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| '''Total functional programming''' (also known as '''strong functional programming''',<ref>This term is due to: {{Citation|last1=Turner|first1=D.A.|author-link=David Turner (computer scientist)|contribution=Elementary Strong Functional Programming|title=First International Symposium on Functional Programming Languages in Education|date=December 1995|journal=Springer LNCS|volume=1022|pages=1–13}}.</ref> to be contrasted with ordinary, or ''weak'' [[functional programming]]) is a [[computer programming|programming]] paradigm that restricts the range of programs to those that are [[Machine that always halts|provably terminating]].<ref name="TFP">{{Citation|last=Turner|first=D.A.|author-link=David Turner (computer scientist)|title=Total Functional Programming|journal=Journal of Universal Computer Science|volume=10|date=2004-07-28|pages=751–768|url=http://www.jucs.org/jucs_10_7/total_functional_programming|doi=10.3217/jucs-010-07-0751|issue=7}}</ref>
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| Termination is guaranteed by the following restrictions:
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| # A restricted form of [[recursion]], which operates only upon ‘reduced’ forms of its arguments, such as [[Walther recursion]], substructural recursion, or "strongly normalizing" as proven by abstract interpretation of code.<ref name="ETinESFP">{{Citation|last=Turner|first=D.A.|author-link=David Turner (computer scientist)|title=Ensuring Termination in ESFP|journal=Journal of Universal Computer Science|volume=6|date=2000-04-28|pages=474–488|url=http://www.jucs.org/jucs_6_4/ensuring_termination_in_esfp|doi=10.3217/jucs-006-04-0474|issue=4}}</ref>
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| # Every function must be a total (as opposed to [[partial function|partial]]) function. That is, it must have a definition for everything inside its domain.
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| #* There are several possible ways to extend commonly used partial functions such as division to be total: choosing an arbitrary result for inputs on which the function is normally undefined (such as <math>\forall x \in \mathbb{N}. x \div 0 = 0</math> for division); adding another argument to specify the result for those inputs; or excluding them by use of type system features such as [[refinement type]]s.<ref name="TFP"/>
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| These restrictions mean that total functional programming is not [[Turing-complete]]. However, the set of algorithms that can be used is still huge. For example, any algorithm for which an [[Upper bound|asymptotic upper bound]] can be calculated (by a program that itself only uses Walther recursion) can be trivially transformed into a provably-terminating function by using the upper bound as an extra argument decremented on each iteration or recursion.
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| For example, [[quicksort]] is not trivially shown to be substructural recursive, but it only recurses to a maximum depth of the length of the vector (in the worst-case O(n^2) case). A quicksort implementation on lists (which would be rejected by a substructural recursive checker) is:
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| <code> | |
| qsort [] = []
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| qsort [a] = [a]
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| qsort (a:as) = let
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| (lesser, greater) = partition a as
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| in qsort lesser ++ [a] ++ qsort greater
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| </code>
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| To make it substructural recursive using the length of the vector as a limit, we could do:
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| <code> | |
| qsort x = qsortSub x x
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| -- minimum case
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| qsortSub [] as = as -- shows termination
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| -- standard qsort cases
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| qsortSub (l:ls) [] = [] -- nonrecursive, so accepted
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| qsortSub (l:ls) [a] = [a] -- nonrecursive, so accepted
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| qsortSub (l:ls) (a:as) = let
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| (lesser, greater) = partition a as
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| -- recursive, but recurses on ls, which is a substructure of
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| -- its first input.
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| in qsortSub ls lesser ++ [a] ++ qsortSub ls greater
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| </code>
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| Some classes of algorithms that have no theoretical upper bound but have a practical upper bound (for example, some heuristic-based algorithms) can be programmed to "give up" after so many recursions, also ensuring termination.
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| Another outcome of total functional programming is that both [[strict evaluation]] and [[lazy evaluation]] result in the same behaviour, in principle; however, one or the other may still be preferable (or even required) for performance reasons.<ref>The differences between lazy and eager evaluation are discussed in: {{cite book|last=Granström|first=J. G.|title=Treatise on Intuitionistic Type Theory|series=Logic, Epistemology, and the Unity of Science|volume=7|year=2011|url=http://www.springer.com/philosophy/book/978-94-007-1735-0|isbn=978-94-007-1735-0}} See in particular pp. 86-91.</ref>
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| In total functional programming, a distinction is made between [[data]] and [[codata]]—the former is [[finitary]], while the latter is potentially infinite. Such potentially infinite data structures are used for applications such as [[I/O]]. Using codata entails the usage of such operations as [[corecursion]]. However, it is possible to do [[I/O]] in a total functional programming language (with [[dependent types]]) also without codata.<ref>{{Citation|last=Granström|first=J. G.|title=A New Paradigm for Component-based Development|journal=Journal of Software|volume=7|date=May 2012|pages= 1136–1148 |url=http://ojs.academypublisher.com/index.php/jsw/article/view/jsw070511361148|doi=10.4304/jsw.7.5.1136-1148|issue=5}}</ref>
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| Both [[Epigram (programming language)|Epigram]] and [[Charity (programming language)|Charity]] could be considered total functional programming languages, even though they don't work in the way Turner specifies in his paper. So could programming directly in plain [[System F]], in [[Martin-Löf type theory]] or the [[Calculus of Constructions]].
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Total Functional Programming}}
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| [[Category:Programming paradigms]]
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| [[Category:Functional programming]]
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| [[Category:Proof assistants]]
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