Lecture #8, Tuesday, January 29th
=================================

- Introducing Racket's `lambda`
- Using Functions as Objects
- Currying
- Using Higher-Order & Anonymous Functions
- Side-note: "Point-Free" combinators
- This is not Runtime Code Generation
- Substitution Caches
- Implementation of Cache Functionality

------------------------------------------------------------------------
# Introducing Racket's `lambda`

`fun` & `lambda`
difference between lambda and simple values
not being able to do recursive functions with `let`
let* as a derived form
let with lambda in Racket --> can be a derived form
how `if` can be used to implement `and` and `or` as derived forms

Newtonian syntax vs. a lambda expression.

Don't be fooled into making a bogus connection between Racket's syntax,
and its `unique` powers...  The fact is that it is not the only language
that has this capability.  For example, this:

    (define (f g) (g 2 3))
    (f +) ==> 5
    (f *) ==> 6
    (f (lambda (x y) (+ (square x) (square y)))) ==> 13

Can be written in JavaScript like this:

    function f(g) { return g(2,3); }
    function square(x) { return x*x; }
    console.log(f(function (x,y) { return square(x) + square(y); }));

or in ES6 JavaScript:

    let f = (g) => g(2,3);
    let square = (x) => x*x;
    console.log(f((x,y) => square(x) + square(y)));

In Perl:

    sub f { my ($g) = @_; return $g->(2,3); }
    sub square { my ($x) = @_; return $x * $x; }
    print f(sub { my ($x, $y) = @_; return square($x) + square($y); });

In Ruby:

    def f(g) g.call(2,3) end
    def square(x) x*x end
    puts f(lambda{|x,y| square(x) + square(y)})

etc.  Even [Java has lambda expressions], and recently
[C++ added them too].

[Java has lambda expressions]:
  http://www.drdobbs.com/jvm/lambda-expressions-in-java-8/240166764
[C++ added them too]:
  http://www.cprogramming.com/c++11/c++11-lambda-closures.html


------------------------------------------------------------------------
# Using Functions as Objects

A very important aspect of Racket --- using "higher order" functions ---
functions that get and return functions.  Here is a very simple example:

    (define (f x) (lambda () x))
    (define a (f 2))
    (a)  -->  2
    (define b (f 3))
    (b)  -->  3

Note: what we get is actually an object that remembers (by the
substitution we're doing) a number.  How about:

    (define aa (f a))
    (aa)    -->  #<procedure> (this is a)
    ((aa))  -->  2

Take this idea to the next level:

    (define (kons x y)
      (lambda (b)
        (if b x y)))
    (define (kar p) (p #t))
    (define (kdr p) (p #f))
    (define a (kons 1 2))
    (define b (kons 3 4))
    (list (kar a) (kdr a))
    (list (kar b) (kdr b))

Or, with types:

    (: kons : (All (A B) A B -> (Boolean -> (U A B))))
    (define (kons x y)
      (lambda (b)
        (if b x y)))
    (: kar : (All (T) (Boolean -> T) -> T))
    (define (kar p) (p #t))
    (: kdr : (All (T) (Boolean -> T) -> T))
    (define (kdr p) (p #f))
    (define a (kons 1 2))
    (define b (kons 3 4))
    (list (kar a) (kdr a))
    (list (kar b) (kdr b))

Even more --- why should the internal function expect a boolean and
choose what to return?  We can simply expect a function that will take
the two values and return one:

    (define (kons x y) (lambda (s) (s x y)))
    (define (kar p) (p (lambda (x y) x)))
    (define (kdr p) (p (lambda (x y) y)))
    (define a (kons 1 2))
    (define b (kons 3 4))
    (list (kar a) (kdr a))
    (list (kar b) (kdr b))

And a typed version, using our own constructor to make it a little less
painful:

    (define-type (Kons A B) = ((A B -> (U A B)) -> (U A B)))

