When a is quadratic, addition in the Ostrowski numeration system based on a is recognizable by a finite automaton. We deduce that a subset of XNn is definable in (N,+,Va), where Va is the function that maps a natural number x to the smallest denominator of a convergent of a that appears in the Ostrowski representation based on a of x with a nonzero coefficient if and only if the set of Ostrowski representations of elements of X is recognizable by a finite automaton. The decidability of the theory of (N,+,Va) follows.


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Philipp Hieronymi. Alonza Terry Jr.. "Ostrowski Numeration Systems, Addition, and Finite Automata." Notre Dame J. Formal Logic 59 (2) 215 - 232, 2018.


Received: 31 July 2014; Accepted: 8 October 2015; Published: 2018
First available in Project Euclid: 23 December 2017

zbMATH: 06870290
MathSciNet: MR3778309
Digital Object Identifier: 10.1215/00294527-2017-0027

Primary: 11A67
Secondary: 03B25 , 68Q25 , 68R15

Keywords: addition , Continued fraction , decidability , expansion of Presburger arithmetic , finite automata , Ostrowski numeration system

Rights: Copyright © 2018 University of Notre Dame


Vol.59 • No. 2 • 2018
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