Model theory of the real exponential function
A branch of model theory studying the elementary theory of the ordered field of real numbers with the real exponential function (cf. Exponential function, real). It is motivated by Tarski's question [a7], p. 45, whether is decidable.
A. Wilkie showed in [a8] that is model complete. Combining this with Khovanskii's finiteness theorem [a5], it follows that this theory is -minimal. In fact, Wilkie first studies expansions (cf. Structure) of by a Pfaffian chain of functions (see also [a2]): Fix and an open set containing the closed unit box . A Pfaffian chain of functions on is a sequence of analytic functions (cf. Analytic function) for which there exist polynomials (for ; ) such that
for all . Wilkie shows that the expansion of by a Pfaffian chain of functions restricted to the closed unit box has a model-complete theory. In particular, the expansion of by the restricted exponential function has a model-complete theory. Wilkie then deduces the model completeness of from this last result. An alternative proof of the model completeness, and an axiomatization of over , was found by J.P. Ressayre in 1991 (see [a3] for a generalization of Ressayre's result).
The theory does not admit elimination of quantifiers. In fact, an expansion of by a family of total real-analytic functions (see [a1]) admits elimination of quantifiers if and only if each function is semi-algebraic, i.e., has a semi-algebraic graph (cf. Semi-algebraic set). However, let denote the family of restricted real-analytic functions, i.e., functions , for all , which are given on by a power series converging on a neighbourhood of and are set equal to outside of . It is shown in [a3] that the expansion admits elimination of quantifiers. The authors also give a complete axiomatization of , and establish that it is -minimal. In [a4] they construct a model of this theory which is not Archimedean and use it to solve a problem raised by G.H. Hardy: they show that the compositional inverse of the function is not asymptotic at to a composition of semi-algebraic functions, and .
|[a1]||L. van den Dries, "Remarks on Tarski's problem concerning " G. Lolli (ed.) G. Longo (ed.) A. Marcja (ed.) , Logic Colloquium '82 , North-Holland (1984) pp. 97–121|
|[a2]||L. van den Dries, "Tarski's problem and Pfaffian functions" J.B. Paris (ed.) A.J. Wilkie (ed.) G.M. Wilmers (ed.) , Logic Colloquium '84 , North-Holland (1986) pp. 59–90|
|[a3]||L. van den Dries, A.J. Macintyre, D. Marker, "The elementary theory of restricted analytic fields with exponentiation" Ann. of Math. , 140 (1994) pp. 183–205|
|[a4]||L. van den Dries, A.J. Macintyre, D. Marker, "Logarithmic-exponential power series" J. London Math. Soc. (to appear)|
|[a5]||A. Khovanskii, "On a class of systems of transcendental equations" Soviet Math. Dokl. , 22 (1980) pp. 762–765 (In Russian)|
|[a6]||A.J. Macintyre, A.J. Wilkie, "On the decidability of the real exponential field" P.G. Odifreddi (ed.) , Kreisel 70th Birthday Volume , CLSI (1995)|
|[a7]||A. Tarski, J.C.C. McKinsey, "A decision method for elementary algebra and geometry" , Univ. California Press (1951)|
|[a8]||A.J. Wilkie, "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function" J. Amer. Math. Soc. , 9 : 4 (1996)|
Model theory of the real exponential function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Model_theory_of_the_real_exponential_function&oldid=13098