Quantifier rank

In mathematical logic, the quantifier rank of a formula is the depth of nesting of its quantifiers. It plays an essential role in model theory.

Notice that the quantifier rank is a property of the formula itself (i.e. the expression in a language). Thus two logically equivalent formulae can have different quantifier ranks, when they express the same thing in different ways.

Definition
Quantifier Rank of a Formula in First-order language (FO)

Let φ be a FO formula. The quantifier rank of φ, written qr(φ), is defined as
 * $$qr(\varphi) = 0$$, if φ is atomic.
 * $$qr(\varphi_1 \land \varphi_2) = qr(\varphi_1 \lor \varphi_2) = max(qr(\varphi_1), qr(\varphi_2))$$.
 * $$qr(\lnot \varphi) = qr(\varphi)$$.
 * $$qr(\exists_x \varphi) = qr(\varphi) + 1$$.
 * $$qr(\forall_x \varphi) = qr(\varphi) + 1$$.

Remarks
 * We write FO[n] for the set of all first-order formulas φ with $$qr(\varphi) \le n$$.
 * Relational FO[n] (without function symbols) is always of finite size, i.e. contains a finite number of formulas
 * Notice that in Prenex normal form the Quantifier Rank of φ is exactly the number of quantifiers appearing in φ.

Quantifier Rank of a higher order Formula
 * For Fixpoint logic, with a least fix point operator LFP: $$qr([LFP_\phi]y) = 1 + qr( \phi )$$

Examples

 * A sentence of quantifier rank 2:
 * $$\forall x\exists y R(x, y)$$


 * A formula of quantifier rank 1:
 * $$\forall x R(y, x) \wedge \exists x R(x, y)$$


 * A formula of quantifier rank 0:
 * $$R(x, y) \wedge x \neq y$$


 * A sentence in prenex normal form of quantifier rank 3:
 * $$\forall x \exists y \exists z ((x \neq y \wedge x R y) \wedge (x \neq z \wedge z R x)) $$


 * A sentence, equivalent to the previous, although of quantifier rank 2:
 * $$\forall x (\exists y (x \neq y \wedge x R y)) \wedge \exists z (x \neq z \wedge z R x )) $$