Ең кіші түйін нүктесі және оның қолданыстары

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 ("poset" for short) to itself is the fixed point which is less than each other fixed point, according to the order of the poset. A function need not have a least fixed point, but if it does then the least fixed point is unique.

With the usual order on the real numbers, the least fixed point of the real function f(x) = x2 is x = 0 (since the only other fixed point is 1 and 0 < 1). In contrast, f(x) = x + 1 has no fixed points at all, so has no least one, and f(x) = x has infinitely many fixed points, but has no least one. Let be a directed graph and be a vertex. The set of vertices accessible from can be defined as the least fixed point of the function , defined as
The set of vertices which are co accessible from is defined by a similar least fix point. The strongly connected component of is the intersection of those two least fixed points. Let be a context free grammar. The set of symbols which produces the empty string can be obtained as the least fixed point of the function , defined as , where denotes the power set of .

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.

The greatest fixed point of a function can be defined analogously to the least fixed point, as the fixed point which is greater than any other fixed point, according to the order of the poset. In computer science, greatest fixed points are much less commonly used than least fixed points. Specifically, the posets found in domain theory usually do not have a greatest element, hence for a given function, there may be multiple, mutually incomparable maximal fixed points, and the greatest fixed point of that function may not exist. To address this issue, the optimal fixed point has been defined as the most defined fixed point compatible with all other fixed points. The optimal fixed point always exists, and is the greatest fixed point if the greatest fixed point exists. The optimal fixed point allows formal study of recursive and corecursive functions that do not converge with the least fixed point. Unfortunately, whereas Kleene's recursion theorem shows that the least fixed point is effectively computable, the optimal fixed point of a computable function may be a non computable function.