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For a relational signature X, FO[PFP](X) is the set of formulas formed from X using first-order connectives and predicates, second-order variables as well as a partial fixed point operator used to form formulas of the form [, ], where is a second-order variable, a tuple of first-order variables, a tuple of terms and the lengths of and coincide with the arity of .
The class of all problems computable in polynomial space, PSPACE, can be characterised by augmenting first-order logic with a more expressive partial fixed-point operator. Partial fixed-point logic, FO[PFP], is the extension of first-order logic with a partial fixed-point operator, which expresses the fixed-point of a formula if there is one ...
The function f(x) = x 2 − 4 has two fixed points, shown as the intersection with the blue line; its least one is at 1/2 − √ 17 /2.. 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 ...
A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition. [1] For example, the Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, fixed-point iteration will always converge to a fixed point.
In theoretical computer science, the modal μ-calculus (Lμ, L μ, sometimes just μ-calculus, although this can have a more general meaning) is an extension of propositional modal logic (with many modalities) by adding the least fixed point operator μ and the greatest fixed point operator ν, thus a fixed-point logic.
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...
Several variations of fixed point logics have been studied. In least fixed point logic, the right hand side of the operator in the defining formula must use the predicate only positively (that is, each appearance should be nested within an even number of negations) in order to make the least fixed point well defined. In another variant with ...
Any fixed-point combinator is also a non-standard one, but not all non-standard fixed-point combinators are fixed-point combinators because some of them fail to satisfy the fixed-point equation that defines the "standard" ones. These combinators are called strictly non-standard fixed-point combinators; an example is the following combinator: