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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Monoidal t-norm logic ： ウィキペディア英語版 Monoidal t-norm logic Monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices;〔Ono (2003).〕 it extends the logic of commutative bounded integral residuated lattices (known as Höhle's monoidal logic, Ono's FLew, or intuitionistic logic without contraction) by the axiom of prelinearity. == Motivation == T-norms are binary functions on the real unit interval () which are often used to represent a conjunction connective in fuzzy logic. Every ''left-continuous'' t-norm $$ * has a unique residuum, that is, a function $\backslash Rightarrow$ such that for all ''x'', ''y'', and ''z'', :$x$ *y\le z if and only if $x\backslash le\; (y\backslash Rightarrow\; z).$ The residuum of a left-continuous t-norm can explicitly be defined as :$(x\backslash Rightarrow\; y)=\backslash sup\backslash .$ This ensures that the residuum is the largest function such that for all ''x'' and ''y'', :$x$ *(x\Rightarrow y)\le y. The latter can be interpreted as a fuzzy version of the modus ponens rule of inference. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold. Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation $\backslash neg\; x=(x\backslash Rightarrow\; 0).$ In this way, the left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives (see the section ''Standard semantics'' below) determine the truth values of complex propositional formulae in (). Formulae that always evaluate to 1 are then called ''tautologies'' with respect to the given left-continuous t-norm $$ *, or ''$$ *\mboxtautologies.'' The set of all $$ *\mboxtautologies is called the ''logic'' of the t-norm $$ *, since these formulae represent the laws of fuzzy logic (determined by the t-norm) which hold (to degree 1) regardless of the truth degrees of atomic formulae. Some formulae are tautologies with respect to ''all'' left-continuous t-norms: they represent general laws of propositional fuzzy logic which are independent of the choice of a particular left-continuous t-norm. These formulae form the logic MTL, which can thus be characterized as the ''logic of left-continuous t-norms.''〔Conjectured by Esteva and Godo who introduced the logic (2001), proved by Jenei and Montagna (2002).〕 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「Monoidal t-norm logic」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.