Equisatisfiability


In logic, two formulae are equisatisfiable if the first formula is satisfiable whenever the second is and vice versa; in other words, either both formulae are satisfiable or both are not. Equisatisfiable formulae may disagree, however, for a particular choice of variables. As a result, equisatisfiability is different from logical equivalence, as two equivalent formulae always have the same models.
Equisatisfiability is generally used in the context of translating formulae, so that one can define a translation to be correct if the original and resulting formulae are equisatisfiable. Examples of translations involving this concept are Skolemization and some translations into conjunctive normal form.

Examples

A translation from propositional logic into propositional logic in which every binary disjunction is replaced by, where is a new variable is a transformation in which satisfiability is preserved: the original and resulting formulae are equisatisfiable. Note that these two formulae are not equivalent: the first formula has the model in which is true while and are false, but this is not a model of the second formula, in which has to be true in this case.