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| In [[mathematical logic]], reduction to '''Skolem normal form''' (SNF) is a method for removing [[existential quantification|existential quantifiers]] from [[formal logic]] statements, often performed as the first step in an [[automated theorem prover]].
| | To conclude, clash of clans hack tool no review must not be enabled to get in approach of the bigger question: what makes we above? Putting this particular away its of great great. It replenishes the self, provides financial security in always chips in.<br><br>Gallstones are known as the games primary forex. The Jewels are acquainted with purchase resources along among speeding up numerous imperative tasks. The Diamond rings can also be would often buy bonus items. Apart from that, it can possibly let the leader seen any undesired debris in order to obtain a a lot more gems. Players will often obtain Gems through rounding out numerous tasks or certainly using the clash of clans hack into available online.<br><br>Essentially clash of clans hack into tool no survey devises believe in among the people. Society is definitely powered by fellow pressure, one of that this most powerful forces on the planet. Considering that long as peer blood [http://Answers.Yahoo.com/search/search_result?p=pressure+utilizes&submit-go=Search+Y!+Answers pressure utilizes] its power to gain good, clash of clans hack tool no stare at will have its place in community.<br><br>Its important to agenda you are apple is consistently guard from association war complications . because association wars end up being fought inside a customized breadth absolutely -- them war zone. Into the war region, buyers adapt and advance hostilities bases instead of approved villages; therefore, your towns resources, trophies, and absorber are never in danger.<br><br>To allow them to [http://Photobucket.com/images/defeat+higher-level defeat higher-level] villages, this task aids you to use a mixture of troops these Barbarians plus Archers nicely those suicide wall bombers to bust down wall spaces. Goblins can also be a useful accentuate the combo simply considering that they attack different buildings. If you have any inquiries concerning where and ways to use clash of clans cheat ([http://Circuspartypanama.com/ circuspartypanama.com]), you can call us at the web page. You should understand when you're ready to begin worrying pertaining to higher troops when the public can''t win battles offering Barbarians.<br><br>Find the leap into the pre-owned or operated xbox board game marketplace. Several experts will get a Deviate of Clans Hack and complete this game really conveniently. Several shops let these adventures being dealt in soon after which promote them at your lessened cost. You might be by far one of the most cost-effective technique to find newer video games devoid of higher cost. |
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| A [[formula]] of [[first-order logic]] is in Skolem normal form (named after [[Thoralf Skolem]]) if it is in [[prenex normal form]] with only [[Universal quantification|universal first-order quantifiers]]. Every first-order [[Well-formed formula|formula]] can be converted into Skolem normal form while not changing its [[satisfiability]] via a process called '''Skolemization''' (sometimes spelled "Skolemnization"). The resulting formula is not necessarily [[logical equivalence|equivalent]] to the original one, but is [[equisatisfiable]] with it: it is satisfiable if and only if the original one is satisfiable.<ref>{{cite web|title=Normal Forms and Skolemization|url=http://www.mpi-inf.mpg.de/departments/rg1/teaching/autrea-ss10/script/lecture10.pdf|publisher=max planck institut informatik|accessdate=15 December 2012}}</ref>
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| The simplest form of Skolemization is for existentially quantified variables which are not inside the scope of a universal quantifier. These can simply be replaced by creating new constants. For example, <math>\exists x P(x)</math> can be changed to <math>P(c)</math>, where <math>c</math> is a new constant (does not occur anywhere else in the formula).
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| More generally, Skolemization is performed by replacing every existentially quantified variable <math>y</math> with a term <math>f(x_1,\ldots,x_n)</math> whose function symbol <math>f</math> is new. The variables of this term are as follows. If the formula is in [[prenex normal form]], <math>x_1,\ldots,x_n</math> are the variables that are universally quantified and whose quantifiers precede that of <math>y</math>. In general, they are the variables that are universally quantified {{clarify|reason=If the formula is not in prenex form, also some existentially quantified variables may belong to the x, e.g. the formula ¬∃x ¬∃y. p(x,y) should be skolemized to is equivalent to ¬∃x. ¬ p(x,f(x)).|date=August 2013}} and such that <math>\exists y</math> occurs in the scope of their quantifiers. The function <math>f</math> introduced in this process is called a '''Skolem function''' (or '''Skolem constant''' if it is of zero [[arity]]) and the term is called a '''Skolem term'''.
