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On Decidable Extensions of Presburger Arithmetic: From A. Bertrand Numeration Systems to Pisot Numbers
Journal of Symbolic Logic 65 (3):1347-1374 (2000)
Abstract
We study extensions of Presburger arithmetic with a unary predicate R and we show that under certain conditions on R, R is sparse and the theory of $\langle\mathbb{N}, +, R\rangle$ is decidable. We axiomatize this theory and we show that in a reasonable language, it admits quantifier elimination. We obtain similar results for the structure $\langle\mathbb{Q},+, R\rangle$.Links
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Citations of this work
Vapnik–Chervonenkis Density in Some Theories without the Independence Property, II.Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson & Sergei Starchenko - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):311-363.
On expansions of.Quentin Lambotte & Françoise Point - 2020 - Annals of Pure and Applied Logic 171 (8):102809.
Elimination of unbounded quantifiers for some poly-regular groups of infinite rank.Philip Scowcroft - 2007 - Annals of Pure and Applied Logic 149 (1-3):40-80.
References found in this work
Weak Second‐Order Arithmetic and Finite Automata.J. Richard Büchi - 1960 - Mathematical Logic Quarterly 6 (1-6):66-92.
Presburger arithmetic and recognizability of sets of natural numbers by automata: New proofs of Cobham's and Semenov's theorems.Christian Michaux & Roger Villemaire - 1996 - Annals of Pure and Applied Logic 77 (3):251-277.
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