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In [[computer science]], '''graph reduction''' implements an efficient version of non-strict evaluation, an [[evaluation strategy]] where the arguments to a function are not immediately evaluated. This form of non-strict evaluation is also known as [[lazy evaluation]] and used in [[functional programming|functional programming languages]]. The technique was first developed by [[Chris Wadsworth]] in 1971.
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== Motivation ==
A simple example of evaluating an arithmetic expression follows:
 
:<math>
\begin{align}
& {} \qquad ((2+2)+(2+2))+(3+3) \\
& {} =((2+2)+(2+2))+ 6 \\
& {} =((2+2)+ 4)+6 \\
& {} =(4+4)+6 \\
& {} =8+6 \\
& {} =14
\end{align}
</math>
 
The above reduction sequence employs a strategy known as [[outermost tree reduction]]. The same expression can be evaluated using [[innermost tree reduction]], yielding the reduction sequence:
 
:<math>
\begin{align}
& {} \qquad ((2+2)+(2+2))+(3+3) \\
& {} = ((2+2)+4)+(3+3) \\
& {} = (4+4)+(3+3) \\
& {} = (4+4)+6 \\
& {} = 8+6 \\
& {} = 14
\end{align}
</math>
 
Notice that the reduction order is made explicit by the addition of parentheses. This expression could also have been simply evaluated right to left, because addition is an [[associative]] operation.
 
Represented as a [[Tree data structure|tree]], the expression above looks like this:
 
[[Image:Expression Tree.svg|300px]]
 
This is where the term tree reduction comes from. When represented as a tree, we can think of innermost reduction as working from the bottom up, while outermost works from the top down.
 
The expression can also be represented as a [[graph (data structure)|graph]], allowing sub-expressions to be shared:
 
[[Image:Expression Graph.svg|300px]]
 
As for trees, outermost and innermost reduction also applies to graphs. Hence we have '''graph reduction'''.
 
Now evaluation with outermost graph reduction can proceed as follows:
 
[[Image:Expression Graph Reduction.svg|200px]]
 
Notice that evaluation now only requires four steps. Outermost graph reduction is referred to as [[lazy evaluation]] and innermost graph reduction is referred to as [[eager evaluation]].
 
== Combinator graph reduction ==
'''Combinator graph reduction''' is a fundamental implementation technique for [[functional programming]] languages, in which a program is converted into a [[combinator]] representation which is mapped to a [[directed graph]] [[data structure]] in computer memory, and program execution then consists of rewriting parts of this graph ("reducing" it) so as to move towards useful results.
 
== History ==
The concept of a graph reduction that allows evaluated values to be shared was first developed by [[Chris Wadsworth]] in his 1971 Ph.D. dissertation.<ref>{{cite journal | last = Hudak | first = Paul | title = Conception, evolution, and application of functional programming languages | journal = [[Association for Computing Machinery|ACM]] Computing Surveys | volume = 21 | issue = 3 | pages = 359–411 |date=September 1989 | id = {{citeseerx|10.1.1.83.6505}} | doi =10.1145/72551.72554 }}</ref> This dissertation was cited by Peter Henderson and James H. Morris Jr. in 1976 page, “A lazy evaluator” [http://portal.acm.org/citation.cfm?id=811543] that introduced the notion of lazy evaluation. In 1976 David Turner incorporated lazy evaluation into [[SASL programming language|SASL]] using combinators.<ref>{{cite conference |last=Hudak |first=Paul |coauthors=Hughes, John; Peyton Jones, Simon; Wadler, Philip |title=A History of Haskell |url =http://haskell.org/haskellwiki/History_of_Haskell |booktitle=History of Programming Languages Conference 2007 }}</ref>
SASL was an early functional programming language first developed by Turner in 1972.
 
==See also==
*[[SECD machine]]
*[[graph reduction machine]]
 
==Notes==
<references/>
 
==References==
*{{cite book
|title=Introduction to Functional Programming using Haskell
|last=Bird|first=Richard
|publisher=Prentice Hall
|year=1998
|isbn=0-13-484346-0
}}
 
==Further reading==
*[[Simon Peyton Jones]], ''The Implementation of Functional Programming Languages'', Prentice Hall, 1987. Full text online.[http://research.microsoft.com/users/simonpj/papers/slpj-book-1987/index.htm]
 
[[Category:Implementation of functional programming languages]]
[[Category:Graph algorithms]]
[[Category:Graph rewriting]]

Revision as of 17:42, 26 February 2014

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