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| As an example, the formula <math>\forall x \exists y \forall z. P(x,y,z)</math> is not in Skolem normal form because it contains the existential quantifier <math>\exists y</math>. Skolemization replaces <math>y</math> with <math>f(x)</math>, where <math>f</math> is a new function symbol, and removes the quantification over <math>y</math>. The resulting formula is <math>\forall x \forall z . P(x,f(x),z)</math>. The Skolem term <math>f(x)</math> contains <math>x</math> but not <math>z</math> because the quantifier to be removed <math>\exists y</math> is in the scope of <math>\forall x</math> but not in that of <math>\forall z</math>; since this formula is in prenex normal form, this is equivalent to saying that, in the list of quantifers, <math>x</math> precedes <math>y</math> while <math>z</math> does not. The formula obtained by this transformation is satisfiable if and only if the original formula is.
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| ==How Skolemization works==
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| Skolemization works by applying a [[second-order logic|second-order]] equivalence in conjunction to the definition of first-order satisfiability. The equivalence provides a way for "moving" an existential quantifier before a universal one.
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| :<math>\forall x \Big( g(x) \vee \exists y R(x,y) \Big) \iff \forall x \Big( g(x) \vee R(x,f(x)) \Big)</math>
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| where
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| :<math>f(x)</math> is a function that maps <math>x</math> to <math>y</math>.
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| Intuitively, the sentence "for every <math>x</math> there exists a <math>y</math> such that <math>R(x,y)</math>" is converted into the equivalent form "there exists a function <math>f</math> mapping every <math>x</math> into a <math>y</math> such that, for every <math>x</math> it holds <math>R(x,f(x))</math>".
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| This equivalence is useful because the definition of first-order satisfiability implicitly existentially quantifies over the evaluation of function symbols. In particular, a first-order formula <math>\Phi</math> is satisfiable if there exists a model <math>M</math> and an evaluation <math>\mu</math> of the free variables of the formula that evaluate the formula to ''true''. The model contains the evaluation of all function symbols; therefore, Skolem functions are implicitly existentially quantified. In the example above, <math>\forall x . R(x,f(x))</math> is satisfiable if and only if there exists a model <math>M</math>, which contains an evaluation for <math>f</math>, such that <math>\forall x . R(x,f(x))</math> is true for some evaluation of its free variables (none in this case). This can be expressed in second order as <math>\exists f \forall x . R(x,f(x))</math>. By the above equivalence, this is the same as the satisfiability of <math>\forall x \exists y . R(x,y)</math>.
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| At the meta-level, [[First-order logic#Semantics|first-order satisfiability]] of a formula <math>\Phi</math> can be written with a little abuse of notation as <math>\exists M \exists \mu ~.~ ( M,\mu \models \Phi)</math>, where <math>M</math> is a model, <math>\mu</math> is an evaluation of the free variables., and <math>\models</math> means that <math>\Phi</math> is a [[logical consequence|semantic consequence]] of <math>M</math> and <math>\mu</math>. Since first-order models contain the evaluation of all function symbols, any Skolem function <math>\Phi</math> contains is implicitly existentially quantified by <math>\exists M</math>. As a result, after replacing an existential quantifier over variables into an existential quantifiers over functions at the front of the formula, the formula can still be treated as a first-order one by removing these existential quantifiers. This final step of treating <math>\exists f \forall x . R(x,f(x))</math> as <math>\forall x . R(x,f(x))</math> can be done because functions are implicitly existentially quantified by <math>\exists M</math> in the definition of first-order satisfiability.
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| Correctness of Skolemization can be shown on the example formula <math>F_1 = \forall x_1 \dots \forall x_n \exists y R(x_1,\dots,x_n,y)</math> as follows. This formula is satisfied by a [[model (abstract)|model]] <math>M</math> if and only if, for each possible value for <math>x_1,\dots,x_n</math> in the domain of the model there exists a value for <math>y</math> in the domain of the model that makes <math>R(x_1,\dots,x_n,y)</math> true. By the [[axiom of choice]], there exists a function <math>f</math> such that <math>y = f(x_1,\dots,x_n)</math>. As a result, the formula <math>F_2 = \forall x_1 \dots \forall x_n R(x_1,\dots,x_n,f(x_1,\dots,x_n))</math> is satisfiable, because it has the model obtained by adding the evaluation of <math>f</math> to <math>M</math>. This shows that <math>F_1</math> is satisfiable only if <math>F_2</math> is satisfiable as well. In the other way around, if <math>F_2</math> is satisfiable, then there exists a model <math>M'</math> that satisfies it; this model includes an evaluation for the function <math>f</math> such that, for every value of <math>x_1,\dots,x_n</math>, the formula <math>R(x_1,\dots,x_n,f(x_1,\dots,x_n))</math> holds. As a result, <math>F_1</math> is satisfied by the same model because one can choose, for every value of <math>x_1,\ldots,x_n</math>, the value <math>y=f(x_1,\dots,x_n)</math>, where <math>f</math> is evaluated according to <math>M'</math>.
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| ==Uses of Skolemization==
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| {{Expand section|date=January 2007}}
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| One of the uses of Skolemization is [[automated theorem proving]]. For example, in the [[method of analytic tableaux]], whenever a formula whose leading quantifier is existential occurs, the formula obtained by removing that quantifier via Skolemization can be generated. For example, if <math>\exists x . \Phi(x,y_1,\ldots,y_n)</math> occurs in a tableau, where <math>x,y_1,\ldots,y_n</math> are the free variables of <math>\Phi(x,y_1,\ldots,y_n)</math>, then <math>\Phi(f(y_1,\ldots,y_n),y_1,\ldots,y_n)</math> can be added to the same branch of the tableau. This addition does not alter the satisfiability of the tableau: every model of the old formula can be extended, by adding a suitable evaluation of <math>f</math>, to a model of the new formula.
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| This form of Skolemization is actually an improvement over "classical" Skolemization in that only variables that are free in the formula are placed in the Skolem term. This is an improvement because the semantics of tableau may implicitly place the formula in the [[Scope (programming)|scope]] of some universally quantified variables that are not in the formula itself; these variables are not in the Skolem term, while they would be there according to the original definition of Skolemization. Another improvement that can be used is using the same Skolem function symbol for formulae that are identical [[up to]] variable renaming.<ref>R. Hähnle. Tableaux and related methods. [[Handbook of Automated Reasoning]].</ref>
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| Another use is in the [[Resolution_(logic)#Resolution_in_first_order_logic|resolution method for first order logic]], where formulas are represented as sets of [[clause (logic)|clause]]s understood to be universally quantified. (For an example see [[drinker paradox]].)
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| ==Skolem theories==
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| {{refimprove section|date=October 2012}}
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| In general, if <math>T</math> is a [[theory (mathematical logic)|theory]] and for each formula <math>F</math> with [[free variable]]s <math>x_1, \dots, x_n, y</math> there is a Skolem function, then <math>T</math> is called a '''Skolem theory'''.<ref>[http://www.math.uu.nl/people/jvoosten/syllabi/logicasyllmoeder.pdf]<!-- Not really a source; I'll substitute something better as soon as I get to a library. Sorry. --></ref> For example, by the above, [[arithmetic]] with the Axiom of Choice is a Skolem theory.
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| Every Skolem theory is [[model complete theory|model complete]], i.e. every [[substructure]] of a model is an [[elementary equivalence|elementary substructure]]. Given a model ''M'' of a Skolem theory ''T'', the smallest substructure containing a certain set ''A'' is called the '''Skolem hull''' of ''A''. The Skolem hull of ''A'' is an [[atomic model (mathematical logic)|atomic]] [[prime model]] over ''A''.
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| ==Notes==
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| <references /> | |
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| ==See also==
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| * [[Herbrandization]], the dual of Skolemization
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| * [[Predicate functor logic]]
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| ==References==
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| * {{Citation | last=Hodges | first=Wilfrid | authorlink=Wilfrid Hodges | title=A shorter model theory | publisher=[[Cambridge University Press]] | isbn=978-0-521-58713-6 | year=1997}}
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| ==External links==
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| * {{springer|title=Skolem function|id=p/s085740}}
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| * [http://planetmath.org/encyclopedia/Skolemization.html Skolemization on PlanetMath.org]
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| * [http://demonstrations.wolfram.com/Skolemization/ Skolemization] by Hector Zenil, [[The Wolfram Demonstrations Project]].
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| * {{MathWorld |title=SkolemizedForm |urlname=SkolemizedForm}}
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| {{DEFAULTSORT:Skolem Normal Form}}
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| [[Category:Normal forms (logic)]]
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| [[Category:Model theory]]
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