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Lambda abstraction

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Haskell theoretical foundations

Mathematics - Category theory
Research - Curry/Howard/Lambek

Lambda calculus:
Alpha conversion - Beta reduction
Eta conversion - Lambda abstraction

Recursion - Combinatory logic
Chaitin's construction - Turing machine
Relational algebra

A lambda abstraction is another name for an anonymous function. It gets its name from the usual notation for writing it: for example, \lambda x \to x^2. (Another common, equivalent notation is: \lambda x . \ x^2.)

In Haskell source code, the Greek letter lambda is replaced by a backslash character ('
') instead, since this is easier to type and requires only the basic 7-bit ASCII character set. Similarly, the arrow is replaced with the ASCII character sequence '
'. So, for example, the lambda abstraction above would be written in Haskell as
  \ x -> x * x

There is actually a whole mathematical theory devoted to expressing computation entirely using lambda abstractions: the lambda calculus. Most functional programming languages (including Haskell) are based upon some extension of this idea.

When a lambda abstraction is applied to a value—for instance, (\lambda x \to x^2 ) \ 7—the result of the expression is determined by replacing every free occurrence of the parameter variable (in this case x) with the parameter value (in this case 7). This is a beta reduction.