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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 T-norm fuzzy logics ： ウィキペディア英語版 T-norm fuzzy logics T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics which takes the real unit interval () for the system of truth values and functions called t-norms for permissible interpretations of conjunction. They are mainly used in applied fuzzy logic and fuzzy set theory as a theoretical basis for approximate reasoning. T-norm fuzzy logics belong in broader classes of fuzzy logics and many-valued logics. In order to generate a well-behaved implication, the t-norms are usually required to be left-continuous; logics of left-continuous t-norms further belong in the class of substructural logics, among which they are marked with the validity of the ''law of prelinearity'', (''A'' → ''B'') ∨ (''B'' → ''A''). Both propositional and first-order (or higher-order) t-norm fuzzy logics, as well as their expansions by modal and other operators, are studied. Logics which restrict the t-norm semantics to a subset of the real unit interval (for example, finitely valued Łukasiewicz logics) are usually included in the class as well. Important examples of t-norm fuzzy logics are monoidal t-norm logic MTL of all left-continuous t-norms, basic logic BL of all continuous t-norms, product fuzzy logic of the product t-norm, or the nilpotent minimum logic of the nilpotent minimum t-norm. Some independently motivated logics belong among t-norm fuzzy logics, too, for example Łukasiewicz logic (which is the logic of the Łukasiewicz t-norm) or Gödel–Dummett logic (which is the logic of the minimum t-norm). == Motivation == As members of the family of fuzzy logics, t-norm fuzzy logics primarily aim at generalizing classical two-valued logic by admitting intermediary truth values between 1 (truth) and 0 (falsity) representing ''degrees'' of truth of propositions. The degrees are assumed to be real numbers from the unit interval (). In propositional t-norm fuzzy logics, propositional connectives are stipulated to be truth-functional, that is, the truth value of a complex proposition formed by a propositional connective from some constituent propositions is a function (called the ''truth function'' of the connective) of the truth values of the constituent propositions. The truth functions operate on the set of truth degrees (in the standard semantics, on the () interval); thus the truth function of an ''n''-ary propositional connective ''c'' is a function ''F''''c'': ()''n'' → (). Truth functions generalize truth tables of propositional connectives known from classical logic to operate on the larger system of truth values. T-norm fuzzy logics impose certain natural constraints on the truth function of conjunction. The truth function $$ *\colon()^2\to() of conjunction is assumed to satisfy the following conditions: * ''Commutativity'', that is, $x$ *y=y *x for all ''x'' and ''y'' in (). This expresses the assumption that the order of fuzzy propositions is immaterial in conjunction, even if intermediary truth degrees are admitted. * ''Associativity'', that is, $(x$ *y) *z = x *(y *z) for all ''x'', ''y'', and ''z'' in (). This expresses the assumption that the order of performing conjunction is immaterial, even if intermediary truth degrees are admitted. * ''Monotony'', that is, if $x\; \backslash le\; y$ then $x$ *z \le y *z for all ''x'', ''y'', and ''z'' in (). This expresses the assumption that increasing the truth degree of a conjunct should not decrease the truth degree of the conjunction. * ''Neutrality of 1'', that is, $1$ *x = x for all ''x'' in (). This assumption corresponds to regarding the truth degree 1 as full truth, conjunction with which does not decrease the truth value of the other conjunct. Together with the previous conditions this condition ensures that also $0$ *x = 0 for all ''x'' in (), which corresponds to regarding the truth degree 0 as full falsity, conjunction with which is always fully false. * ''Continuity'' of the function $$ * (the previous conditions reduce this requirement to the continuity in either argument). Informally this expresses the assumption that microscopic changes of the truth degrees of conjuncts should not result in a macroscopic change of the truth degree of their conjunction. This condition, among other things, ensures a good behavior of (residual) implication derived from conjunction; to ensure the good behavior, however, ''left''-continuity (in either argument) of the function $$ * is sufficient.〔Esteva & Godo (2001)〕 In general t-norm fuzzy logics, therefore, only left-continuity of $$ * is required, which expresses the assumption that a microscopic ''decrease'' of the truth degree of a conjunct should not macroscopically decrease the truth degree of conjunction. These assumptions make the truth function of conjunction a left-continuous t-norm, which explains the name of the family of fuzzy logics (''t-norm based''). Particular logics of the family can make further assumptions about the behavior of conjunction (for example, Gödel logic requires its idempotence) or other connectives (for example, the logic IMTL requires the involutiveness of negation). All left-continuous t-norms $$ * have a unique residuum, that is, a binary function $\backslash Rightarrow$ such that for all ''x'', ''y'', and ''z'' in (), :$x$ *y\le z if and only if $x\backslash le\; y\backslash Rightarrow\; z.$ The residuum of a left-continuous t-norm can explicitly be defined as :$(x\backslash Rightarrow\; y)=\backslash sup\backslash .$ This ensures that the residuum is the pointwise largest function such that for all ''x'' and ''y'', :$x$ *(x\Rightarrow y)\le y. The latter can be interpreted as a fuzzy version of the modus ponens rule of inference. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold. Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation $\backslash neg\; x=(x\backslash Rightarrow\; 0)$ or bi-residual equivalence $x\backslash Leftrightarrow\; y\; =\; (x\backslash Rightarrow\; y)$ *(y\Rightarrow x). Truth functions of propositional connectives may also be introduced by additional definitions: the most usual ones are the minimum (which plays a role of another conjunctive connective), the maximum (which plays a role of a disjunctive connective), or the Baaz Delta operator, defined in () as $\backslash Delta\; x\; =\; 1$ if $x=1$ and $\backslash Delta\; x\; =\; 0$ otherwise. In this way, a left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives determine the truth values of complex propositional formulae in (). Formulae that always evaluate to 1 are called ''tautologies'' with respect to the given left-continuous t-norm $$ *, or ''$$ *\mboxtautologies.'' The set of all $$ *\mboxtautologies is called the ''logic'' of the t-norm $$ *, as 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 a larger class of left-continuous t-norms; the set of such formulae is called the logic of the class. Important t-norm logics are the logics of particular t-norms or classes of t-norms, for example: * Łukasiewicz logic is the logic of the Łukasiewicz t-norm $x$ *y = \max(x+y-1,0) * Gödel–Dummett logic is the logic of the minimum t-norm $x$ *y = \min(x,y) * Product fuzzy logic is the logic of the product t-norm $x$ *y = x\cdot y * Monoidal t-norm logic MTL is the logic of (the class of) ''all'' left-continuous t-norms * Basic fuzzy logic BL is the logic of (the class of) all ''continuous'' t-norms It turns out that many logics of particular t-norms and classes of t-norms are axiomatizable. The completeness theorem of the axiomatic system with respect to the corresponding t-norm semantics on () is then called the ''standard completeness'' of the logic. Besides the standard real-valued semantics on (), the logics are sound and complete with respect to general algebraic semantics, formed by suitable classes of prelinear commutative bounded integral residuated lattices. 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「T-norm fuzzy logics」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.