Diff: LambdaCalculus

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 Newer page: version 11 Last edited on Friday, January 13, 2006 11:02:16 pm by ToniMarsh Older page: version 1 Last edited on Friday, October 24, 2003 11:49:11 am by StuartYeates Revert
@@ -1,3 +1,38 @@
-A conceptual model of computers based on defining and evaluating functions. Of current ProgrammingLanguages , [LISP], [Scheme] and [Haskell] best expose this conceptual model.
+A conceptual model of computers based on defining and evaluating functions. Of the current [ProgrammingLanguage]s , [LISP], [Scheme], [Haskell] and other Category:FunctionalProgrammingLanguages best expose this conceptual model.

-One of the fundamental proofs in ComputerScience is that any result calculatable in LambdaCalculus is calculatable on a FiniteStateMachine.
+One of the fundamental proofs in ComputerScience is that any result calculable in LambdaCalculus is calculable on a FiniteStateMachine.
+
+(From memory:)
+
+LambdaCalculus is based on the idea that you have two types of rewrite rules, one where you change the names of variables, and one where you replace a variable with its expansion. And that's it.
+
+For instance, if we define "one" as a function which outputs its argument once, we have:
+
+ one = λx(x)
+
+then the successor function might be defined as
+ successor = λx.y(x y y)
+
+so "successor one" is
+ two = successor one
+ two = λx.y(x y y) λx(x)
+now we do an alpha reduction (rename some variables)
+ two = λx.y(x y y) λz(z)
+now we do a beta reduction (replace x with its value (λz(z))
+ two = λy(λz(z) y y)
+and a beta reduction on z
+ two = λy(y y)
+we now have a function that outputs its argument twice.
+
+Three is obvious
+ three = sucessor two
+which eventually expands to
+ three = λx(x x x)
+