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In model theory, Tarski's exponential function problem asks whether the 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

Th⁡(R)⊨φ.{displaystyle 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.

Conditional and equivalent results

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 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 Th(R_exp) is what they dubbed the Weak Schanuel's Conjecture. This conjecture states that there is an effective procedure that, given n ≥ 1 and exponential polynomials in n variables with integer coefficients f₁,..., f_n, g, produces an integer η ≥ 1 that depends on n, f₁,..., f_n, g, and such that if α ∈ Rⁿ is a non-singular solution of the system

f1(x1,…,xn,ex1,…,exn)=…=fn(x1,…,xn,ex1,…,exn)=0{displaystyle f_{1}(x_{1},ldots ,x_{n},e^{x_{1}},ldots ,e^{x_{n}})=ldots =f_{n}(x_{1},ldots ,x_{n},e^{x_{1}},ldots ,e^{x_{n}})=0}

then either g(α) = 0 or |g(α)| > η⁻¹.

Workaround

Recently there are attempts at handling the theory of the real numbers with functions such as exp, sin by relaxing decidability to the weaker notion of quasi-decidability. A theory is quasi-decidable if there is a procedure that decides satisfiability but that may run forever for inputs that are not robust in a certain, well-defined sense. Such a procedure exists for systems of n equations in n variables (Franek, Ratschan & Zgliczynski (2011)).

See also

• Decidability of first-order theories of the real numbers

References

• 
  Kuhlmann, S. (2001) [1994], "Model theory of the real exponential function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  Macintyre, A.J.; Wilkie, A.J. (1995), "On the decidability of the real exponential field", in Odifreddi, P.G., Kreisel 70th Birthday Volume, CLSI
  


• 
  Franek, Peter; Ratschan, Stefan; Zgliczynski, Piotr (2011), "Satisfiability of Systems of Equations of Real Analytic Functions Is Quasi-decidable", Mathematical Foundations of Computer Science 2011, LNCS, 6907, Springer, doi:10.1007/978-3-642-22993-0_30