Skolemization

Skolemization is a way of removing existential quantifiers from a formula. Variables bound by existential quantifiers which are not inside the scope of universal quantifiers can simply be replaced by constants: $\\exists x[x<3]$ can be changed to $c<3$ , with $c$ a suitable constant.

When the existential quantifier is inside a universal quantifier, the bound variable must be replaced by a Skolem function of the variables bound by universal quantifiers. Thus $\\forall x[x=0\\vee\\exists y[x=y+1]]$ becomes $\\forall x[x=0\\vee x=f(x)+1]$ .

In general, the functions and constants symbols are new ones added to the language for the purpose of satisfying these formulas, and are often denoted by the formula they realize, for instance $c_{\\exists x\\phi(x)}$ .

This is used in second order logic to move all existential quantifiers outside the scope of first order universal quantifiers. This can be done since second order quantifiers can quantify over functions. For instance $\\forall^{1}x\\forall^{1}y\\exists^{1}z\\phi(x,y,z)$ is equivalent to $\\exists^{2}F\\forall^{1}x\\forall^{1}y\\phi(x,y,F(x,y))$ .

单词	Skolemization
释义	Skolemization Skolemization is a way of removing existential quantifiers from a formula. Variables bound by existential quantifiers which are not inside the scope of universal quantifiers can simply be replaced by constants: $\\exists x[x<3]$ can be changed to $c<3$ , with $c$ a suitable constant. When the existential quantifier is inside a universal quantifier, the bound variable must be replaced by a Skolem function of the variables bound by universal quantifiers. Thus $\\forall x[x=0\\vee\\exists y[x=y+1]]$ becomes $\\forall x[x=0\\vee x=f(x)+1]$ . In general, the functions and constants symbols are new ones added to the language for the purpose of satisfying these formulas, and are often denoted by the formula they realize, for instance $c_{\\exists x\\phi(x)}$ . This is used in second order logic to move all existential quantifiers outside the scope of first order universal quantifiers. This can be done since second order quantifiers can quantify over functions. For instance $\\forall^{1}x\\forall^{1}y\\exists^{1}z\\phi(x,y,z)$ is equivalent to $\\exists^{2}F\\forall^{1}x\\forall^{1}y\\phi(x,y,F(x,y))$ .
随便看	Extension extension ExtensionAndRestrictionOfStates extension and restriction of states ExtensionByLocalization extension by localization ExtensionField extension field ExtensionOfAFunction extension of a function ExtensionOfAPoset extension of a poset ExtensionOfKrullValuation extension of Krull valuation ExtensionOfValuationFromCompleteBaseField extension of valuation from complete base field ExtensionsWithoutUnramifiedSubextensionsAndClassNumberDivisibility extensions without unramified subextensions and class number divisibility ExtensiveFormGame extensive form game ExteriorAlgebra exterior algebra ExteriorAnglesOfTriangle exterior angles of triangle ExternalDirectProductOfGroups