Home

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

## 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.

## References

Hide All
[1]Balcázar, José Luis, Díaz, Josep, and Gabarró, Joaquim, Structural complexity, Springer-Verlag, Berlin, 1988.
[2]Beth, Evert Willem, On Padoa's method in the theory of definition, Koningklijke Nederlandse Akademie van wetenschappen, Proceedings of the section of sciences, series A, Mathematical sciences, no. 56, 1953, pp. 330339; also Indagationes mathematicae, vol. 15, pp. 330–339.
[3]Cegielski, Patrick, La théorie élémentaire de la multiplication, Model theory and arithmetic (Berline, , McAloon, , Ressayre, , editors), Lecture Notes in Mathematics, Number 890, Springer-Verlag, pp. 4489.
[4]Cegielski, Patrick, La théorie élémentaire de la divisibilité est finiment axiomatisable, Comptes Rendus de l'Académie des Sciences, Paris t. 299, vol. I (1984), pp. 367369.
[5]Cegielski, Patrick, The Elementary Theory of the natural Lattice is Finitely Axiomatizable, Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 138150.
[6]Cegielski, Patrick, Definability, decidability and complexity, to appear in Mathematics and Computer Science, (Nivat, Maurice and Grigorieff, Serge, editors), Annals of Mathematics on Artificial Intelligence, Baltzer, Suisse, 1996.
[7]Cegielski, Patrick, and Richard, Denis, Indécidabilité de la théorie des entiers naturels munis d'une énumération des premiers et de la divisibilité, Comptes Rendus de l'Académie des Scinces, Paris t. 315, vol. I (1992), pp. 14311434.
[8]Compton, Kevin J. and Henson, C. Ward, A uniform method for proving lower bounds on the computational complexity of logical theories, Annals of Pure and Applied Logic, vol. 48 (1990), pp. 179.
[9]Ebbinghaus, H. D., Flum, J., and Thomas, W., Mathematical Logic, Springer-Verlag, Berlin, 1984.
[10]Erdös, Paul, How many pairs of products of consecutive integers have the same prime factors?, American Mathematical Monthly, vol. 87 (1980), pp. 391392.
[11]Goodstein, Reuben Louis, Hilbert's Tenth Problem and the independence of recursive difference, The Journal of the London Mathematical Society (Second Series), vol. 10, no. 2, pp. 175176.
[12]Langevin, Michel, Approximations diophantiennes et nombres transcendants, Luminy 1990 (Philippon, Patrice, editor), de Gruyter, Berlin, 1992, pp. 203214.
[13]Mendelson, Elliott, Introduction to mathematical logic, Van Nostrand, 1964; 2nd edition, 1979; 3rd edition, Wadsworth, California, 1987.
[14]Michel, Pascal, Borne supérieure de la complexité de la théorie de ℕ muni de la relation de divisibilité, Model Theory and Arithmetic (Berline, , McAloon, , and Ressayre, , editors), Lecture Notes in Mathematics, Number 890, Springer-Verlag, Berlin, 1981, pp. 242250.
[15]More, Malika, Images de spectres binaires par des polynômes, Comptes Rendus de l'Académie des Sciences, Paris, t. 315, vol. I (1992), pp. 343348.
[16]Padoa, Alessandro, Essai d'une théorie algébrique des nombres entiers, précédé d'une introduction logique à une théorie deductive quelconque, Bibliothèque du Congrès international de philosophie, Paris, 1900, vol. 3, Armand Colin, 1901, pp. 309365; partial English translation in van Heijenoort, From Frege to Gödei, Harvard University Press, 1967, pp. 118–123.
[17]Poizat, Bruno, Cours de théorie des modèles, Offilib, Paris, 1985.
[18]Richard, Denis, first publication in [19] and [22].
[19]Poizat, Bruno, Définissabilité en arithmétique et méthode de codage Z.B.V. appliquée à des langages avec successeurs et coprimarité, Thèse de doctorat d'État, Université de Lyon-I, 1985.
[20]Poizat, Bruno, Answer to a problem raised by J. Robinson: the arithmetic of positive or negative integers is definable from successor and divisibility, this Journal, vol. 50 (1985), pp. 927935.
[21]Poizat, Bruno, All arithmetical sets ofpowers ofprimes are first-order definable in terms of the successor function and the coprimeness predicate, Discrete Mathematics, vol. 53 (1985), pp. 221247.
[22]Poizat, Bruno, Equivalence of some questions in mathematical logic with some conjectures in number theory, Number Theory and Applications, (Mollin, , editor), NATO Asi Series, Series C: Mathematical and Physical Sciences, vol. 265 (1988), pp. 529545.
[23]Robinson, Julia, Definability and decision problems in arithmetic, this Journal, vol. 14 (1949), pp. 98114.
[24]Svenonius, Lars, A theorem on permutations in models, Theoria, vol. 25 (1959), pp. 173178.
[25]Woods, Alan, Some problems in logic and number theory and their connections, Thesis, University of Manchester, 1981.
[26]Younger, D. H., Recognition and parsing of context-free languages in time n 3, Information and Control, vol. 10 (1967), pp. 189208.

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *