X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week1.mdwn;h=1bc2309e1e9c79eea7118b9f6e72045a90b3317d;hp=c9b52d1d52acead984f8a6149cdb62db2a39c111;hb=c86596d41ed6ca32fff882b468eade84bc13fb07;hpb=fceb99fda0c4287f9f3c476a26c6e202cf5e26c5 diff --git a/week1.mdwn b/week1.mdwn index c9b52d1d..1bc2309e 100644 --- a/week1.mdwn +++ b/week1.mdwn @@ -62,6 +62,13 @@ Some authors reserve the term "term" for just variables and abstracts. We'll pro Examples of expressions: x + (y x) + (x x) + (\x y) + (\x x) + (\x (\y x)) + (x (\x x)) + ((\x (x x)) (\x (x x))) The lambda calculus has an associated proof theory. For now, we can regard the proof theory as having just one rule, called the rule of **beta-reduction** or @@ -278,18 +285,16 @@ It's possible to enhance the lambda calculus so that functions do get identified It's often said that dynamic systems are distinguished because they are the ones in which **order matters**. However, there are many ways in which order can matter. If we have a trivalent boolean system, for example---easily had in a purely functional calculus---we might choose to give a truth-table like this for "and": -
-true and true = true
-true and true = true
-true and * = *
-true and false = false
-* and true = *
-* and * = *
-* and false = *
-false and true = false
-false and * = false
-false and false = false
-
+ true and true = true
+ true and true = true
+ true and * = *
+ true and false = false
+ * and true = *
+ * and * = *
+ * and false = *
+ false and true = false
+ false and * = false
+ false and false = false
And then we'd notice that `* and false` has a different intepretation than `false and *`. (The same phenomenon is already present with the material conditional in bivalent logics; but seeing that a non-symmetric semantics for `and` is available even for functional languages is instructive.)
@@ -596,7 +601,6 @@ Here's how it looks to say the same thing in various of these languages.
It's easy to be lulled into thinking this is a kind of imperative construction. *But it's not!* It's really just a shorthand for the compound "let"-expressions we've already been looking at, taking the maximum syntactically permissible scope. (Compare the "dot" convention in the lambda calculus, discussed above.)
-
9. Some shorthand
OCaml permits you to abbreviate:
@@ -667,9 +671,8 @@ Here's how it looks to say the same thing in various of these languages.
and there's no more mutation going on there than there is in:
-
- ∀x. (F x or ∀x (not (F x)))
-
+ ∀x. (F x or ∀x (not (F x)))
+
When a previously-bound variable is rebound in the way we see here, that's called **shadowing**: the outer binding is shadowed during the scope of the inner binding.