Denis Richard

Decidability of the theory of the natural integers with the cantor pairing function and the successor

Abstract The binary Cantor pairing function C from N × N into N is defined by C(x,y) = ( 1 2 )(x+y)(x+y+1)+y . We consider the theory of natural integers equipped with the Cantor pairing function and an extra relation or function X on N . When X is equal either to multiplication, or coprimeness, or divisibility, or addition or natural ordering, it can be proved that the theory Th( N ,C,X) is undecidable. Let S be the successor function. We provide an algorithm solving the decision problem for Th( N ,C,S) .

On arithmetical first-order theories allowing encoding and decoding of lists

Abstract In Computer Science, n -tuples and lists are usual tools; we investigate both notions in the framework of first-order logic within the set of nonnegative integers. Godel had firstly shown that the objects which can be defined by primitive recursion schema, can also be defined at first-order, using natural order and some coding devices for lists. Second he had proved that this encoding can be defined from addition and multiplication. We show this can be also done with addition and a weaker predicate, namely the coprimeness predicate. The theory of integers equipped with a pairing function can be decidable or not. The theory of decoding of lists (under some natural condition) is always undecidable. We distinguish the notions encoding of n -tuples and encoding of lists via some properties of decidability-undecidability. At last, we prove it is possible in some structure to encode lists although neither addition nor multiplication are definable in this structure.

Answer to a problem raised by J. Robinson: the arithmetic of positive or negative integers is definable from successor and divisibility

In this paper we give a positive answer to Julia Robinsons 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]).

What are weak arithmetics

It is amusing, indeed astonishing, that no-one among a community of about 100 computer scientists, logicians and mathematicians organizing meetings twice a year for almost 10 years 1 has thought it advisable properly and precisely to de-ne the -eld of research one usually calls Weak Arithmetics. In my opinion, everybody, within this group, brought to it his own interest and wondered at not having to justify the relevance of Weak Arithmetics. In discussions by ourselves, it appears that this relevance is intuitively founded on a common -eld of mathematical interest, a common set of questions and logical methods to investigate problems, and a general culture within computer science. Basically, a scientist interested in Weak Arithmetics knows some mathematical logic, like Peano arithmetic and the two G4 odel Theorems, works or has been working on decision problems, on algorithms and their complexities, and uses all kinds of abstract machines. Through these machines Weak Arithmetics are strongly in6uenced by the computer-dominated modern world. The Weak Arithmetics scientist is not a professional mathematician who studies numbers (using such tools as algebraic methods, complex analysis and algebraic geometry) but is often (or always in some areas) in contact with Number Theory. Therefore, it is di8cult to give a precise de-nition of Weak Arithmetics as a discipline in the same way as, say, Model Theory. Nevertheless we can nowadays consider the list of lectures and talks given from JAF1 to JAF17, in order to determine the main directions and themes provided by the participants at those events. One can distinguish four groups of lectures which the reader can -nd in the Annex.

On the amplitude of intervals of natural numbers whose every element has a common prime divisor with at least an extremity

An interval [a, a + d] of natural numbers verifies the property of no coprimeness if and only if every element a + 1, a + 2,..., a + d - 1 has a common prime divisor with extremity a or a + d. We show the set of such a and the set of such d are recursive. The computation of the first d leads to rise a lot of open problems.

La théorie élémentaire de la fonction de couplage de Cantor des entiers naturels est décidable

The Cantor pairing function C from N N into N is defined by C(x;y )=( 1=2) (x + y)(x + y +1 ) +y. The first order theory of natural integers equipped with the Cantor pairing function is decidable.© 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS