Using x to Generate Values

Cf. jp-reprod.html


In [1]:
from notebook_preamble import J, V, define

Consider the x combinator x == dup i:

[a B] x
[a B] a B

Let B swap the a with the quote and run some function [C] on it.

[a B] a B
[a B] a swap [C] dip
a [a B]      [C] dip
a C [a B]

Now discard the quoted a with rest and cons the result of C on a whatever that is:

aC [a B] rest cons
aC [B] cons
[aC B]

Altogether, this is the definition of B:

B == swap [C] dip rest cons

We can create a quoted program that generates the Natural numbers (integers 0, 1, 2, ...) by using 0 for a and [dup ++] for [C]:

[0 swap [dup ++] dip rest cons]

Let's try it:


In [2]:
V('[0 swap [dup ++] dip rest cons] x')


                                           . [0 swap [dup ++] dip rest cons] x
           [0 swap [dup ++] dip rest cons] . x
           [0 swap [dup ++] dip rest cons] . 0 swap [dup ++] dip rest cons
         [0 swap [dup ++] dip rest cons] 0 . swap [dup ++] dip rest cons
         0 [0 swap [dup ++] dip rest cons] . [dup ++] dip rest cons
0 [0 swap [dup ++] dip rest cons] [dup ++] . dip rest cons
                                         0 . dup ++ [0 swap [dup ++] dip rest cons] rest cons
                                       0 0 . ++ [0 swap [dup ++] dip rest cons] rest cons
                                       0 1 . [0 swap [dup ++] dip rest cons] rest cons
       0 1 [0 swap [dup ++] dip rest cons] . rest cons
         0 1 [swap [dup ++] dip rest cons] . cons
         0 [1 swap [dup ++] dip rest cons] . 

After one application of x the quoted program contains 1 and 0 is below it on the stack.


In [3]:
J('[0 swap [dup ++] dip rest cons] x x x x x pop')


0 1 2 3 4

direco


In [4]:
define('direco == dip rest cons')

In [5]:
V('[0 swap [dup ++] direco] x')


                                    . [0 swap [dup ++] direco] x
           [0 swap [dup ++] direco] . x
           [0 swap [dup ++] direco] . 0 swap [dup ++] direco
         [0 swap [dup ++] direco] 0 . swap [dup ++] direco
         0 [0 swap [dup ++] direco] . [dup ++] direco
0 [0 swap [dup ++] direco] [dup ++] . direco
0 [0 swap [dup ++] direco] [dup ++] . dip rest cons
                                  0 . dup ++ [0 swap [dup ++] direco] rest cons
                                0 0 . ++ [0 swap [dup ++] direco] rest cons
                                0 1 . [0 swap [dup ++] direco] rest cons
       0 1 [0 swap [dup ++] direco] . rest cons
         0 1 [swap [dup ++] direco] . cons
         0 [1 swap [dup ++] direco] . 

Generating Generators

We want to go from:

a [C] G

to:

[a swap [C] direco]

Working in reverse:

[a swap   [C] direco] cons
a [swap   [C] direco] concat
a [swap] [[C] direco] swap
a [[C] direco] [swap]
a [C] [direco] cons [swap]

Reading from the bottom up:

G == [direco] cons [swap] swap concat cons
G == [direco] cons [swap] swoncat cons

We can try it out:

0 [dup ++] G

In [6]:
define('G == [direco] cons [swap] swoncat cons')

In [7]:
V('0 [dup ++] G')


                           . 0 [dup ++] G
                         0 . [dup ++] G
                0 [dup ++] . G
                0 [dup ++] . [direco] cons [swap] swoncat cons
       0 [dup ++] [direco] . cons [swap] swoncat cons
       0 [[dup ++] direco] . [swap] swoncat cons
0 [[dup ++] direco] [swap] . swoncat cons
0 [[dup ++] direco] [swap] . swap concat cons
0 [swap] [[dup ++] direco] . concat cons
  0 [swap [dup ++] direco] . cons
  [0 swap [dup ++] direco] . 

In [8]:
V('0 [dup ++] G x')


                                    . 0 [dup ++] G x
                                  0 . [dup ++] G x
                         0 [dup ++] . G x
                         0 [dup ++] . [direco] cons [swap] swoncat cons x
                0 [dup ++] [direco] . cons [swap] swoncat cons x
                0 [[dup ++] direco] . [swap] swoncat cons x
         0 [[dup ++] direco] [swap] . swoncat cons x
         0 [[dup ++] direco] [swap] . swap concat cons x
         0 [swap] [[dup ++] direco] . concat cons x
           0 [swap [dup ++] direco] . cons x
           [0 swap [dup ++] direco] . x
           [0 swap [dup ++] direco] . 0 swap [dup ++] direco
         [0 swap [dup ++] direco] 0 . swap [dup ++] direco
         0 [0 swap [dup ++] direco] . [dup ++] direco
0 [0 swap [dup ++] direco] [dup ++] . direco
0 [0 swap [dup ++] direco] [dup ++] . dip rest cons
                                  0 . dup ++ [0 swap [dup ++] direco] rest cons
                                0 0 . ++ [0 swap [dup ++] direco] rest cons
                                0 1 . [0 swap [dup ++] direco] rest cons
       0 1 [0 swap [dup ++] direco] . rest cons
         0 1 [swap [dup ++] direco] . cons
         0 [1 swap [dup ++] direco] . 

Powers of 2


In [9]:
J('1 [dup 1 <<] G x x x x x x x x x')


1 2 4 8 16 32 64 128 256 [512 swap [dup 1 <<] direco]

n [x] times

If we have one of these quoted programs we can drive it using times with the x combinator.

Let's define a word n_range that takes a starting integer and a count and leaves that many consecutive integers on the stack. For example:


In [10]:
J('23 [dup ++] G 5 [x] times pop')


23 24 25 26 27

We can use dip to untangle [dup ++] G from the arguments.


In [11]:
J('23 5 [[dup ++] G] dip [x] times pop')


23 24 25 26 27

Now that the givens (arguments) are on the left we have the definition we're looking for:


In [12]:
define('n_range == [[dup ++] G] dip [x] times pop')

In [13]:
J('450 10 n_range')


450 451 452 453 454 455 456 457 458 459

This is better just using the times combinator though...


In [14]:
J('450 9 [dup ++] times')


450 451 452 453 454 455 456 457 458 459

Generating Multiples of Three and Five

Look at the treatment of the Project Euler Problem One in Developing a Program.ipynb and you'll see that we might be interested in generating an endless cycle of:

3 2 1 3 1 2 3

To do this we want to encode the numbers as pairs of bits in a single int:

    3  2  1  3  1  2  3
0b 11 10 01 11 01 10 11 == 14811

And pick them off by masking with 3 (binary 11) and then shifting the int right two bits.


In [15]:
define('PE1.1 == dup [3 &] dip 2 >>')

In [16]:
V('14811 PE1.1')


                  . 14811 PE1.1
            14811 . PE1.1
            14811 . dup [3 &] dip 2 >>
      14811 14811 . [3 &] dip 2 >>
14811 14811 [3 &] . dip 2 >>
            14811 . 3 & 14811 2 >>
          14811 3 . & 14811 2 >>
                3 . 14811 2 >>
          3 14811 . 2 >>
        3 14811 2 . >>
           3 3702 . 

If we plug 14811 and [PE1.1] into our generator form...


In [17]:
J('14811 [PE1.1] G')


[14811 swap [PE1.1] direco]

In [18]:
J('[14811 swap [PE1.1] direco] x')


3 [3702 swap [PE1.1] direco]

...we get a generator that works for seven cycles before it reaches zero:


In [19]:
J('[14811 swap [PE1.1] direco] 7 [x] times')


3 2 1 3 1 2 3 [0 swap [PE1.1] direco]

Reset at Zero

We need a function that checks if the int has reached zero and resets it if so.


In [20]:
define('PE1.1.check == dup [pop 14811] [] branch')

In [21]:
J('[14811 swap [PE1.1.check PE1.1] direco] 21 [x] times')


3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 [0 swap [PE1.1.check PE1.1] direco]

Run 466 times

In the PE1 problem we are asked to sum all the multiples of three and five less than 1000. It's worked out that we need to use all seven numbers sixty-six times and then four more.


