Lambda abstraction

Haskell theoretical foundations General: Mathematics - Category theory Research - Curry/Howard/Lambek Lambda calculus: Alpha conversion - Beta reduction Eta conversion - Lambda abstraction Other: 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 but 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 much more ugly (but strictly 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.