Quantifier elimination

Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "${\displaystyle \exists x}$ such that ${\displaystyle \ldots }$" can be viewed as a question "When is there an ${\displaystyle x}$ such that ${\displaystyle \ldots }$?", and the statement without quantifiers can be viewed as the answer to that question.

One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest. A theory has quantifier elimination if for every formula ${\displaystyle \alpha }$, there exists another formula ${\displaystyle \alpha _{QF}}$ without quantifiers that is equivalent to it (modulo this theory).