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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 Atomic formula ： ウィキペディア英語版 Atomic formula In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives. The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formula are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model. ==Atomic formula in first-order logic== The well-formed terms and propositions of ordinary first-order logic have the following syntax: Terms: * $t\; \backslash equiv\; c\; \backslash mid\; x\; \backslash mid\; f\; (t\_,\; ...,\; t\_)$, that is, a term is recursively defined to be a constant ''c'' (a named object from the domain of discourse), or a variable ''x'' (ranging over the objects in the domain of discourse), or an ''n''-ary function ''f'' whose arguments are terms ''t''''k''. Functions map tuples of objects to objects. Propositions: * $A,\; B,\; ...\; \backslash equiv\; P\; (t\_,\; ...,\; t\_)\; \backslash mid\; A\; \backslash wedge\; B\; \backslash mid\; \backslash top\; \backslash mid\; A\; \backslash vee\; B\; \backslash mid\; \backslash bot\; \backslash mid\; A\; \backslash supset\; B\; \backslash mid\; \backslash forall\; x.\backslash \; A\; \backslash mid\; \backslash exists\; x.\backslash \; A$, that is, a proposition is recursively defined to be an ''n''-ary predicate ''P'' whose arguments are terms ''t''''k'', or an expression composed of logical connectives (and, or) and quantifiers (for-all, there-exists) used with other propositions. An atomic formula or atom is simply a predicate applied to a tuple of terms; that is, an atomic formula is a formula of the form ''P'' (''t''1, …, ''t''''n'') for ''P'' a predicate, and the ''t''''n'' terms. All other well-formed formulae are obtained by composing atoms with logical connectives and quantifiers. For example, the formula ∀''x. P'' (''x'') ∧ ∃''y. Q'' (''y'', ''f'' (''x'')) ∨ ∃''z. R'' (''z'') contains the atoms * $P\; (x)$ * $Q\; (y,\; f\; (x))$ * $R\; (z)$ When all of the terms in an atom are ground terms, then the atom is called a ground atom or ''ground predicate''. 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「Atomic formula」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.