    (: kons : (All (A B) A B -> (Kons A B)))
    (define (kons x y) (lambda (s) (s x y)))
    (: kar : (All (A B) (Kons A B) -> (U A B)))
    (define (kar p) (p (lambda (x y) x)))
    (: kdr : (All (A B) (Kons A B) -> (U A B)))
    (define (kdr p) (p (lambda (x y) y)))
    (define a (kons 1 2))
    (define b (kons 3 4))
    (list (kar a) (kdr a))
    (list (kar b) (kdr b))

Note that the `Kons` type definition is the same as:

    (define-type Kons = (All (A B) (A B -> (U A B)) -> (U A B)))

so `All` is to polymorphic type definitions what `lambda` is for
function definitions.

Finally in JavaScript:

    function kons(x,y) { return function(s) { return s(x, y); } }
    function kar(p) { return p(function(x,y){ return x; }); }
    function kdr(p) { return p(function(x,y){ return y; }); }
    a = kons(1,2);
    b = kons(3,4);
    console.log('a = <' + kar(a) + ',' + kdr(a) + '>' );
    console.log('b = <' + kar(b) + ',' + kdr(b) + '>' );

Or with ES6 *arrow functions*, the function definitionss become:

    var kons = (x,y) => s => s(x,y);
    var kar  = p => p((x,y) => x);
    var kdr  = p => p((x,y) => y);


------------------------------------------------------------------------
# Currying

A *curried* function is a function that, instead of accepting two (or
more) arguments, accepts only one and returns a function that accepts
the rest.  For example:

    (: plus : Number -> (Number -> Number))
    (define (plus x)
      (lambda (y)
        (+ x y)))

It's easy to write functions for translating between normal and curried
versions.

    (define (currify f)
      (lambda (x)
        (lambda (y)
          (f x y))))

Typed version of that, with examples:

    (: currify : (All (A B C) (A B -> C) -> (A -> (B -> C))))
    ;; convert a double-argument function to a curried one
    (define (currify f)
      (lambda (x) (lambda (y) (f x y))))

    (: add : Number Number -> Number)
    (define (add x y) (+ x y))

    (: plus : Number -> (Number -> Number))
    (define plus (currify add))

    (test ((plus 1) 2) => 3)
    (test (((currify add) 1) 2) => 3)
    (test (map (plus 1) '(1 2 3)) => '(2 3 4))
    (test (map ((currify add) 1) '(1 2 3)) => '(2 3 4))
    (test (map ((currify +) 1) '(1 2 3)) => '(2 3 4))

Usages --- common with H.O. functions like map, where we want to *fix*
one argument.

When dealing with such higher-order code, the types are very helpful,
since every arrow corresponds to a function:

    (: currify : (All (A B C) (A B -> C) -> (A -> (B -> C))))

It is common to make the `->` function type associate to the right, so
you can find this type written as:

    currify : (A B -> C) -> (A -> B -> C)

or even as

    currify : (A B -> C) -> A -> B -> C

but that can be a little confusing...


------------------------------------------------------------------------
# Using Higher-Order & Anonymous Functions

Say that we have a function for estimating derivatives of a function at
a specific point:

    (define dx 0.01)

    (: deriv : (Number -> Number) Number -> Number)
    ;; compute the derivative of `f' at the given point `x'
    (define (deriv f x)
      (/ (- (f (+ x dx)) (f x)) dx))

    (: integrate : (Number -> Number) Number -> Number)
    ;; compute an integral of `f' at the given point `x'
    (define (integrate f x)
      (: loop : Number Number -> Number)
      (define (loop y acc)
        (if (> y x)
          (* acc dx)
          (loop (+ y dx) (+ acc (f y)))))
      (loop 0 0))

And say that we want to try out various functions given some `plot`
function that draws graphs of numeric functions, for example:

    (plot sin)

The problem is that `plot` expects a single `(Number -> Number)`
function --- if we want to try it with a derivative, we can do this:

    (: sin-deriv : Number -> Number)
    ;; the derivative of sin
    (define sin-deriv (lambda (x) (deriv sin x)))
    (plot sin-deriv)

