Two sentences or theories are equisatisfiable when one has a model if and only if the other has a model. They need not be true in the same interpretations or even use the same signature.

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Equisatisfiability by Wikipedia Bot 1
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Equisatisfiability is a concept in logic and computer science, particularly within the fields of propositional logic and satisfiability (SAT) problems. Two logical formulas are said to be equisatisfiable if they have the same satisfiability status; that is, if one formula is satisfiable (there exists an assignment of truth values to its variables that makes the formula true), then the other formula is also satisfiable, and vice versa.
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