BCK algebra: Difference between revisions

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In [[model theory]], '''[[Alfred Tarski|Tarski]]'s exponential function problem''' asks whether the [[Theory (mathematical logic)|theory]] of the [[real number]]s together with the [[exponential function]] is [[Decidability (logic)#Decidability of a theory|decidable]]. Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable.
 
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==The problem==
 
The ordered real field '''R''' is a structure over the language of [[ordered ring]]s ''L''<sub>or</sub>&nbsp;=&nbsp;(+,·,&minus;,<,0,1), with the usual interpretation given to each symbol. It was proved by Tarski that the theory of the [[Real closed field|real field]], Th('''R'''), is decidable. That is, given any ''L''<sub>or</sub>-sentence ''φ'' there is an effective procedure for determining whether
 
:<math>\operatorname{Th}(\R)\models\varphi.</math>
 
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'''<sub>exp</sub>.
 
==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 '''R'''<sup>''n''</sup>. {{harvtxt|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'''<sub>exp</sub>, 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 and sufficient condition|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'''<sub>exp</sub>) is what they dubbed the Weak Schanuel's Conjecture. This conjecture states that there is an effective procedure that, given ''n''&nbsp;&ge;&nbsp;1 and exponential polynomials in ''n'' variables with integer coefficients ''f''<sub>1</sub>,..., ''f''<sub>''n''</sub>, ''g'', produces an integer ''η''&nbsp;&ge;&nbsp;1 that depends on ''n'', ''f''<sub>1</sub>,..., ''f''<sub>''n''</sub>, ''g'', and such that if ''α''&nbsp;&isin;&nbsp;'''R'''<sup>''n''</sup> is a [[Singular point of a curve|non-singular]] solution of the system
 
:<math>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</math>
 
then either ''g''(''α'')&nbsp;=&nbsp;0 or |''g''(''α'')|&nbsp;>&nbsp;''η''<sup>&minus;1</sup>.
 
==References==
 
*{{springer|id=M/m110160|first=S.|last= Kuhlmann|title=Model theory of the real exponential function}}
*{{citation|first=A.J. |last=Macintyre|first2=  A.J. |last2=Wilkie|chapter=On the decidability of the real exponential field|editor-first= P.G.|editor-last= Odifreddi |title= Kreisel 70th Birthday Volume |publisher= CLSI  |year=1995}}
 
[[Category:Model theory]]
[[Category:Unsolved problems in mathematics]]

Latest revision as of 10:05, 28 July 2014

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