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Inmathematical logic, thequantifier rank of aformula is the depth of nesting of itsquantifiers. It plays an essential role inmodel theory.

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

Let${\displaystyle \phi }$ be afirst-order formula. The quantifier rank of${\displaystyle \phi }$, written${\displaystyle \mathrm{qr}(\phi )}$, is defined as:

• ${\displaystyle \mathrm{qr}(\phi )=0}$, if${\displaystyle \phi }$ is atomic.
• ${\displaystyle \mathrm{qr}({\phi }_1\wedge {\phi }_2)=\mathrm{qr}({\phi }_1\vee {\phi }_2)=max(\mathrm{qr}({\phi }_1),\mathrm{qr}({\phi }_2))}$.
• ${\displaystyle \mathrm{qr}(\mathrm{\neg }\phi )=\mathrm{qr}(\phi )}$.
• ${\displaystyle \mathrm{qr}({\mathrm{\exists }}_x\phi )=\mathrm{qr}(\phi )+1}$.
• ${\displaystyle \mathrm{qr}({\mathrm{\forall }}_x\phi )=\mathrm{qr}(\phi )+1}$.

Remarks

• We write${\displaystyle \mathrm{FO}[n]}$ for the set of allfirst-order formulas${\displaystyle \phi }$ with${\displaystyle \mathrm{qr}(\phi )\le n}$.
• Relational${\displaystyle \mathrm{FO}[n]}$ (without function symbols) is always of finite size, i.e. contains a finite number of formulas.
• Inprenex normal form, the quantifier rank of${\displaystyle \phi }$ is exactly the number of quantifiers appearing in${\displaystyle \phi }$.

Forfixed-point logic, with a least fixed-point operator${\displaystyle \mathrm{LFP}}$:${\displaystyle \mathrm{qr}([{\mathrm{LFP}}_{\varphi }]y)=1+\mathrm{qr}(\varphi )}$.

• A sentence of quantifier rank 2:

${\displaystyle \mathrm{\forall }x\mathrm{\exists }yR(x,y)}$

• A formula of quantifier rank 1:

${\displaystyle \mathrm{\forall }xR(y,x)\wedge \mathrm{\exists }xR(x,y)}$

• A formula of quantifier rank 0:

${\displaystyle R(x,y)\wedge x\ne y}$

• A sentence inprenex normal form of quantifier rank 3:

${\displaystyle \mathrm{\forall }x\mathrm{\exists }y\mathrm{\exists }z((x\ne y\wedge xRy)\wedge (x\ne z\wedge zRx))}$

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

${\displaystyle \mathrm{\forall }x(\mathrm{\exists }y(x\ne y\wedge xRy))\wedge \mathrm{\exists }z(x\ne z\wedge zRx))}$

• Ehrenfeucht–Fraïssé game

• 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.

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

• Finite model theory
• Model theory
• Predicate logic
• Quantifier (logic)

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