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. 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.