Skolemisaatiolla

Skolemisaatiolla is a concept in formal logic, specifically within the field of automated theorem proving and logic programming. It refers to the process of transforming a logical formula into an equivalent one where all existential quantifiers appear at the beginning of the formula and are not nested. This transformed formula is then free of existential quantifiers in the inner part of the structure.

The transformation is achieved by replacing existentially quantified variables with Skolem functions or Skolem constants. A Skolem constant is introduced for an existentially quantified variable that does not depend on any other variables. A Skolem function is used when the existentially quantified variable's existence depends on other variables. The arguments of the Skolem function are the variables that the existentially quantified variable depends on.

The Skolemization process is crucial because many automated reasoning systems, particularly those based on resolution, can only handle formulas in clausal form. Clausal form is a conjunctive normal form where all quantifiers have been removed. Skolemization is a key step in converting first-order logic formulas into a form suitable for these resolution-based theorem provers. By eliminating existential quantifiers, Skolemization preserves the satisfiability of the original formula, meaning that if the original formula has a model, the Skolemized formula also has a model, and vice-versa. It is a fundamental technique for making logical reasoning computationally tractable.