But this will get very tedious very fast --- it is much simpler to use
an anonymous function:

    (plot (lambda (x) (deriv sin x)))

we can even verify that our derivative is correct by comparing a known
function to its derivative

    (plot (lambda (x) (- (deriv sin x) (cos x))))

But it's still not completely natural to do these things --- you need to
explicitly combine functions, which is not too convenient.  Instead of
doing this, we can write H.O. functions that will work with functional
inputs and outputs.  For example, we can write a function to subtract
functions:

    (: fsub : (Number -> Number) (Number -> Number)
              -> (Number -> Number))
    ;; subtracts two numeric 1-argument functions
    (define (fsub f g)
      (lambda (x) (- (f x) (g x))))

and the same for the derivative:

    (: fderiv : (Number -> Number) -> (Number -> Number))
    ;; compute the derivative function of `f'
    (define (fderiv f)
      (lambda (x) (deriv f x)))

Now we can try the same in a much easier way:

    (plot (fsub (fderiv sin) cos))

More than that --- our `fderiv` could be created from `deriv`
automatically:

    (: currify : (All (A B C) (A B -> C) -> (A -> B -> C)))
    ;; convert a double-argument function to a curried one
    (define (currify f)
      (lambda (x) (lambda (y) (f x y))))

    (: fderiv : (Number -> Number) -> (Number -> Number))
    ;; compute the derivative function of `f'
    (define fderiv (currify deriv))

Same principle with `fsub`: we can write a function that converts a
binary arithmetical function into a function that operates on unary
numeric function.  But to make things more readable we can define new
types for unary and binary numeric functions:

    (define-type UnaryFun  = (Number -> Number))
    (define-type BinaryFun = (Number Number -> Number))

    (: binop->fbinop : BinaryFun -> (UnaryFun UnaryFun -> UnaryFun))
    ;; turns an arithmetic binary operator to a function operator
    (define (binop->fbinop op)
      (lambda (f g)
        (lambda (x) (op (f x) (g x)))))

    (: fsub : UnaryFun UnaryFun -> UnaryFun)
    ;; functional pointwise subtraction
    (define fsub (binop->fbinop -))

We can do this with anything --- developing a rich library of functions
and functionals (functions over functions) is extremely easy...  Here's
a pretty extensive yet very short library of functions:

    #lang pl untyped

    (define (currify f)
      (lambda (x) (lambda (y) (f x y))))
    (define (binop->fbinop op)
      (lambda (f g)
        (lambda (x) (op (f x) (g x)))))
    (define (compose f g)
      (lambda (x) (f (g x))))

    (define dx 0.01)
    (define (deriv f x)
      (/ (- (f (+ x dx)) (f x)) dx))
    (define (integrate f x)
      (define over? (if (< x 0) < >))
      (define step  (if (< x 0) - +))
      (define add   (if (< x 0) - +))
      (define (loop y acc)
        (if (over? y x)
          (* acc dx)
          (loop (step y dx) (add acc (f y)))))
      (loop 0 0))

    (define fadd (binop->fbinop +))
    (define fsub (binop->fbinop -))
    (define fmul (binop->fbinop *))
    (define fdiv (binop->fbinop /))
    (define fderiv     (currify deriv))
    (define fintegrate (currify integrate))
    ;; ...

This is written in the "untyped dialect" of the class language, but it
should be easy now to add the types.

Examples:

    ;; want to verify that `integrate' is the opposite of `deriv':
    ;;   take a function, subtract it from its derivative's integral
    (plot (fsub sin (fintegrate (fderiv sin))))

    ;; want to magnify the errors? -- here's how you magnify:
    (plot (compose ((currify *) 5) sin))

    ;; so:
    (plot (compose ((currify *) 20)
                   (fsub sin (fintegrate (fderiv sin)))))


------------------------------------------------------------------------
## Side-note: "Point-Free" combinators

> Forming functions without using `lambda` (or an implicit `lambda`
> using a `define` syntactic sugar) is called *point-free style*.  It's
> especially popular in Haskell, where it is easier to form functions
> this way because of implicit currying and a large number of higher
> level function combinators.  If used too much, it can easily lead to
> obfuscated code.

