Elimination of Variables In Parallel

where , the existential quantifier, denotes “there exists’’ and is the conjunction that denotes “and.’’ Notice that while both sides of the equivalence, , are in fact equivalent, the right hand side does not involve y, i.e., the variable y has been eliminated. In fact, the quantifier has been also eliminated; for this reason, the procedure is also called “quantifier elimination.’’ A serious deficiency of Gaussian elimination is that it works only for an existentially quantified conjunction of linear equations. However, many real-life scientific and engineering problems deal with nonlinear equations, inequalities ( , >, . . .), disjunctions ( , which stands for “or’’), implications ( ), and the universal quantifier (∀ , which denotes “for all”). In studying the stability of a numerical method for solving a certain PDE, we encounter the following expression: