Бірінші реттік логикадағы Сколем нормалық түрі

Formalism of first order logic
In mathematical logic, a formula of first order logic is in Skolem normal form if it is in prenex normal form with only universal first order quantifiers. Every first order formula may 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 equivalent to the original one, but is equisatisfiable with it: it is satisfiable if and only if the original one is satisfiable. Reduction to Skolem normal form is a method for removing existential quantifiers from formal logic statements, often performed as the first step in an automated theorem prover.

The simplest form of Skolemization is for existentially quantified variables that are not inside the scope of a universal quantifier. These may be replaced simply by creating new constants. For example, may be changed to , where is a new constant (does not occur anywhere else in the formula). More generally, Skolemization is performed by replacing every existentially quantified variable with a term whose function symbol is new. The variables of this term are as follows. If the formula is in prenex normal form, then are the variables that are universally quantified and whose quantifiers precede that of In general, they are the variables that are quantified universally (we assume we get rid of existential quantifiers in order, so all existential quantifiers before have been removed) and such that occurs in the scope of their quantifiers. The function 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. As an example, the formula is not in Skolem normal form because it contains the existential quantifier Skolemization replaces with , where is a new function symbol, and removes the quantification over The resulting formula is The Skolem term contains , but not , because the quantifier to be removed is in the scope of , but not in that of ; since this formula is in prenex normal form, this is equivalent to saying that, in the list of quantifiers, precedes while does not. The formula obtained by this transformation is satisfiable if and only if the original formula is.

One of the uses of Skolemization is within 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 may be generated. For example, if occurs in a tableau, where are the free variables of , then may 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 may be extended, by adding a suitable interpretation of , to a model of the new formula. This form of Skolemization is 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 tableaux may implicitly place the formula in the 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 may be used is applying the same Skolem function symbol for formulae that are identical up to variable renaming. Another use is in the resolution method for first order logic, where formulas are represented as sets of clauses understood to be universally quantified. (For an example see drinker paradox.) An important result in model theory is the Löwenheim–Skolem theorem, which can be proven via Skolemizing the theory and closing under the resulting Skolem functions.

In general, if is a theory and for each formula with free variables there is an n ary function symbol that is provably a Skolem function for , then is called a Skolem theory. Every Skolem theory is model complete, i. e. every substructure of a model is an 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 prime model over A.

Skolem normal form is named after the late Norwegian mathematician Thoralf Skolem.