In [22]:
J('7 66 * 4 +')


466

If we drive our generator 466 times and sum the stack we get 999.


In [23]:
J('[14811 swap [PE1.1.check PE1.1] dip rest cons] 466 [x] times')


3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 [57 swap [PE1.1.check PE1.1] dip rest cons]

In [24]:
J('[14811 swap [PE1.1.check PE1.1] dip rest cons] 466 [x] times pop enstacken sum')


999

Project Euler Problem One


In [25]:
define('PE1.2 == + dup [+] dip')

Now we can add PE1.2 to the quoted program given to times.


In [26]:
J('0 0 [0 swap [PE1.1.check PE1.1] direco] 466 [x [PE1.2] dip] times popop')


233168

Or using G we can write:


In [27]:
J('0 0 0 [PE1.1.check PE1.1] G 466 [x [PE1.2] dip] times popop')


233168

A generator for the Fibonacci Sequence.

Consider:

[b a F] x
[b a F] b a F

The obvious first thing to do is just add b and a:

[b a F] b a +
[b a F] b+a

From here we want to arrive at:

b [b+a b F]

Let's start with swons:

[b a F] b+a swons
[b+a b a F]

Considering this quote as a stack:

F a b b+a

We want to get it to:

F b b+a b

So:

F a b b+a popdd over
F b b+a b

And therefore:

[b+a b a F] [popdd over] infra
[b b+a b F]

And lastly:

[b b+a b F] uncons
b [b+a b F]

Done.

Putting it all together:

F == + swons [popdd over] infra uncons

And:

fib_gen == [1 1 F]

In [28]:
define('fib == + swons [popdd over] infra uncons')

In [29]:
define('fib_gen == [1 1 fib]')

In [30]:
J('fib_gen 10 [x] times')


1 2 3 5 8 13 21 34 55 89 [144 89 fib]

Project Euler Problem Two

By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.

Now that we have a generator for the Fibonacci sequence, we need a function that adds a term in the sequence to a sum if it is even, and pops it otherwise.


In [31]:
define('PE2.1 == dup 2 % [+] [pop] branch')

And a predicate function that detects when the terms in the series "exceed four million".


In [32]:
define('>4M == 4000000 >')

Now it's straightforward to define PE2 as a recursive function that generates terms in the Fibonacci sequence until they exceed four million and sums the even ones.


In [33]:
define('PE2 == 0 fib_gen x [pop >4M] [popop] [[PE2.1] dip x] primrec')

In [34]:
J('PE2')


4613732

Here's the collected program definitions:

fib == + swons [popdd over] infra uncons
fib_gen == [1 1 fib]

even == dup 2 %
>4M == 4000000 >

PE2.1 == even [+] [pop] branch
PE2 == 0 fib_gen x [pop >4M] [popop] [[PE2.1] dip x] primrec

Even-valued Fibonacci Terms

Using o for odd and e for even:

o + o = e
e + e = e
o + e = o

So the Fibonacci sequence considered in terms of just parity would be:

o o e o o e o o e o o e o o e o o e
1 1 2 3 5 8 . . .

Every third term is even.


In [35]:
J('[1 0 fib] x x x')  # To start the sequence with 1 1 2 3 instead of 1 2 3.


1 1 2 [3 2 fib]

Drive the generator three times and popop the two odd terms.


In [36]:
J('[1 0 fib] x x x [popop] dipd')


2 [3 2 fib]

In [37]:
define('PE2.2 == x x x [popop] dipd')

In [38]:
J('[1 0 fib] 10 [PE2.2] times')


2 8 34 144 610 2584 10946 46368 196418 832040 [1346269 832040 fib]

Replace x with our new driver function PE2.2 and start our fib generator at 1 0.


In [39]:
J('0 [1 0 fib] PE2.2 [pop >4M] [popop] [[PE2.1] dip PE2.2] primrec')


4613732

How to compile these?

You would probably start with a special version of G, and perhaps modifications to the default x?