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In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them.〔van Oosten 2000〕 Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another. Realizability can be seen as a formalization of the BHK interpretation of intuitionistic logic; in realizability the notion of "proof" (which is left undefined in the BHK interpretation) is replaced with a formal notion of "realizer". Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable. The realizer, however, usually gives more information about the formula than a formal proof would directly provide. Beyond giving insight into intuitionistic provability, realizability can be applied to prove the disjunction and existence properties for intuitionistic theories and to extract programs from proofs, as in proof mining. It is also related to topos theory via the realizability topos. == Example: realizability by numbers == Kleene's original version of realizability uses natural numbers as realizers for formulas in Heyting arithmetic. The following clauses are used to define a relation "''n'' realizes ''A''" between natural numbers ''n'' and formulas ''A'' in the language of Heyting arithmetic. A few pieces of notation are required: first, an ordered pair (''n'',''m'') is treated as a single number using a fixed effective pairing function; second, for each natural number ''n'', φ''n'' is the computable function with index ''n''. * A number ''n'' realizes an atomic formula ''s''=''t'' if and only if ''s''=''t'' is true. Thus every number realizes a true equation, and no number realizes a false equation. * A pair (''n'',''m'') realizes a formula ''A''∧''B'' if and only if ''n'' realizes ''A'' and ''m'' realizes ''B''. Thus a realizer for a conjunction is a pair of realizers for the conjuncts. * A pair (''n'',''m'') realizes a formula ''A''∨''B'' if and only if the following hold: ''n'' is 0 or 1; and if ''n'' is 0 then ''m'' realizes ''A''; and if ''n'' is 1 then ''m'' realizes ''B''. Thus a realizer for a disjunction explicitly picks one of the disjuncts (with ''n'') and provides a realizer for it (with ''m''). * A number ''n'' realizes a formula ''A''→''B'' if and only if, for every ''m'' that realizes ''A'', φ''n''(''m'') realizes ''B''. Thus a realizer for an implication is a computable function that takes a realizer for the hypothesis and produces a realizer for the conclusion. * A pair (''n'',''m'') realizes a formula (∃ ''x'')''A''(''x'') if and only if ''m'' is a realizer for ''A''(''n''). Thus a realizer for an existential formula produces an explicit witness for the quantifier along with a realizer for the formula instantiated with that witness. * A number ''n'' realizes a formula (∀ ''x'')''A''(''x'') if and only if, for all ''m'', φ''n''(''m'') is defined and realizes ''A''(''m''). Thus a realizer for a universal statement is a computable function that produces, for each ''m'', a witness for the formula instantiated with ''m''. With this definition, the following theorem is obtained:〔van Oosten 2000, p. 7〕 :Let ''A'' be a sentence of Heyting arithmetic (HA). If HA proves ''A'' then there is an ''n'' such that ''n'' realizes ''A''. On the other hand, there are formulas that are realized but which are not provable in HA, a fact first established by Rose.〔Rose 1953〕 Further analysis of the method can be used to prove that HA has the "disjunction and existence properties":〔van Oosten 2000, p. 6〕 * If HA proves a sentence (∃ ''x'')''A''(''x''), then there is an ''n'' such that HA proves ''A''(''n'') * If HA proves a sentence ''A''∨''B'', then HA proves ''A'' or HA proves ''B''. 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「Realizability」の詳細全文を読む スポンサード リンク
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