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.

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

In first-order logic

Let $\varphi$ be a first-order formula. The quantifier rank of $\varphi$, written $\operatorname {qr} (\varphi )$, is defined as:

• $\operatorname {qr} (\varphi )=0$, if $\varphi$ is atomic.
• $\operatorname {qr} (\varphi _{1}\land \varphi _{2})=\operatorname {qr} (\varphi _{1}\lor \varphi _{2})=\max(\operatorname {qr} (\varphi _{1}),\operatorname {qr} (\varphi _{2}))$.
• $\operatorname {qr} (\lnot \varphi )=\operatorname {qr} (\varphi )$.
• $\operatorname {qr} (\exists _{x}\varphi )=\operatorname {qr} (\varphi )+1$.
• $\operatorname {qr} (\forall _{x}\varphi )=\operatorname {qr} (\varphi )+1$.

Remarks

• We write $\operatorname {FO} [n]$ for the set of all first-order formulas $\varphi$ with $\operatorname {qr} (\varphi )\leq n$.
• Relational $\operatorname {FO} [n]$ (without function symbols) is always of finite size, i.e. contains a finite number of formulas.
• In prenex normal form, the quantifier rank of $\varphi$ is exactly the number of quantifiers appearing in $\varphi$.

In higher-order logic

For fixed-point logic, with a least fixed-point operator $\operatorname {LFP}$: $\operatorname {qr} ([\operatorname {LFP} _{\phi }]y)=1+\operatorname {qr} (\phi )$.

Examples

• A sentence of quantifier rank 2:

$\forall x\exists yR(x,y)$

• A formula of quantifier rank 1:

$\forall xR(y,x)\wedge \exists xR(x,y)$

• A formula of quantifier rank 0:

$R(x,y)\wedge x\neq y$

• A sentence in prenex normal form of quantifier rank 3:

$\forall x\exists y\exists z((x\neq y\wedge xRy)\wedge (x\neq z\wedge zRx))$

• A sentence, equivalent to the previous, although of quantifier rank 2:

$\forall x(\exists y(x\neq y\wedge xRy))\wedge \exists z(x\neq z\wedge zRx))$

See also

• Ehrenfeucht–Fraïssé game

References

• Ebbinghaus, Heinz-Dieter; Flum, Jörg (1995), Finite Model Theory, Springer, ISBN 978-3-540-60149-4.
• Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Maarten, Marx; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (2007), Finite model theory and its applications, Texts in Theoretical Computer Science. An EATCS Series, Berlin: Springer-Verlag, p. 133, ISBN 978-3-540-00428-8, Zbl 1133.03001.

External links

• Quantifier Rank Spectrum of L-infinity-omega BA Thesis, 2000