Definitions

# Completeness

[kuhm-pleet]
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.
• 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, $models_\left\{mathrm L\right\} A$ $vdash_\left\{mathrm S\right\} A$.

## 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.

## 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

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