Skolemization

In mathematical logic, reduction to Skolem normal form (SNF) 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 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.

The simplest form of Skolemization is for existentially quantified variables which are not inside the scope of a universal quantifier. These may be replaced simply by creating new constants. For example, ${\displaystyle \exists xP(x)}$ may be changed to ${\displaystyle P(c)}$, where ${\displaystyle c}$ is a new constant (does not occur anywhere else in the formula).

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