Skolem normal form


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.[1]

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.

Examples Edit

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.

How Skolemization works Edit

Skolemization works by applying a second-order equivalence together with the definition of first-order satisfiability. The equivalence provides a way for "moving" an existential quantifier before a universal one.



  is a function that maps   to  .

Intuitively, the sentence "for every   there exists a   such that  " is converted into the equivalent form "there exists a function   mapping every   into a   such that, for every   it holds that  ".

This equivalence is useful because the definition of first-order satisfiability implicitly existentially quantifies over functions interpreting the function symbols. In particular, a first-order formula   is satisfiable if there exists a model   and an evaluation   of the free variables of the formula that evaluate the formula to true. The model contains the interpretation of all function symbols; therefore, Skolem functions are implicitly existentially quantified. In the example above,   is satisfiable if and only if there exists a model  , which contains an interpretation for  , such that   is true for some evaluation of its free variables (none in this case). This may be expressed in second order as  . By the above equivalence, this is the same as the satisfiability of  .

At the meta-level, first-order satisfiability of a formula   may be written with a little abuse of notation as  , where   is a model,   is an evaluation of the free variables, and   means that   is true in   under  . Since first-order models contain the interpretation of all function symbols, any Skolem function that   contains is implicitly existentially quantified by  . As a result, after replacing existential quantifiers over variables by existential quantifiers over functions at the front of the formula, the formula still may be treated as a first-order one by removing these existential quantifiers. This final step of treating   as   may be completed because functions are implicitly existentially quantified by   in the definition of first-order satisfiability.

Correctness of Skolemization may be shown on the example formula   as follows. This formula is satisfied by a model   if and only if, for each possible value for   in the domain of the model, there exists a value for   in the domain of the model that makes   true. By the axiom of choice, there exists a function   such that  . As a result, the formula   is satisfiable, because it has the model obtained by adding the interpretation of   to  . This shows that   is satisfiable only if   is satisfiable as well. Conversely, if   is satisfiable, then there exists a model   that satisfies it; this model includes an interpretation for the function   such that, for every value of  , the formula   holds. As a result,   is satisfied by the same model because one may choose, for every value of  , the value  , where   is evaluated according to  .

Uses of Skolemization Edit

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.[2]

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.[3]

Skolem theories Edit

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.[4]

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.

History Edit

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

See also Edit

Notes Edit

  1. ^ "Normal Forms and Skolemization" (PDF). Max-Planck-Institut für Informatik. Retrieved 15 December 2012.
  2. ^ Reiner Hähnle. Tableaux and related methods. Handbook of Automated Reasoning.
  3. ^ Scott Weinstein, The Lowenheim-Skolem Theorem, lecture notes (2009). Accessed 6 January 2023.
  4. ^ "Sets, Models and Proofs" (3.3) by I. Moerdijk and J. van Oosten

References Edit

External links Edit