Quantifier elimination

Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. One way of classifying formulas is by the amount of quantification. Formulae with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulae as the simplest. A theory has quantifier elimination if for every formula $\alpha$, there exists another formula $\alpha_{QF}$ without quantifiers that is equivalent to it (modulo the theory).

Examples

Examples of theories that have been shown decidable using quantifier elimination are Presburger arithmetic,[1] algebraically closed fields, real closed fields,[1][2] atomless Boolean algebras, term algebras, dense linear orders,[1] random graphs, Feature trees, 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.[1]

Quantifier elimination can also be used to show that "combining" decidable theories leads to new decidable theories. Such constructions include the Feferman-Vaught theorem and Term Powers.

Algorithms and decidability

If a theory has quantifier elimination, then a specific question can be addressed: Is there a method of determining $\alpha_{QF}$ for each $\alpha$? If there is such a method we call it a quantifier elimination algorithm. If there is such an algorithm, then decidability for the theory reduces to deciding the truth of the quantifier-free sentences. 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.

Related concepts

Various model theoretic ideas are related to quantifier elimination, and there are various equivalent conditions.

Every theory with quantifier elimination is model complete.

A first-order theory T has quantifier elimination if and only if for any two models B and C of T and for any common substructure A of B and C, B and C are elementarily equivalent in the language of T augmented with constants from A. In fact, it is sufficient here to show that any sentence with only existential quantifiers have the same truth value in B and C.

Basic ideas

To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of literals, that is, show that each formula of the form:

$\exists x. \bigwedge_{i=1}^n L_i$

where each $L_i$ is a literal, is equivalent to a quantifier-free formula. Indeed, suppose we know how to eliminate quantifiers from conjunctions of formulae, then if $F$ is a quantifier-free formula, we can write it in disjunctive normal form

$\bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij},$

and use the fact that

$\exists x. \bigvee_{j=1}^m \bigwedge_{i=1}^n L_{ij}$

is equivalent to

$\bigvee_{j=1}^m \exists x. \bigwedge_{i=1}^n L_{ij}.$

Finally, to eliminate a universal quantifier

$\forall x. F$

where $F$ is quantifier-free, we transform $\lnot F$ into disjunctive normal form, and use the fact that $\forall x. F$ is equivalent to $\lnot \exists x. \lnot F.$

History

In early model theory, quantifier elimination was used to demonstrate that various theories possess certain model-theoretic properties like decidability and completeness. 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. This technique is used to show that Presburger arithmetic, i.e. the theory of the additive natural numbers, is decidable.

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 in a countable language is decidable, it is possible to extend its language with countably many relations to ensure that it admits quantifier elimination (for example, one can introduce a relation symbol for each formula).

Example: Nullstellensatz in ACF and DCF.