Tarski's exponential function problem

In mathematics, Tarski's exponential function problem, of mathematician and logician Alfred Tarski, asks whether the usual theory of the real numbers together with the exponential function is decidable. Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable.

The problem

The ordered real field R is a structure over the language of ordered rings L_or = (+,·,−,<,0,1), with the usual interpretation given to each symbol. It was proved by Tarski that the theory of the real field, Th(R), is decidable. That is, given any L_or-sentence φ there is an effective procedure for determining whether

$\operatorname {Th} (\mathbb {R} )\models \varphi .$

He then asked whether this was still the case if one added a unary function exp to the language that was interpreted as the exponential function on R, to get the structure R_exp.

The problem can be reduced to finding an effective procedure for determining whether any given exponential polynomial in n variables and with coefficients in Z has a solution in Rⁿ. Macintyre & Wilkie (1995) showed that Schanuel's conjecture implies such a procedure exists, and hence gave a conditional solution to Tarski's problem. Schanuel's conjecture deals with all complex numbers so would be expected to be a stronger result than the decidability of R_exp, and indeed, Macintyre and Wilkie proved that the decidability of this structure is equivalent to a weakened version of Schanuel's conjecture that only deals with real numbers.

References

• Kuhlmann, S. (2001) [1994], "Model theory of the real exponential function", Encyclopedia of Mathematics, EMS Press
• Macintyre, A.J.; Wilkie, A.J. (1995), "On the decidability of the real exponential field", in Odifreddi, P.G. (ed.), Kreisel 70th Birthday Volume, CLSI