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Since the iterated predicates involved in calculating the partial fixed point are not in general monotone, the fixed-point may not always exist. FO(LFP,X), least fixed-point logic , is the set of formulas in FO(PFP,X) where the partial fixed point is taken only over such formulas φ that only contain positive occurrences of P (that is ...
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 ...
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.
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.
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 ...
In mathematics, Lawvere's fixed-point theorem is an important result in category theory. [1] It is a broad abstract generalization of many diagonal arguments in mathematics and logic, such as Cantor's diagonal argument, Cantor's theorem, Russell's paradox, Gödel's first incompleteness theorem, Turing's solution to the Entscheidungsproblem, and Tarski's undefinability theorem.
The sequence F k used in this proof corresponds to the Kleene chain in the proof of the Kleene fixed-point theorem. The second part of the first recursion theorem follows from the first part. The assumption that Φ is a recursive operator is used to show that the fixed point of Φ is the graph of a partial function.