Objective
Weak arithmetic is the study of problems of Number theory and Computer Science using methods of mathematical logic; just as Algebraic Number Theory or Analytic Number Theory use Algebra and Analysis. Five of the main sources of results in wreaks arithmetic's are undecidability of the field of rational numbers, Matiyasevich-Davis-Robinson-Putnam theorem, solving Hilbert's tenth problem, Erdös-Woods conjecture, Buss arithmetic and study of the real exponential field.
The proposed project is constituted of nine teams from university of Paris, Steklov Institute of Mathematics at Saint-Petersburg (Russia), Udmurt University (Russia), Institute for Informatics and Automaton Problems at Yerevan (Armenia), Kiev University (Ukraine), University of Naples (Italy), university of Mons-Hainaut (Belgium), university of Clermont-Ferrand (France), and University of Athens (Greece).
Objectives for the three years were :
to construct Nonstandard Models of Buss Arithmetic to establish some bounds on the class NP inter co-NP ;
to solve the problem of existence of end extensions of countable models of bounded collection ;
to explore further the additive theory of infinite sets of prime numbers, both with absolute results and its links with number theory via the Schinzel's hypothesis ;
to code natural numbers, complex numbers, and quadratic integers by automata accepting numbers (written in non-classical systems) ;
to prove results for ultra-linear unary recursive schemata (with individual constants);
to obtain decidability results for S2S[P] theories ;
to build formal constructive theories with (from Grzegorczyk hierarchy) with applications to databases ;
to investigate the power of counting in very small complexity classes and in the corresponding logical or arithmetical settings ;
to explore the relation between Infinite games, automata, and arithmetic ;
to study the fine structure of the BSS recursive enumerable sets ;
to investigate concurrent processes in distributed conveyer systems and their relation with corresponding weak arithmetics;
to construct Diophantine representations of recursively enumerable sets of particular interest ;
to show the cofinality of primes in D0 (P), where P is the function which counts the primes below x ;
to study subsystems of Goodstein's Arithmetic (without quantifiers) ;
to develop families of formalized languages for presenting arithmetical texts.
Main exchanges are by e-mail but it is planning two workshops (a local one and a general) by year, two co-ordination meeting and exchanges of scientists.
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Coordinator
77300 Fontainebleau
France
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