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In model theory, a branch of mathematical logic, a complete theory ''T'' is said to satisfy NIP (or "not the independence property") if none of its formulae satisfy the independence property, that is if none of its formulae can pick out any given subset of an arbitrarily large finite set. ==Definition== Let ''T'' be a complete ''L''-theory. An ''L''-formula φ(''x'',''y'') is said to have the independence property (with respect to ''x'', ''y'') if in every model ''M'' of ''T'' there is, for each ''n'' = < ω, a family of tuples ''b''0,…,''b''''n''−1 such that for each of the 2''n'' subsets ''X'' of ''n'' there is a tuple ''a'' in ''M'' for which : The theory ''T'' is said to have the independence property if some formula has the independence property. If no ''L''-formula has the independence property then ''T'' is called dependent, or said to satisfy NIP. An ''L''-structure is said to have the independence property (respectively, NIP) if its theory has the independence theory (respectively, NIP). The terminology comes from the notion of independence in the sense of boolean algebras. In the nomenclature of Vapnik–Chervonenkis theory, we may say that a collection ''S'' of subsets of ''X'' ''shatters'' a set ''B'' ⊆ ''X'' if every subset of ''B'' is of the form ''B'' ∩ ''S'' for some ''S'' ∈ ''S''. Then ''T'' has the independence property if in some model ''M'' of ''T'' there is a definable family (''S''''a'' | ''a''∈''M''''n'') ⊆ ''M''''k'' that shatters arbitrarily large finite subsets of ''M''''k''. In other words, (''S''''a'' | ''a''∈''M''''n'') has infinite Vapnik–Chervonenkis dimension. 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「NIP (model theory)」の詳細全文を読む スポンサード リンク
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