Arithmetization of the field of reals with exponentiation extended abstract
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 105-119.

(1) Shepherdson proved that a discrete unitary commutative semi-ring A + satisfies IE 0 (induction scheme restricted to quantifier free formulas) iff A is integral part of a real closed field; and Berarducci asked about extensions of this criterion when exponentiation is added to the language of rings. Let T range over axiom systems for ordered fields with exponentiation; for three values of T we provide a theory T in the language of rings plus exponentiation such that the models (A, exp A ) of T are all integral parts A of models M of T with A + closed under exp M and exp A =exp M A + . Namely T = EXP, the basic theory of real exponential fields; T = EXP+ the Rolle and the intermediate value properties for all 2 x -polynomials; and T = T exp , the complete theory of the field of reals with exponentiation. (2) T exp is recursively axiomatizable iff T exp is decidable. T exp implies LE 0 (x y ) (least element principle for open formulas in the language <,+,×,-1,x y ) but the reciprocal is an open question. T exp satisfies “provable polytime witnessing”: if T exp proves xy:|y|<|x| k )R(x,y) (where |y|:= log(y) , k<ω and R is an NP relation), then it proves xR(x,f(x)) for some polynomial time function f. (3) We introduce “blunt” axioms for Arithmetics: axioms which do as if every real number was a fraction (or even a dyadic number). The falsity of such a contention in the standard model of the integers does not mean inconsistency; and bluntness has both a heuristic interest and a simplifying effect on many questions - in particular we prove that the blunt version of T exp is a conservative extension of T exp for sentences in Δ 0 (x y ) (universal quantifications of bounded formulas in the language of rings plus x y ). Blunt Arithmetics - which can be extended to a much richer language - could become a useful tool in the non standard approach to discrete geometry, to modelization and to approximate computation with reals.

DOI : https://doi.org/10.1051/ita:2007048
Classification : 03H15
Mots clés : computation with reals, exponentiation, model theory, o-minimality
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     title = {Arithmetization of the field of reals with exponentiation extended abstract},
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Boughattas, Sedki; Ressayre, Jean-Pierre. Arithmetization of the field of reals with exponentiation extended abstract. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 105-119. doi : 10.1051/ita:2007048. http://www.numdam.org/articles/10.1051/ita:2007048/

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