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For example, ''x'' ∈ {1,2,3} (where ''x'' is the value of the [[Variable (programming)|variable]] ''x'' after an operation) could be refined to ''x'' ∈ {1,2}, then ''x'' ∈ {1}, and implemented as ''x'' := 1. Implementations of ''x'' := 2 and ''x'' := 3 would be equally acceptable in this case, using a different route for the refinement. However, we must be careful not to refine to ''x'' ∈ {} (equivalent to ''false'') since this is unimplementable; it is impossible to select a [[Element (mathematics)|member]] from the [[empty set]].
In [[type theory]], a '''refinement type'''<ref>{{cite conference|first1=T.|last1=Freeman|first2=F.|last2=Pfenning|title=Refinement types for ML|booktitle=Proceedings of the ACM Conference on Programming Language Design and Implementation|pages=268–277|year=1991}}</ref><ref>{{cite conference|first=S.|last=Hayashi|title=Logic of refinement types|booktitle=Proceedings of the Workshop on Types for Proofs and Programs|pages=157–172|year=1993}}</ref><ref>{{cite conference|first=E.|last=Denney|title=Refinement types for specification|booktitle=Proceedings of the IFIP International Conference on Programming Concepts and Methods|volume=125|pages=148–166|publisher=Chapman & Hall|year=1998}}</ref> is a type endowed with a predicate which is assumed to hold for any element of the refined type. Refinement types can express [[precondition]]s when used as [[function argument]]s or [[postcondition]]s when used as [[return type]]s: for instance, the type of a function which accepts natural numbers and returns natural numbers greater than 5 may be written as <math>f: \mathbb{N} \rarr \{n: \mathbb{N} | n > 5\}</math>. Refinement types are thus related to [[behavioral subtyping]].
The term [[Reification (computer science)|reification]] is also sometimes used (coined by [[Cliff Jones (computer scientist)|Cliff Jones]]). [[Retrenchment (computing)|Retrenchment]] is an alternative technique when formal refinement is not possible. The opposite of refinement is [[Abstraction (computer science)|abstraction]].
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[[Refinement calculus]] is one method for program refinement. The [[FermaT Transformation System]] is an industrial-strength implementation of refinement.
== References ==
[[Category:Formal methods]]▼
<references/>
{{soft-eng-stub}}
▲[[Category:Formal methods]]
[[fr:Raffinement]]
[[ja:詳細化]]
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