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Definability and decidability issues in extensions of the integers with the divisibility predicate

Published online by Cambridge University Press:  12 March 2014

Patrick Cegielski ,
Yuri Matiyasevich  and
Denis Richard
Patrick Cegielski
Affiliation:
L.I.T.P., I.B.P., U.R.A. 248, Université Paris XII, I.U.T de Fontainebleau, Route Hurtault, F-77300 Fontainebleau, France, E-mail: cep@litp.ibp.fr
Yuri Matiyasevich
Affiliation:
Steklov Institute of Mathematics, 27 Fontanka, Saint-Petersbourg, 191011, Russie, E-mail: yumat@pomi.ras.ru
Denis Richard
Affiliation:
Laboratoire de Logique, Algorithmique, et Informatique de Clermont i (LLAIC 1), I.U.T. Informatique, B.P. 86, F-63172 Aubière Cedex, France, E-mail: richard@llaic.univ-bpclermont.fr
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Abstract

Let be a first-order structure; we denote by DEF() the set of all first-order definable relations and functions within . Let π be any one-to-one function from ℕ into the set of prime integers.

Let ∣ and • be respectively the divisibility relation and multiplication as function. We show that the sets DEF(ℕ, π, ∣) and DEF(ℕ, π, •) are equal. However there exists function π such that the set DEF(ℕ, +, ∣), or, equivalently, DEF(ℕ, π, •) is not equal to DEF(ℕ, +, •). Nevertheless, in all cases there is an {π, •}-definable and hence also {π, |}-definable structure over π which is isomorphic to 〈ℕ, +, •〉. Hence theories TH(ℕ, π, ∣) and TH(ℕ, π, •) are undecidable.

The binary relation of equipotence between two positive integers saying that they have equal number of prime divisors is not definable within the divisibility lattice over positive integers. We prove it first by comparing the lower bound of the computational complexity of the additive theory of positive integers and of the upper bound of the computational complexity of the theory of the mentioned lattice.

The last section provides a self-contained alternative proof of this latter result based on a decision method linked to an elimination of quantifiers via specific tables.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 2 , June 1996 , pp. 515 - 540
Copyright
Copyright © Association for Symbolic Logic 1996

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