Differences

This shows you the differences between two versions of the page.

--- sav08:untyped_lambda_calculus [2009/03/05 12:45]
vkuncak
+++ sav08:untyped_lambda_calculus [2015/04/21 17:30]
@@ Line 1: / Line 1: @@
-====== Untyped Lambda Calculus ======
-Lambda calculus is a simple and powerful notation for defining functions.  Among its uses are
-  * foundation of programming languages, especially functional languages such as Ocaml, Haskell, and Scala
-  * basis of higher order logic
-===== Intuition =====
-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*}
-Concrete syntax for $(\lambda x.x+2)5$:
-  * Isabelle (and Jahob): '(% x.x+2)5' or '(\<lambda> x.x+2)5'
-  * [[http://caml.inria.fr|ocaml]]: '(function x -> x+2)5'
-  * [[http://www.haskell.org|Haskell]]: '(\ x -> x+2)5'
-  * [[http://www.scala-lang.org|Scala]]: '((x:int) => x+2)(5)'
-===== Lambda expressions =====
-Untyped lambda expressions are given by following grammar:
-\[\begin{array}{l}
-  T ::= v \mid \lambda v.T \mid T\, T \\
-  v ::= v_1 \mid v_2 \mid ...
-\end{array}\]
-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\ \mathop{\to}\limits^{\beta}\ s[x:=t]
-\end{equation*}
-where we assume capture-avoiding substitution.  In general, we identify terms that differ only by renaming of fresh bound variables.
-===== Non-termination and Difficulties with Semantics =====
-Let $\omega$ denote term
-\[
-    (\lambda x.xx)(\lambda x.xx)
-\]
-Note that $\omega$ beta reduces to itself.  It represents infinite loop.
-Note that the set of total functions is not a model of untyped lambda calculus.  Consider the self-application expression $x x$. Suppose the interpretation of variable $x$ ranges over a domain $D$.  This domain should be a set of total functions $f : A \to B$ to enable application.  But all possible values of $x$ are an argument, so we have that
-\[
-    D = [D \to D]
-\]
-that is, the set $D$ is equal to the set of all functions from $D$ to $D$.  But this is not possible in set theory, because we can prove that the set $D^D$ has larger cardinality (even if it is infinite).  For example, if $D$ is the set of natural numbers (which is countably infinite), then $D^D$ would be uncountably infinite.
-The models of untyped lambda calculus do exist, but they usually consider only certain 'continuous', partial, functions.  These domains are studied in //domain theory//.  Roughly speaking, in domain theory, a solution to equation such as $D \simeq [D \leadsto D]$ is obtained by finding a fixpoint of a monotonic operator $F(D) = [D \leadsto D]$.
-Thus, untyped lambda calculus is easy to understand syntactically (as a set of rules for manipulating formal expressions), rather than a system where each expression has some intuitive meaning.
-References:
-  * Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984)
-  * [[http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf|Domain Theory]] and {{sav08:domaintheory.pdf|pdf file}}