lambda abstraction


lambda abstraction

A term in lambda-calculus denoting a function. A lambdaabstraction begins with a lower-case lambda (represented as"\\" in this document), followed by a variable name (the "boundvariable"), a full stop and a lambda expression (the body).The body is taken to extend as far to the right as possibleso, for example an expression,

\\ x . \\ y . x+y

is read as

\\ x . (\\ y . x+y).

A nested abstraction such as this is often abbreviated to:

\\ x y . x + y

The lambda expression (\\ v . E) denotes a function which takesan argument and returns the term E with all free occurrencesof v replaced by the actual argument. Application isrepresented by juxtaposition so

(\\ x . x) 42

represents the identity function applied to the constant 42.

A lambda abstraction in Lisp is written as the symbollambda, a list of zero or more variable names and a list ofzero or more terms, e.g.

(lambda (x y) (plus x y))

Lambda expressions in Haskell are written as a backslash,"\\", one or more patterns (e.g. variable names), "->" and anexpression, e.g. \\ x -> x.