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The Classic FizzBuzz program implemented in pure Ruby-Flavored Lambda Calculus

branch: master
README.textile

FizzBuzz in Lambda-Calculus

This file is inspired by Tom Stuart’s “Programming with Nothing”
article
(http://experthuman.com/programming-with-nothing).
Most of the code is taken directly from there, although I do vary
from it in a few places.

Files of Interest:

  • fizzbuzz.rb — The original code file that generated this readme.
  • cfb.rb — The “compiled” version of the fizzbuzz program, with all the constants expanded inline.
  • cfb2.rb — Same as cfb.rb, but with the code nicely formatted and indented.
  • interface.rb — The full Ruby interface functions used to translate the lambda calculus data structures to something that is human readable.

The Rules:

We are only allowed to write procs and to call procs. No other Ruby
operation is allowed in the main program.

That’s it! No numbers, strings, methods, classes, mixins,
operators, or any other Ruby construct other than creating procs and
calling procs.

We do relax that rule to allow defining constants. This allows us
to name useful fragments of code. Since strict replacement is the
key, we don’t allow any self-reference (this become important when
dealing with recursive functions).

Although the main program is completely written using nothing but lambda
expressions, we do allow some standard Ruby helper functions to
translate from the lambda calculus representations to something human
readable. You will find the helper functions in the interface.rb
file.

Numbers

We will start with some Church numerals. A Church numeral is a
function that takes a function f and a value v, and applies f
to v the appropriate number of times. For example, ONE applies
f once. THREE applies f 3 times. And ZERO applies f zero
times.

ZERO  = ->(f) { ->(v) {                v      } }
ONE   = ->(f) { ->(v) {             f.(v)     } }
THREE = ->(f) { ->(v) {       f.(f.(f.(v)))   } }
FIVE  = ->(f) { ->(v) { f.(f.(f.(f.(f.(v))))) } }

TEN   = ->(f) { ->(v) { f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(v)))))))))) } }

FIFTEEN   = ->(f) { ->(v) { f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(v))))))))))))))) } }

HUNDRED = ->(f) {
  ->(v) {
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
    f.(f.(f.(f.(f.(f.(f.(f.(f.(f.(
     v
    ))))))))))
    ))))))))))
    ))))))))))
    ))))))))))
    ))))))))))
    ))))))))))
    ))))))))))
    ))))))))))
    ))))))))))
    ))))))))))
  }
}

Boolean Values and Functions

Boolean values are functions that take two arguments and return one
of them. True returns the first argument and false returns the
second.

BTRUE  = ->(a) { ->(b) { a } }
BFALSE = ->(a) { ->(b) { b } }

Side note about multiple arguments. We could write functions that
take two arguments like this:

        # Example
        BTRUE = ->(a, b) { a }

But it turns out that it is convenient to represent functions of
multiple arguments as nested functions, so we will write all of our
functions in this manner. So a boolean function (BTRUE or
BFALSE) written in this manner will be called like this:

        # Example
        bool.(TRUE_VALUE).(FALSE_VALUE)

The logical not operation is interesting. It is a function that
takes a boolean, and returns the opposite value of the boolean as a
result.

NOT = ->(bool) { bool.(BFALSE).(BTRUE) }

Is a number zero? Give the number a function that always returns
false, and feed it an initial value of true. If the number is zero,
then the initial value (true) is returned. If the number is
anything else, then then false (from the supplied function) will be
returned.

IS_ZERO = ->(n) { n.(->(x) { BFALSE }).(BTRUE) }

An interesting result of defining boolean values as functions is
that we don’t need an explicit if statement. If I want to test
number n for being zero and return ONE if it is zero and THREE
if is not, it would be written like this:

        # Example
        IS_ZERO.(n).(ONE).(THREE)

Pairs

The pair is going to be our basic construct for building data
structures. The pair constructure takes two values (a right side
value and a left side value) and returns a closure that binds the
two values together.

        # Example
        P = PAIR.(ONE).(THREE)  # returns a pair containing 1 and 3

Here’s the definition of PAIR:

PAIR  = ->(left) { ->(right){ ->(f) { f.(left).(right) } } }

To extract the paired values, call the pair function with a function
that takes the left and right values as arguments and returns the
one you need. For example, given P above, LEFT and RIGHT
work like this:

        # Example
       LEFT.(P)   # returns ONE
       RIGHT.(P)  # returns THREE

Here’s the defintions of LEFT and RIGHT:

LEFT  = ->(p) { p.(->(left) { ->(right) { left } } ) }
RIGHT = ->(p) { p.(->(left) { ->(right) { right } } ) }

Math Functions

To increment a number, we just return a number-like function (one
taking a function argument and a value) and arrange to have the
given function called one more time that the number.

