Completeness

In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.

Logical completeness

In logic, completeness is the converse of soundness for formal systems. A formal system has "completeness" when all tautologies are theorems whereas a formal system has "soundness" when all theorems are tautologies. Kurt Gödel, Leon Henkin, and Post all published proofs of completeness. (See History of the Church-Turing thesis.) A system is consistent if a proof never exists for both P and not P. The proof of Gödel's incompleteness theorem proves that no recursive system that is sufficiently powerful, such as the Peano axioms, can be both consistent and complete.

A language is expressively complete if it can express the subject matter for which it is intended.

A formal system is complete with respect to a property iff every sentence that has the property is a theorem.

A formal system is functionally complete if it has adequate logical connectives to express all of the theorems of the language.

A formal system is strongly complete or complete in the strong sense iff no sentence which is not a theorem can become a theorem through the addition of a new basic rule to the deductive apparatus of the formal system (a rule of inference or an axiom) without the system becoming unsound. First-order sentential calculus is strongly complete.

A formal system is extremely complete or complete in the extreme sense iff every sentence is a theorem.

A formal system is deductively complete iff there are no formulas constructed on the base of the system (the axioms) which are derivable by the rules of the system as theorems and which are not tautologies.

In one sense, a formal system S is syntactically complete or maximally complete iff for each formula A of the language of the system either A or ~A is a theorem of S. This is also called negation completeness. In another sense, a formal system is syntactically complete iff no unprovable schema can be added to it as an axiom schema without inconsistency. Truth-functional propositional logic is semantically, and syntactically complete. First-order predicate logic is semantically complete, but not syntactically or negation complete.^[1]

For a formal system S in formal language L, S is semantically complete iff every logically valid formula of L (every formula which is true under every interpretation of L) is a theorem of S. That is, $\models _{\mathrm {L} }A$ $\to$ $\ \vdash _{\mathrm {S} }A$ .^[1]

Mathematical completeness

In mathematics, "complete" is 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 or compactification.

In mathematical logic, a theory is complete, if it contains either $\scriptstyle S$ or $\scriptstyle \neg S$ for every sentence $\scriptstyle S$ in the language.

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, $\scriptstyle G\ \models \ P$ implies $\scriptstyle G\ \vdash \ P$ . The calculus is complete, if the same holds for the empty set of premises $\scriptstyle G\,=\,\varnothing$ (i.e., if all semantically true sentences of the logic can be proved).

A metric space (or uniform space) is complete if every Cauchy sequence in it converges. See complete space.

In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. If V is separable, it follows that any vector in V can be written as a (possibly infinite) linear combination of vectors from S. In the particular case of Hilbert spaces (or more generally, inner product spaces), an orthonormal basis is a set that is both complete and orthonormal.

A measure space is complete if every subset of every null set is measurable. See complete measure.

In commutative algebra, a commutative ring can be completed at an ideal (in the topology defined by the powers of the ideal). See Completion (ring theory).

More generally, any topological group can be completed at a decreasing sequence of open subgroups.

In statistics, a statistic is called complete if it does not allow an unbiased estimator of zero. See completeness (statistics).

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 diagram from a small category to C has a limit; it is cocomplete if every such functor has a colimit.

In order theory and related fields such as lattice and domain theory, completeness generally refers to the existence of certain suprema or infima of some partially ordered set. Notable special usages of the term include the concepts of complete Boolean algebra, complete lattice, and complete partial order (cpo). Furthermore, an ordered field is complete if every non-empty subset of it that has an upper bound within the field has a least upper bound within the field, which should be compared to the (slightly different) order-theoretical notion of bounded completeness. Up to isomorphism there is only one complete ordered field: the field of real numbers (but note that this complete ordered field, which is also a lattice, is not a complete lattice).

In algebraic geometry, an algebraic variety is complete if it satisfies an analog of compactness. See complete algebraic variety.

Computing

In algorithms, the notion of completeness refers to the ability of the algorithm to find a solution if one exists, and if not, reports that no solution is possible.
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.
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.
In software testing, completeness has for goal the functional verification of call graph (between software item) and control graph (inside each software item).

Economics, finance, and industry

Complete market
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.
Oil or gas well completion, the process of making a well ready for production.

References

^ ^a ^b Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971

[metalogic-1] Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971

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