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In order theory, a branch of mathematics, the least fixed point (lfp or LFP, sometimes also smallest fixed point) of a function from a partially ordered set to itself is the fixed point which is less than each other fixed point, according to the set's order. A function need not have a least fixed point, and cannot have more than one. For example, with the usual order on the real numbers, the least fixed point of the real function ''f''(''x'') = ''x''2 is ''x'' = 0 (since the only other fixed point is 1 and 0 < 1). In contrast, ''f''(''x'') = ''x''+1 has no fixed point at all, let alone a least one, and ''f''(''x'')=''x'' has infinitely many fixed points, but no least one. ==Applications== Many fixed-point theorems yield algorithms for locating the least fixed point. Least fixed points often have desirable properties that arbitrary fixed points do not. In mathematical logic and computer science, the least fixed point is related to making recursive definitions (see domain theory and/or denotational semantics for details). Immerman 〔N. Immerman, Relational queries computable in polynomial time, Information and Control 68 (1–3) (1986) 86–104.〕〔 Revised version in ''Information and Control'', 68 (1986), 86–104.〕 and Vardi independently showed the descriptive complexity result that the polynomial-time computable properties of linearly ordered structures are definable in FO(LFP), i.e. in first-order logic with a least fixed point operator. However, FO(LFP) is too weak to express all polynomial-time properties of unordered structures (for instance that a structure has even size). 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「least fixed point」の詳細全文を読む スポンサード リンク
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