<strong>Application</strong>: <code>(M N)</code>
</blockquote>
-Some authors reserve the term "term" for just variables and abstracts. We won't participate in that convention; we'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms.
+Some authors reserve the term "term" for just variables and abstracts. We'll probably just say "term" and "expression" indiscriminately for expressions of any of these three forms.
Examples of expressions:
(x (\x x))
((\x (x x)) (\x (x x)))
-<p>
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
"beta-contraction". Suppose you have some expression of the form:
- ((\a M) N)
+ ((\ a M) N)
that is, an application of an abstract to some other expression. This compound form is called a **redex**, meaning it's a "beta-reducible expression." `(\a M)` is called the **head** of the redex; `N` is called the **argument**, and `M` is called the **body**.
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":
-<pre><code>
-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
-</code></pre>
+ 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.)
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:
and there's no more mutation going on there than there is in:
- <pre>
- <code>∀x. (F x or ∀x (not (F x)))</code>
- </pre>
+ <pre><code>∀x. (F x or ∀x (not (F x)))
+ </code></pre>
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.