{
  "id": "enwiki-001323636442",
  "slug": "quantifier-rank",
  "title": "Quantifier rank",
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  "lead": "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.",
  "body": "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== \\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) \\le 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 . 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 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 )) ==See also== Ehrenfeucht–Fraïssé game ==References== . . ==External links== Quantifier Rank Spectrum of L-infinity-omega BA Thesis, 2000",
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    "title": "Quantifier rank",
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    "publisher-place": "Dubai, United Arab Emirates",
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