Editing Monad (functional programming) (section)

{{short description|Design pattern in functional programming to build generic types}}
{{for|the concept in category theory|Monad (category theory)}}

In [[functional programming]], '''monads''' are a way to structure computations as a sequence of steps, where each step not only produces a value but also some extra information about the computation, such as a potential failure, non-determinism, or side effect. More formally, a monad is a [[type constructor]] M equipped with two operations, {{Code|return : <A>(a : A) -> M(A)|typescript}} which lifts a value into the monadic context, and {{Code|bind : <A,B>(m_a : M(A), f : A -> M(B)) -> M(B)|typescript}} which chains monadic computations. In simpler terms, monads can be thought of as [[Interface (computing)|interfaces]] implemented on type constructors, that allow for functions to abstract over various type constructor variants that implement monad (e.g. {{Code|Option}}, {{Code|List}}, etc.).<ref name="RealWorldHaskell">{{cite book|last1=O'Sullivan|first1=Bryan|url=http://book.realworldhaskell.org/|title=Real World Haskell|last2=Goerzen|first2=John|last3=Stewart|first3=Don|publisher=O'Reilly Media|year=2009|isbn=978-0596514983|location=Sebastopol, California|at=chapter 14|chapter=Monads|chapter-url=http://book.realworldhaskell.org/read/monads.html}}</ref><ref name="Wadler1990">{{cite conference|last=Wadler|first=Philip|author-link=Philip Wadler|date=June 1990|title=Comprehending Monads|conference=ACM Conference on LISP and Functional Programming|location=Nice, France|citeseerx=10.1.1.33.5381}}</ref>

Both the concept of a monad and the term originally come from [[category theory]], where a monad is defined as an [[endofunctor]] with additional structure.{{efn|More formally, a monad is a [[monoid (category theory)|monoid]] in the category of [[endofunctor]]s.}}{{efn|Due to the fact that functions on multiple [[free variable]]s are common in programming, monads as described in this article are technically what category theorists would call [[strong monad]]s.<ref name="Moggi1991" />}} Research beginning in the late 1980s and early 1990s established that monads could bring seemingly disparate computer-science problems under a unified, functional model. Category theory also provides a few formal requirements, known as the '''[[#Definition|monad laws]]''', which should be satisfied by any monad and can be used to [[formal verification|verify]] monadic code.<ref name="Moggi1991">{{cite journal | author-link = Eugenio Moggi | last = Moggi | first = Eugenio | year = 1991 | title = Notions of computation and monads | journal = Information and Computation | volume = 93 | issue = 1 | pages = 55–92 | url = http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf | citeseerx = 10.1.1.158.5275 | doi=10.1016/0890-5401(91)90052-4}}</ref><ref name="Wadler1992">{{cite conference | author-link = Philip Wadler | last = Wadler | first = Philip | title = The essence of functional programming | conference = 19th Annual ACM Symposium on Principles of Programming Languages | location = Albuquerque, New Mexico | date = January 1992 | citeseerx = 10.1.1.38.9516}}</ref>

Since monads make [[semantics (computer science)|semantics]] explicit for a kind of computation, they can also be used to implement convenient language features. Some languages, such as [[Haskell (programming language)|Haskell]], even offer pre-built definitions in their core [[library (computing)|libraries]] for the general monad structure and common instances.<ref name="RealWorldHaskell" /><ref name="GentleIntroHaskell">{{cite book | author-link1 = Paul Hudak | last1 = Hudak | first1 = Paul | last2 = Peterson | first2 = John | last3 = Fasel | first3 = Joseph | title = A Gentle Introduction to Haskell 98 | year = 1999 | chapter = About Monads | at = chapter 9 | chapter-url = https://www.haskell.org/tutorial/monads.html | url = https://www.haskell.org/tutorial/index.html}}</ref>