The field of *Weak arithmetics* is application of logical methods
to
Number Theory, developed by mathematicians, philosophers, and
Theoretical Computer Scientists. In this volume, after a general
presentation of weak arithmetics, the following topics are studied:
the properties of integers of a real closed field equipped
with exponentiation; conservation results for the induction
schema restricted to first-order formulas with a finite number
of alternations of quantifiers; a survey on a class of tools,
called *pebble games,* used in finite model theory; the fact
that
reals *e* and *π* have approximations expressed by first-order
formulas using bounded quantifiers; properties on infinite
*pictures* depending on the universe of sets used;
a language that simulates in a sufficiently nice manner
all algorithms of a certain restricted class; the logical
complexity of the axiom of infinity in some variants of set theory
without the axiom of foundation; and the complexity to determine
whether a trace is included in another one.

Patrick Cégielski
is professor at Université Paris-Est Créteil IUT de Sénart-Fontainebleau

December 2009

Foreword
by Simone Bonnafous vii
Introduction
by Patrick CÂ´egielski 1
1 On the Arithmetization of Real Fields with Exponentiation
by Sedki Boughattas and Jean-Pierre Ressayre 13
2 On Conservation Results for Parameter-Free n-Induction
by A. CordÂ´onâ€“Franco, A. FernÂ´andezâ€“Margarit, and F.F. Laraâ€“MartÂ´Ä±n 49
3 Pebble Games for Logics with Counting and Rank
by Anuj Dawar and Bjarki Holm 99
4 Rudimentary of Two Famous Real Numbers
by Henri-Alex Esbelin 121
5 Decision Problems for Recognizable Languages of Infinite Pictures
by Olivier Finkel 127
6 A Total Functional Programming Language Computing APRA
by David Michel and Pierre Valarcher 153
7 Statements of Ill-Founded Infinity in Set Theory
by Eugenio Omodeo, Alberto Policriti, and Alexandru Tomescu 173
8 On some Matching Problems in Trace Monoids
by Karine Shahbazyan and Yuri Shoukourian 201