Look up completeness in Wiktionary, the free dictionary.
In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. This is made precise in various ways, several of which have a related notion of completion. "Complete" here is just a term that takes on specific meanings in specific situations, and not every situation in which a type of "completion" occurs is called a "completion". See, for example, algebraically closed field, compactification, Gödel's completeness theorem or Gödel's incompleteness theorem. Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ... In mathematics, compactification is the process or result of enlarging a topological space to make it compact. ... Gödels completeness theorem is an important theorem in mathematical logic which was first proved by Kurt Gödel in 1929. ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...
More generally, any topological group can be completed at a decreasing sequence of open subgroups.
In auditing, completeness is one of the financial statement assertions that have to be ensured. For example, auditing classes of transactions. Rental expense which includes 12-month or 52-week payments should be all booked according to the terms agreed in the tenancy agreement.
In graph theory, a complete graph is an undirected graph in which every pair of vertices has exactly one edge connecting them.
In category theory, a category C is complete if every functor from a small category to C has a limit; it is cocomplete if every such functor has a colimit. For more information, see the given article on limits in category theory.
In mathematical logic, a theory is complete, if it contains either S or for every sentenceS in the language. A system is consistent if a proof never exists for both P and not P. Gödel's incompleteness theorem says that no system as powerful as the Peano axioms can be both consistent and complete. See also below for another notion of completeness in logic.
In mathematical logic, a formal calculus for a logic L is strongly complete with respect to a certain semantics of L, if every statement P that follows semantically from a set of premises G can be derived syntactically from these premises within the calculus. Formally, implies . The calculus is complete, if the same holds for the empty set of premises (i.e., if all tautologies of the logic can be proven). First-order logic is strongly complete in this sense. Even when working with classical logic, this is not equivalent to the notion of completeness introduced above (both a statement and its negation might not be tautologies with respect to the logic). The reverse implication is called soundness.
In computational complexity theory, a problem P is complete for a complexity class C, under a given type of reduction, if P is in C, and every problem in C reduces to P using that reduction. For example, each problem in the class NP-complete is complete for the class NP, under polynomial-time, many-one reduction.
A decision procedure (or decision algorithm, see decision problem) is complete if, whenever the answer is "yes", the algorithm finds it correctly. It is sound if every time the algorithm answers "yes", it is the correct answer.
In computing, a data-entry field can autocomplete the entered data based on the prefix typed into the field; that capability is known as autocompletion.
If this completion procedure is applied to a normed vector space, one obtains a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, one obtains a Hilbert space containing the original space as a dense subspace.
Note that completeness is a property of the metric and not of the topology, meaning that a completemetric space can be homeomorphic to a non-complete one.
Completely metrizable spaces can be characterized as those spaces which can be written as an intersection of countably many open subsets of some completemetric space.
In complexity theory, the complexity class P-complete is a set of decision problems and is useful in the analysis of which problems can be efficiently solved on parallel computers.
A decision problem is in P-complete if it is complete for P, meaning that it is in P, and that every problem in P can be reduced to it in polylogarithmic time on a parallel computer with a polynomial number of processors.
In other words, a problem A is in P-complete if, for each problem B in P, there are constants c and k such that B can be reduced to A in time O((log n)
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