Reduction
---------
-Find "normal forms" for the following---that is, reduce them until no more reductions are possible. We'll write λ`x` as `\x`.
+Find "normal forms" for the following---that is, reduce them until no more reductions are possible. We'll write <code>λx</code> as `\x`.
1. `(\x \y. y x) z`
2. `(\x (x x)) z`
Recall our definitions of true and false.
-> "true" defined to be `\t \f. t`
-> "false" defined to be `\t \f. f`
+> **true** is defined to be `\t \f. t`
+> **false** is defined to be `\t \f. f`
In Racket, these can be defined like this:
(define false (lambda (t) (lambda (f) f)))
<OL start=8>
-<LI>Define a "neg" operator that negates "true" and "false".
+<LI>Define a `neg` operator that negates `true` and `false`.
Expected behavior:
evaluates to 10.
-<LI>Define an "and" operator.
+<LI>Define an `and` operator.
-<LI>Define an "xor" operator.
-
-(If you haven't seen this term before, here's a truth table:
+<LI>Define an `xor` operator. If you haven't seen this term before, here's a truth table:
true xor true = false
true xor false = true
false xor true = true
false xor false = false
-)
<LI>Inspired by our definition of boolean values, propose a data structure
capable of representing one of the two values `black` or `white`.
one of those values, call it a "black-or-white value", we should be able to
write:
- the-value if-black if-white
+ the-value if-black if-white
(where `if-black` and `if-white` are anything), and get back one of `if-black` or
`if-white`, depending on which of the black-or-white values we started with. Give
Recall our definitions of ordered pairs.
-> the pair (x,y) is defined as `\f. f x y`
+> the pair **(**x**,**y**)** is defined to be `\f. f x y`
To extract the first element of a pair p, you write:
- p (\fst \snd. fst)
+ p (\fst \snd. fst)
Here are some definitions in Racket:
- (define make-pair (lambda (fst) (lambda (snd) (lambda (f) ((f fst) snd)))))
- (define get-first (lambda (fst) (lambda (snd) fst)))
- (define get-second (lambda (fst) (lambda (snd) snd)))
+ (define make-pair (lambda (fst) (lambda (snd) (lambda (f) ((f fst) snd)))))
+ (define get-first (lambda (fst) (lambda (snd) fst)))
+ (define get-second (lambda (fst) (lambda (snd) snd)))
Now we can write:
- (define p ((make-pair 10) 20))
- (p get-first) ; will evaluate to 10
- (p get-second) ; will evaluate to 20
+ (define p ((make-pair 10) 20))
+ (p get-first) ; will evaluate to 10
+ (p get-second) ; will evaluate to 20
-If you're bothered by having the pair to the left and the function that
+If you're puzzled by having the pair to the left and the function that
operates on it come second, think about why it's being done this way: the pair
-is a package that takes a function for operating on its elements as an
-argument, and returns the result of operating on its elemens with that
-function. In other words, the pair is also a function. (Of course, in the
-untyped lambda calculus, absolutely *everything* is a function: functors,
-arguments, abstracts, redexes, values---everything.)
+is a package that takes a function for operating on its elements *as an
+argument*, and returns *the result of* operating on its elements with that
+function. In other words, the pair is a higher-order function. (Consider the similarities between this definition of a pair and a generalized quantifier.)
If you like, you can disguise what's going on like this:
- (define lifted-get-first (lambda (p) (p get-first)))
- (define lifted-get-second (lambda (p) (p get-second)))
+ (define lifted-get-first (lambda (p) (p get-first)))
+ (define lifted-get-second (lambda (p) (p get-second)))
Now you can write:
- (lifted-get-first p)
+ (lifted-get-first p)
instead of:
- (p get-first)
+ (p get-first)
-However, the latter is still what's going on under the hood.
+However, the latter is still what's going on under the hood. (Remark: `(lifted-f ((make-pair 10) 20))` stands to `(((make-pair 10) 20) f)` as `(((make-pair 10) 20) f)` stands to `((f 10) 20)`.)
<OL start=13>
-<LI>Define a `swap` function that reverses the elements of a pair.
-
-Expected behavior:
+<LI>Define a `swap` function that reverses the elements of a pair. Expected behavior:
- (define p ((make-pair 10) 20))
- ((p swap) get-first) ; evaluates to 20
- ((p swap) get-second) ; evaluates to 10
+ (define p ((make-pair 10) 20))
+ ((p swap) get-first) ; evaluates to 20
+ ((p swap) get-second) ; evaluates to 10
-Write out the definition of swap in Racket.
+Write out the definition of `swap` in Racket.
<LI>Define a `dup` function that duplicates its argument to form a pair
whose elements are the same.
Expected behavior:
- ((dup 10) get-first) ; evaluates to 10
- ((dup 10) get-second) ; evaluates to 10
+ ((dup 10) get-first) ; evaluates to 10
+ ((dup 10) get-second) ; evaluates to 10
<LI>Define a `sixteen` function that makes
sixteen copies of its argument (and stores them in a data structure of
<LI>Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered triples. That is,
- (((make-triple M) N) P)
+ (((make-triple M) N) P)
should return an object that behaves in a reasonable way to serve as a triple. In addition to defining the `make-triple` function, you have to show how to extract elements of your triple. Write a `get-first-of-triple` function, that does for triples what `get-first` does for pairs. Also write `get-second-of-triple` and `get-third-of-triple` functions.
You can help yourself to the following definition:
- (define add (lambda (x) (lambda (y) (+ x y))))
+ (define add (lambda (x) (lambda (y) (+ x y))))
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