Notes (TILC 05/05): Projects for the Reader

5 Projects for the reader

[N.B.: I have not tested any of this code. I plan on writing a Lambda Calculus interpreter at some point and when I do, I will test this code and add a link back here.]

1 “less than” and “greater than”

See section 3.4

2 Positive and Negative Integers

Let (\x. x np nn) be the pair of natural numbers np, and nn, the integer n will then be the difference np - nn. In other words, np is the positive component of n and nn is the negative component.

However, in this representation there will be many possible expressions that signify the same integer. Let us then define the case where either np or nn equals 0 as the canonical or simplified representation.

To make things easier let’s define some aliases for handling pairs:

pair = \a b. (\x . x a b)
fst = \p. p true
snd = \p. p false

pair makes a pair out of two elements. fst returns the first element of a pair. snd returns the second element of a pair.

We can then define a function simplifyInt as:

simplifyInt = (\x. ( \ p n. (gte p n)
                            (pair (n pred p) 0)
                            (pair 0 (p pred n))
                   (fst x)
                   (snd x)

If p is greater than or equal to n, the number is positive, so we apply pred n times to p, yielding an integer (p - n, 0). Otherwise, we apply pred n times to p, yielding an integer (0, n - p).

In either case, the integer is now in “lowest terms”, so to speak.

3 Addition and Subtraction of Integers

Adding two integers (p, n) and (q, m) is the same as adding their components and simplifying:

addInt = \ x y. simplifyInt
                (pair (add (fst x) (fst y))
                      (add (snd x) (snd y))

Subtracting is the same as adding with sign-flip function, called negate:

negateInt  = \ x . pair (snd x) (fst x)
subtractInt = \ x y. addInt x (negateInt y)

4 Division of positive integers recursively

The positive integers are the natural numbers, so we’ll have our div function return a natural number.

If in div x d either x or d are negative, we’ll return 0 to signify an error.

loop = \ test f next start. Y (\r s. test s ((f s) (r (next s)))) start

divUnsafe = \ x d. loop (\s . (lth s d)) 0) (\s. succ) (\s . d pred s) x

div = \x d. (\x d . (and (gte x 0) (gte d 0) (divUnsafe x d) 0))
            (fst (simplify x))
            (fst (simplify d))

5 Factorial

factorial = \n . loop (\m. isZero m 1) mult pred n

6 Rational Numbers

Our rational number representation will be a pair of integers: a numerator x and a divisor y:

pair x y

7 Addition, Subtraction, Multiplication, and Division of Rationals

First we’re going to need a way to multiply integers, if we have two integers x = (p - n), and y = (q - m), then x multiplied by y is

(p - n) (q - m) = (pq + nm) - (nq + pm)

Since according to our integer representation, p, n, q, and m are all natural numbers, we can do this:

multiplyInt = \x y. pair (add (multiply (fst x) (fst y))
                              (multiply (snd x) (snd y))
                         (add (multiply (snd x) (fst y))
                              (multiply (fst x) (snd y))

For adding rationals, since a/b + c/d = (ad + cb)/bd, and a,b,c,d are all integers:

addR = \x y. pair (addInt (multiplyInt (fst x) (snd y))
                          (multiplyInt (fst y) (snd x))
                  (multiplyInt (snd x) (snd y))

For subtraction, a/b - c/d = (ad - cb)/bd:

subtractR = \x y. pair (subtractInt (multiplyInt (fst x) (snd y))
                                    (multiplyInt (fst y) (snd x))
                  (multiplyInt (snd x) (snd y))

Multiplication, (a/b) * (c/d) = (ac)/(bd):

multiplyR = \x y. pair (multiplyInt (fst x) (fst y))
                       (multiplyInt (snd x) (snd y))

Division, (a/b) / (c/d) = (ad)/(cb):

divideR = \x y. pair (multiplyInt (fst x) (snd y))
                     (multiplyInt (snd x) (fst y))

8 Lists of Numbers

A list can be thought of as either a nil element or a cons of head and a tail, where the head is an expression and the tail is a list.

If x,y,z are list elments then lists of lengths 0 to 3 are, respectively:

0-list = nil
1-list = cons x nil
2-list = cons x (cons y nil))
3-list = cons x (cons y (cons z nil))

Let’s turn the above into abstractions on the variables x,y,z,c,n where c stands for cons and n stands for nil:

0-list = \ c n. n
1-list = \ x c n. c x n
2-list = \ x y c n. c x (c y n)
3 list = \ x y z c n. c x (c y (c z n))

If we pass in elements to the above list constructors (so that x_, y_, z_ are now representing expressions rather than variables):

0-list = \ c n. n
1-list = \ c n. c x_ n
2-list = \ c n. c x_ (c y_ n)
3 list = \ c n. c x_ (c y_ (c z_ n))

This suggests a nice list encoding as the right-fold of some function c over whatever we want our list elements to be, (with n as the base argument).

We already have our nil from the above as \ c n . n, now we need our cons function.

At first glance the function 2-list looks like a decent definition for cons, since it combines two elements into a list.

2-list = \ x y c n. c x (c y n)

But our definition of cons is that it combines an element (the head) and a list (the tail) into a list, not two elements.

Watch what happens if we pass an element x_ and a list into 2-list

\ c n . c x_ (c (\ c n. c a_ n) n)

In Haskell list notation, the above is [x, [a]], not [x, a], which is what we want.

But since a list is a function of c and n, we can modify 2-list slightly like so:

cons = \ x y c n. c x (y c n)

Now if we call cons with x_ and \ c n. c a_ n we get

cons x_ (\ c n. c a_ n) =>
\ c n. c x_ ((\c n. c a_ n) c n)
\ c n. c x_ (c a_ n)

Which does what we want. I like to think of cons as folding some function c over the tail with n and then adding one more fold layer of c with the head.

9 List Head

Since our list is a fold of a function c over the elements of a list, we call our list with true, or \ x y. x to throw away the tail. As an example:

(\ c n. c x_ (c y_ n)) true =>
true x_ (c y_ n) =>

But if the list is nil \ c n. n we want to return nil, so our head function is actually:

head = \ l. l true nil

10 List Length

We fold (\ x. succ) over the list with nil as 0_:

length = \ l. l (\ x. succ) 0_

As an example,

length (\ c n. c x_ (c y_ n)) =>
(\ c n. c x_ (c y_ n)) succ 0_ =>
(\ x. succ) x_ ((\ x. succ) y_ 0_)) =>
(succ (succ 0_)) =>

We add the extra abstraction over succ to throw away the list elements.

11 Turing Machine

So the rest of this document is fairly self-contained, but this question is introducting a pretty big conceptual dependency. Namely, what precisely is a Turing Machine?

I’ll cover this in a future post and link back here.

Also, if I’m going to write a Turing Machine in Lambda Calculus, I absolutely want to write proper executable code. Writing massive walls of pseudocode that don’t do anything is no fun. Code is meant to run!

So I’m going to go write a Lambda Calculus interpreter, vivify all the dead notation in this document, and then come back to this question.