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--- lambda_calculus [2007/05/28 21:18]
vkuncak
+++ lambda_calculus [2015/04/21 17:32]
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-====== Lambda calculus ======
-This is a very brief introduction to lambda calculus.
-Lambda calculus is a notation for defining functions.  It is very simple and powerful.  Two of its main uses are
-  * foundation of programming languages, especially functional languages such as Ocaml, Haskell, and Scala
-  * higher order logic notations, such as Isabelle, HOL, PVS
-For example, a function which takes its argument and increments it by 2 is given by lambda expression $\lambda x.x+2$.  The $\lambda$ operator is a binder used to indicate that a variable is to be treated as an argument of a function.  We indicate function application by juxtaposition.  Here are some example expressions and their values:
-\begin{eqnarray*}
-  (\lambda x. x + 2)\, 7 &=& 9 \\
-  ((\lambda x. (\lambda y. x - y))\, 5)\, 3 &=& (\lambda y. 5 - y) 3 = 5 - 3 = 2 \\
-  (((\lambda f. (\lambda x. f(f x))) (\lambda y. y^2))) 3 &=& (\lambda x. (\lambda y. y^2) ((\lambda y.y^2) x))) 3 = (\lambda y. y^2) ((\lambda y.y^2) 3) = (\lambda y.y^2) 9 = 81
-\end{eqnarray*}
-Note: in Isabelle and Jahob notation we use '% x.F' or '\<lambda> x.F' to denote $\lambda$.F binder.  In [[http://caml.inria.fr|ocaml]], one can use 'function x -> F', and in [[http://www.haskell.org|Haskell]] one can use '\ x -> F'.
-===== Lambda expressions =====
-Untyped lambda expressions are given by following grammar:
-\begin{equation*}
-  T ::= v \mid \lambda v.T \mid T\, T \\
-  v ::= v_1 \mid v_2 \mid ...
-\end{equation*}
-That is, an expression is a variable, a lambda expression defining a function, or an application of one expression to another.
-===== Beta reduction =====
-The main rule of lambda calculus is beta reduction, which formalizes function application:
-\begin{equation*}
-  (\lambda x. s) t = s[x:=t]
-\end{equation*}
-where we assume capture-avoiding substitution.
-One simple use of lambda expressions to introduce shorthands: instead of writing
-\begin{equation*}
-  \mbox{let}\ x = E\ \mbox{in}\ F
-\end{equation*}
-we can write
-\begin{equation*}
-  (\lambda x.F)E
-\end{equation*}
-References:
-  * Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984) ([[http://www.andrew.cmu.edu/user/cebrown/notes/barendregt.html|Chad Brown's notes on this book]])