Quantifier Elimination for Real Algebra - the Quadratic Case and Beyond

Abstract

. We present a new, "elementary" quantifier elimination method for various special cases of the general quantifier elimination problem for the first--order theory of real numbers. These include the elimination of one existential quantifier 9x in front of quantifier--free formulas restricted by a non-trivial quadratic equation in x (the case considered also in [7]), and more generally in front of arbitrary quantifier--free formulas involving only polynomials that are quadratic in x. The method generalizes the linear quantifier elimination method by virtual substitution of test terms in [9]. It yields a quantifier elimination method for an arbitrary number of quantifiers in certain formulas involving only linear and quadratic occurences of the quantified variables. Moreover, for existential formulas ' of this kind it yields sample answers to the query represented by '. The method is implemented in reduce as part of the redlog package (see [4, 5]). Experiments show that the method is appl..