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In mathematical logic, satisfiability and validity are elementary concepts of semantics. A formula is ''satisfiable'' if it is possible to find an interpretation (model) that makes the formula true.〔See, for example, Boolos and Jeffrey, 1974, chapter 11.〕 A formula is ''valid'' if all interpretations make the formula true. The opposites of these concepts are unsatisfiability and invalidity, that is, a formula is ''unsatisfiable'' if none of the interpretations make the formula true, and ''invalid'' if some such interpretation makes the formula false. These four concepts are related to each other in a manner exactly analogous to Aristotle's square of opposition. The four concepts can be raised to apply to whole theories: a theory is satisfiable (valid) if one (all) of the interpretations make(s) each of the axioms of the theory true, and a theory is unsatisfiable (invalid) if all (one) of the interpretations make(s) each of the axioms of the theory false. It is also possible to consider only interpretations that make all of the axioms of a second theory true. This generalization is commonly called satisfiability modulo theories. The question whether a sentence in propositional logic is satisfiable is a decidable problem. In general, the question whether sentences in first-order logic are satisfiable is not decidable. In universal algebra and equational theory, the methods of term rewriting, congruence closure and unification are used to attempt to decide satisfiability. Whether a particular theory is decidable or not depends whether the theory is variable-free or on other conditions. ==Reduction of validity to satisfiability== For classical logics, it is generally possible to reexpress the question of the validity of a formula to one involving satisfiability, because of the relationships between the concepts expressed in the above square of opposition. In particular φ is valid if and only if ¬φ is unsatisfiable, which is to say it is not true that ¬φ is satisfiable. Put another way, φ is satisfiable if and only if ¬φ is invalid. For logics without negation, such as the positive propositional calculus, the questions of validity and satisfiability may be unrelated. In the case of the positive propositional calculus, the satisfiability problem is trivial, as every formula is satisfiable, while the validity problem is co-NP complete. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Satisfiability」の詳細全文を読む スポンサード リンク
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