Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science.
An example from mathematics says that a single-variable quadratic polynomial has a real root if and only if its discriminant is non-negative:[1] Here the sentence on the left-hand side involves a quantifier
Examples of theories that have been shown decidable using quantifier elimination are Presburger arithmetic,[2][3][4][5][6] algebraically closed fields, real closed fields,[7][8] atomless Boolean algebras, term algebras, dense linear orders,[7] abelian groups,[9] random graphs, as well as many of their combinations such as Boolean algebra with Presburger arithmetic, and term algebras with queues.
Quantifier eliminator for the theory of the real numbers as an ordered additive group is Fourier–Motzkin elimination; for the theory of the field of real numbers it is the Tarski–Seidenberg theorem.
If a theory has quantifier elimination, then a specific question can be addressed: Is there a method of determining
If there is such a method we call it a quantifier elimination algorithm.
Quantifier-free sentences have no variables, so their validity in a given theory can often be computed, which enables the use of quantifier elimination algorithms to decide validity of sentences.
Various model-theoretic ideas are related to quantifier elimination, and there are various equivalent conditions.
Every first-order theory with quantifier elimination is model complete.
[10] The theory of linear orders does not have quantifier elimination.
is a quantifier-free formula, we can write it in disjunctive normal form and use the fact that is equivalent to Finally, to eliminate a universal quantifier where
A common technique was to show first that a theory admits elimination of quantifiers and thereafter prove decidability or completeness by considering only the quantifier-free formulas.
Theories could be decidable yet not admit quantifier elimination.
Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable.
Whenever a theory is decidable, and the language of its valid formulas is countable, it is possible to extend the theory with countably many relations to have quantifier elimination (for example, one can introduce, for each formula of the theory, a relation symbol that relates the free variables of the formula).