+[[!toc]]
+
+##More on evaluation strategies##
+
+Here are notes on [[evaluation order]] that make the choice of which
+lambda to reduce next the selection of a route through a network of
+links.
+
+
##Computing the length of a list##
How could we compute the length of a list? Without worrying yet about what lambda-calculus implementation we're using for the list, the basic idea would be to define this recursively:
and the initial `(\x. x x)` is just what we earlier called the <code>ω</code> combinator (lower-case omega, not the non-terminating <code>Ω</code>). So the self-application of `H` can be written:
-<pre><code>ω (\h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst))))</code></pre>
+<pre><code>ω (\h \lst. (isempty lst) zero (add one ((h h) (extract-tail lst))))
+</code></pre>
and this will indeed implement the recursive function we couldn't earlier figure out how to define.
##Fixed-point Combinators Are a Bit Intoxicating##
-![tatoo](/y-combinator.jpg)
+![tatoo](/y-combinator-fixed.jpg)
There's a tendency for people to say "Y-combinator" to refer to fixed-point combinators generally. We'll probably fall into that usage ourselves. Speaking correctly, though, the Y-combinator is only one of many fixed-point combinators.
the identity function. Here's a fixed point for the identity
function:
-<pre>
- Y I
- (\f. (\h. f (h h)) (\h. f (h h))) I
- (\h. I (h h)) (\h. I (h h)))
- (\h. (h h)) (\h. (h h)))
- ω ω
- &Omega
-</pre>
+<pre><code>Y I
+(\f. (\h. f (h h)) (\h. f (h h))) I
+(\h. I (h h)) (\h. I (h h)))
+(\h. (h h)) (\h. (h h)))
+ω ω
+&Omega
+</code></pre>
Oh. Well! That feels right. The meaning of *This sentence is true*
in a context in which *this sentence* refers to the sentence in which
-it occurs is Ω, our prototypical infinite loop...
+it occurs is <code>Ω</code>, our prototypical infinite loop...
What about the liar paradox?
truth and circularity](http://tinyurl.com/2db62bk) for an approach
that is similar, but expressed in terms of non-well-founded sets
rather than recursive functions.
+
+##However...##
+
+You should be cautious about feeling too comfortable with
+these results. Thinking again of the truth-teller paradox, yes,
+<code>Ω</code> is *a* fixed point for `I`, and perhaps it has
+some a privileged status among all the fixed points for `I`, being the
+one delivered by Y and all (though it is not obvious why Y should have
+any special status).
+
+But one could ask: look, literally every formula is a fixed point for
+`I`, since
+
+ X <~~> I X
+
+for any choice of X whatsoever.
+
+So the Y combinator is only guaranteed to give us one fixed point out
+of infinitely many---and not always the intuitively most useful
+one. (For instance, the squaring function has zero as a fixed point,
+since 0 * 0 = 0, and 1 as a fixed point, since 1 * 1 = 1, but `Y
+(\x. mul x x)` doesn't give us 0 or 1.) So with respect to the
+truth-teller paradox, why in the reasoning we've
+just gone through should we be reaching for just this fixed point at
+just this juncture?
+
+One obstacle to thinking this through is the fact that a sentence
+normally has only two truth values. We might consider instead a noun
+phrase such as
+
+(3) the entity that this noun phrase refers to
+
+The reference of (3) depends on the reference of the embedded noun
+phrase *this noun phrase*. It's easy to see that any object is a
+fixed point for this referential function: if this pen cap is the
+referent of *this noun phrase*, then it is the referent of (3), and so
+for any object.
+
+The chameleon nature of (3), by the way (a description that is equally
+good at describing any object), makes it particularly well suited as a
+gloss on pronouns such as *it*. In the system of
+[Jacobson 1999](http://www.springerlink.com/content/j706674r4w217jj5/),
+pronouns denote (you guessed it!) identity functions...
+
+Ultimately, in the context of this course, these paradoxes are more
+useful as a way of gaining leverage on the concepts of fixed points
+and recursion, rather than the other way around.