Tarski's theorem says that the first-order theory of reals with ,
, , and allows quantifier
elimination. Algorithmic quantifier elimination implies decidability
assuming that the truth values of sentences involving only constants can be computed.
However, the converse is not true. For example, the first-order theory of reals with
, , and is decidable, but does not allow quantifier
elimination.
Tarski's theorem means that the solution set of a quantified system of real algebraic equations and inequations is a semialgebraic
set (Tarski 1951, Strzebonski 2000).
Collins, G. E. "Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition." In Proc. 2nd GI Conf.
Automata Theory and Formal Languages. New York: Springer-Verlag, pp. 134-183,
1975.Davenport, J. and Heintz, J. "Real Quantifier Elimination
Is Doubly Exponential." J. Symb. Comput.5, 29-35, 1988.Marker,
D. "Model Theory and Exponentiation." Not. Amer. Math. Soc.43,
753-759, 1996.Strzebonski, A. "Solving Algebraic Inequalities."
Mathematica J.7, 525-541, 2000.Tarski, A. "Sur les
ensembles définissables de nombres réels." Fund. Math.17,
210-239, 1931.Tarski, A. A Decision Method for Elementary Algebra
and Geometry. Manuscript. Santa Monica, CA: RAND Corp., 1948. Republished as
A
Decision Method for Elementary Algebra and Geometry, 2nd ed. Berkeley, CA:
University of California Press, 1951.
,
, 
, and 
allows quantifier
elimination. Algorithmic quantifier elimination implies decidability
assuming that the truth values of sentences involving only constants can be computed.
However, the converse is not true. For example, the first-order theory of reals with
, 
, and 
is decidable, but does not allow quantifier
elimination.
