In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable.
Alfred Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable.
[1] The ordered real field
is a structure over the language of ordered rings
, with the usual interpretation given to each symbol.
It was proved by Tarski that the theory of the real field,
there is an effective procedure for determining whether He then asked whether this was still the case if one added a unary function
to the language that was interpreted as the exponential function on
The problem can be reduced to finding an effective procedure for determining whether any given exponential polynomial in
Macintyre & Wilkie (1996) showed that Schanuel's conjecture implies such a procedure exists, and hence gave a conditional solution to Tarski's problem.
[2] Schanuel's conjecture deals with all complex numbers so would be expected to be a stronger result than the decidability of
, and indeed, Macintyre and Wilkie proved that only a real version of Schanuel's conjecture is required to imply the decidability of this theory.
Even the real version of Schanuel's conjecture is not a necessary condition for the decidability of the theory.
In their paper, Macintyre and Wilkie showed that an equivalent result to the decidability of
is what they dubbed the weak Schanuel's conjecture.
This conjecture states that there is an effective procedure that, given
variables with integer coefficients
is a non-singular solution of the system then either