Quantifier elimination

Algorithms and decidability

If a theory has quantifier elimination, then a specific question can be addressed: Is there a method of determining ${\displaystyle {\alpha }_{QF}}$ for each ${\displaystyle \alpha }$? 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.

Related concepts

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. Conversely, a model-complete theory, whose theory of universal consequences has the amalgamation property, has quantifier elimination.

The models of the theory of the universal consequences of a theory ${\displaystyle T}$ are precisely the substructures of the models of ${\displaystyle T}$. The theory of linear orders does not have quantifier elimination. However the theory of its universal consequences has the amalgamation property.

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:

$${\displaystyle \mathrm{\exists }x.\underset{i=1}{\overset{n}{\bigwedge }}{L}_i}$$

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

$${\displaystyle \underset{j=1}{\overset{m}{\bigvee }}\underset{i=1}{\overset{n}{\bigwedge }}{L}_{ij},}$$

and use the fact that

$${\displaystyle \mathrm{\exists }x.\underset{j=1}{\overset{m}{\bigvee }}\underset{i=1}{\overset{n}{\bigwedge }}{L}_{ij}}$$

is equivalent to

$${\displaystyle \underset{j=1}{\overset{m}{\bigvee }}\mathrm{\exists }x.\underset{i=1}{\overset{n}{\bigwedge }}{L}_{ij}.}$$

Finally, to eliminate a universal quantifier

$${\displaystyle \mathrm{\forall }x.F}$$

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

Relationship with decidability

In early model theory, quantifier elimination was used to demonstrate that various theories possess 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 can be used to show that Presburger arithmetic 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 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).

Example: Nullstellensatz for algebraically closed fields and for differentially closed fields.