Skolem Normal Form

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.

A formula of first-order logic is in Skolem normal form (named after Thoralf Skolem) if it is in conjunctive prenex normal form with only universal first-order quantifiers. Every first-order 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 equivalent to the original one, but is equisatisfiable with it: it is satisfiable if and only if the original one is satisfiable.

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, can be changed to P(c), where c is a new constant.

More generally, Skolemization is performed by replacing every existentially quantified variable with a term whose function symbol is new (does not occur anywhere else in the formula). The variables of this term are as follows. If the formula is in prenex normal form, are the variables that are universally quantified and whose quantifiers precede that of . In general, they are the variables that are universally quantified 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 quantifers, precedes while does not. The formula obtained by this transformation is satisfiable if and only if the original formula is.