Equisatisfiability

In mathematical 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 neither is.The truth values of two equisatisfiable formulae may nevertheless disagree for a particular assignment of variables. As a result, equisatisfiability differs from logical equivalence, since two equivalent formulae always have the same models, whereas equisatisfiable ones need only share satisfiability status. More formally, the equisatisfiability meta formula is true if either the two subformulae are both satisfiable or if they both are not:
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 that preserve equisatisfiability are Skolemization and some translations into conjunctive normal form such as the Tseytin transformation.

Examples

A translation from propositional logic into propositional logic in which every binary disjunction is replaced by, where is a fresh variable is a transformation in which satisfiability is preserved: the original and resulting formulae are equisatisfiable. 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 when is false.