# Lambda abstraction

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 Haskell theoretical foundations Lambda calculus:Alpha conversion - Beta reductionEta conversion - Lambda abstraction

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.