------------------------------------------------------------------------
------------------------------------------------------------------------
## This is not Runtime Code Generation

All of this is similar to run-time code generation, but not really.  The
only thing that `fderiv` does is take a function and store it somewhere
in the returned function, then when that function receives a number, it
uses the stored function and send it to deriv with the number.  We could
simply write deriv as what `fderiv` is --- which is the *real*
derivative function:

    (define (deriv f)
      (lambda (x)
        (/ (- (f (+ x dx)) (f x)) dx)))

but again, this is not faster or slower than the plain `deriv`.
However, there are some situations where we can do some of the
computation on the first-stage argument, saving work from the second
stage.  Here is a cooked-to-exaggeration example --- we want a function
that receives two inputs `x`, `y` and returns `fib(x)*y`, but we must
use a stupid `fib`:

    (define (fib n)
      (if (<= n 1)
        n
        (+ (fib (- n 1)) (fib (- n 2)))))

The function we want is:

    (define (bogus x y)
      (* (fib x) y))

If we currify it as usual (or just use `currify`), we get:

    (define (bogus x)
      (lambda (y)
        (* (fib x) y)))

And try this several times:

    (define bogus24 (bogus 24))
    (map bogus24 '(1 2 3 4 5))

But in the definition of `bogus`, notice that `(fib x)` does not depend
on `y` --- so we can rewrite it a little differently:

    (define (bogus x)
      (let ([fibx (fib x)])
        (lambda (y)
          (* fibx y))))

and trying the above again is much faster now:

    (define bogus24 (bogus 24))
    (map bogus24 '(1 2 3 4 5))

This is therefore not doing any kind of runtime code generation, but it
*enables* doing similar optimizations in our code.  A proper RTCG
facility would recompile the curried function for a given first input,
and (hopefully) automatically achieve the optimization that we did in a
manual way.


------------------------------------------------------------------------
# Substitution Caches

> [PLAI §5] (called "deferred substitutions" there)

Evaluating using substitutions is very inefficient --- at each scope, we
copy a piece of the program AST.  This includes all function calls which
implies an impractical cost (function calls should be *cheap*!).

To get over this, we want to use a cache of substitutions.

Basic idea: we begin evaluating with no cached substitutions, then
collect them as we encounter bindings.

Implies another change for our evaluator: we don't really substitute
cache at that point.


------------------------------------------------------------------------
# Implementation of Cache Functionality

First, we need a type for a substitution cache.  For this we will use a
list of lists of two elements each --- a name and its value FLANG:

    ;; a type for substitution caches:
    (define-type SubstCache = (Listof (List Symbol FLANG)))

We need to have an empty substitution cache, a way to extend it, and a
way to look things up:

    (: empty-subst : SubstCache)
    (define empty-subst null)

    (: extend : Symbol FLANG SubstCache -> SubstCache)
    ;; extend a given substitution cache with a new mapping
    (define (extend id expr sc)
      (cons (list id expr) sc))

    (: lookup : Symbol SubstCache -> FLANG)
    ;; lookup a symbol in a substitution cache, return the value it is
    ;; bound to (or throw an error if it isn't bound)
    (define (lookup name sc)
      (cond [(null? sc) (error 'lookup "no binding for ~s" name)]
            [(eq? name (first (first sc))) (second (first sc))]
            [else (lookup name (rest sc))]))

Actually, the reason to use such list of lists is that Racket has a
built-in function called `assq` that will do this kind of search (`assq`
is a search in an association list using `eq?` for the key comparison).
This is a version of `lookup` that uses `assq`:

    (define (lookup name sc)
      (let ([cell (assq name sc)])
        (if cell
          (second cell)
          (error 'lookup "no binding for ~s" name))))