## Works by Denis Richard

5 found
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1. Undecidable extensions of Skolem arithmetic.Alexis Bès & Denis Richard - 1998 - Journal of Symbolic Logic 63 (2):379-401.
Let $be the restriction of usual order relation to integers which are primes or squares of primes, and let ⊥ denote the coprimeness predicate. The elementary theory of$\langle\mathbb{N};\bot, , is undecidable. Now denote by $the restriction of order to primary numbers. All arithmetical relations restricted to primary numbers are definable in the structure$\langle\mathbb{N};\bot, . Furthermore, the structures $\langle\mathbb{N};\mid, and$\langle\mathbb{N};=,+,x\rangle$are interdefinable. Direct download (8 more) Export citation Bookmark 2 citations 2. Definability and decidability issues in extensions of the integers with the divisibility predicate.Patrick Cegielski, Yuri Matiyasevich & Denis Richard - 1996 - Journal of Symbolic Logic 61 (2):515-540. Let M be a first-order structure; we denote by DEF(M) the set of all first-order definable relations and functions within M. Let π be any one-to-one function from N into the set of prime integers. Let ∣ and$\bullet$be respectively the divisibility relation and multiplication as function. We show that the sets DEF(N,π,∣) and$\mathrm{DEF}(\mathbb{N},\pi,\bullet)$are equal. However there exists function π such that the set DEF(N,π,∣), or, equivalently,$\mathrm{DEF}(\mathbb{N},\pi,\bullet)$is not equal to$\mathrm{DEF}(\mathbb{N},+,\bullet)$. Nevertheless, in all cases (...) Direct download (8 more) Export citation Bookmark 1 citation 3. Preface.Patrick Cegielski, Leszek Pacholski, Denis Richard, Jerzy Tomasik & Alex Wilkie - 1997 - Annals of Pure and Applied Logic 89 (1):1. Direct download (3 more) Export citation Bookmark 4. Answer to a problem raised by J. Robinson: The arithmetic of positive or negative integers is definable from successor and divisibility.Denis Richard - 1985 - Journal of Symbolic Logic 50 (4):927-935. In this paper we give a positive answer to Julia Robinson's question whether the definability of + and · from S and ∣ that she proved in the case of positive integers is extendible to arbitrary integers (cf. [JR, p. 102]). Direct download (8 more) Export citation Bookmark 5. Definability in terms of the successor function and the coprimeness predicate in the set of arbitrary integers.Denis Richard - 1989 - Journal of Symbolic Logic 54 (4):1253-1287. Using coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate$\perp$a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are$\{S, \perp\}\$ -definable over Z. Applications to definability over Z and N (...)