Quantifier rank
From HandWiki
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
- [math]\displaystyle{ qr(\varphi) = 0 }[/math], if φ is atomic.
- [math]\displaystyle{ qr(\varphi_1 \land \varphi_2) = qr(\varphi_1 \lor \varphi_2) = max(qr(\varphi_1), qr(\varphi_2)) }[/math].
- [math]\displaystyle{ qr(\lnot \varphi) = qr(\varphi) }[/math].
- [math]\displaystyle{ qr(\exists_x \varphi) = qr(\varphi) + 1 }[/math].
Remarks
- We write FO[n] for the set of all first-order formulas φ with [math]\displaystyle{ qr(\varphi) \le n }[/math].
- 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: [math]\displaystyle{ qr([LFP_\phi]y) = 1 + qr( \phi ) }[/math]
Examples
- A sentence of quantifier rank 2:
- [math]\displaystyle{ \forall x\exists y R(x, y) }[/math]
- A formula of quantifier rank 1:
- [math]\displaystyle{ \forall x R(y, x) \wedge \exists x R(x, y) }[/math]
- A formula of quantifier rank 0:
- [math]\displaystyle{ R(x, y) \wedge x \neq y }[/math]
- A sentence in prenex normal form of quantifier rank 3:
- [math]\displaystyle{ \forall x \exists y \exists z ((x \neq y \wedge x R y) \wedge (x \neq z \wedge z R x)) }[/math]
- A sentence, equivalent to the previous, although of quantifier rank 2:
- [math]\displaystyle{ \forall x (\exists y (x \neq y \wedge x R y)) \wedge \exists z (x \neq z \wedge z R x )) }[/math]
See also
- Prenex normal form
- Ehrenfeucht game
- Quantifier
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.
External links
- Quantifier Rank Spectrum of L-infinity-omega BA Thesis, 2000
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