Differences between version 4 and predecessor to the previous major change of LambdaCalculus.
Other diffs: Previous Revision, Previous Author, or view the Annotated Edit History
| Newer page: | version 4 | Last edited on Friday, October 24, 2003 12:56:22 pm | by PerryLorier | Revert |
| 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.
One of the fundamental proofs in ComputerScience is that any result calculatable in LambdaCalculus is calculatable 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 it's expansion. And thats it.
+
+For instance, if we define "one" as a function which outputs it's 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 it's 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 it's argument twice.
+
+Three is obvious
+ three = sucessor two
+which eventually expands to
+ three = Λx(x x x)
+
+Addiction is easy
+ addition = Λx.y.z(x z y z)
+
+ five = add two three
+ five = Λx.y.z(x z y z) Λa(a a) Λa(a a a)
+ five = Λz(Λa(a a) z Λa(a a a) z)
+ five = Λz(z z z z z)