INC = ->(n) { ->(f) { ->(v) { f.(n.(f).(v)) } } }

Decrementing turns out to be quite a bit more difficult than
incrementing. Tom Stuart uses a decrementing function that he
doesn’t explain (and I can’t quite figure it out). I’m using a
different technique that seems pretty straight forward to me.

It helps to think of decrementing as finding the predecessor of a
given number. Imagine you have a pair of numbers (n, n-1), where n-1
is the predecessor of n. The PHI function takes this pair of
numbers and returns a new pair that contains (n+1, n).

PHI = ->(pair){
  PAIR.(INC.(LEFT.(pair))).(LEFT.(pair))
}

So, to implement decrement, all we need to do is start with the pair
(0, 0) and apply PHI n times, yielding (n, n-1). We then just
choose the right hand side of the resulting pair to get the
predecessor of n. Notice that this definition of DEC will yield 0
as the predecessor of 0. A strange result, but one we will use to
advantage later.

DEC = ->(n) { RIGHT.(n.(PHI).(PAIR.(ZERO).(ZERO))) }

Now that we can increment and decrement, writing some basic math
functions is pretty straight forward.

ADD = ->(a) { ->(b) { a.(INC).(b) } }
SUB = ->(a) { ->(b) { b.(DEC).(a) } }
MUL = ->(a) { ->(b) { a.(ADD.(b)).(ZERO) } }

Numeric Comparisons

To tell if a is less than or equal to b, subtract b from a
(a-b), and if the result is less than or equal to zero, then a <= b. Since our number representation does not handle negative
numbers, we have a weird case where a-b is 0 if b is larger than
a. This works out for us because then all we need to do is test
for zero.

IS_LEQ = ->(a) { ->(b) { IS_ZERO.(SUB.(a).(b)) } }

I actually want to use “<” in the next function. We can derive
the less than function from “<=” by reversing the arguments and
applying a logical NOT.

IS_LT = ->(a) { ->(b) { NOT.(IS_LEQ.(b).(a)) } }

Modulus (and recursive functions)

If we were to write a modulus function in straight Ruby, it might
look something like this

        # Example
        def mod(a,b)
          a<b ? a : mod(a-b,b)
        end

There are two problems with translating this definition to our
rather limited lambda-calculus. First, mod as defined above is
recursive, calling itself by name from within its own body. Our rule
about no self-references means that we can’t write MOD as a
straight translation of the above code.

No worries. We will use the Y-Combinator to construct our recursive
definition. Here’s the Applicative Order version of the
Y-Combinator (also called the Z-Combinator).

Y = ->(f) {
  ->(g) { ->(n) { f.(g.(g)).(n) } }.(
  ->(g) { ->(n) { f.(g.(g)).(n) } } )
}

To create a recursive function using Y, we need to define what I
call an improver function. An improver function takes a partial
function defintion and improves it just a little bit.

Here’s the improver function for modulus:

MOD_IMPROVER = ->(partial) {
  ->(a) {
    ->(b) {
      IS_LT.(a).(b).
      ( a ).
      ( ->(x) { partial.(SUB.(a).(b)).(b).(x) } )
    }
  }
}

Suppose you have a partial definition of modulus that only works for
cases where a is less than b. If you feed that function into
the improver, you will get a function that works for a < 2*b.
Feed that back into the improver and you get a modulus function that
works for a < 3*b. Each new version is slightly better than the
next. The Y-Combinator takes the improver and turns it into the
full version of modulus.

MOD = Y.(MOD_IMPROVER)

Let’s talk about the ->(x) { } function that wraps the call to
partial in the definition of MOD. Since arguments are always
evaluated before passed to functions, attempting to pass the partial
call directly will result in infinte recursion and stack overflow.
By wrapping it in a function, we pass a function instead. The
function is called and the partial invoked only when needed.

Lists

We use pairs to build lists of arbitrary length. Two pairs are used
to hold a single element of the list. The pairs are

        # Example
        pair 1
          left => false (flag indicating not the end of the list)
          right => pair 2
        pair 2
          left => element
          right => rest of the list

So the empty list is a single pair where the left value is true
(ie. end of list)

EMPTY = PAIR.(BTRUE).(BTRUE)
IS_EMPTY = LEFT

CONS adds an item to the begining of a list.

CONS = ->(item) { ->(lst) { PAIR.(BFALSE).(PAIR.(item).(lst)) } }

HEAD returns the first item of a list. TAIL returns a list of
all the remaining elements after the first.

HEAD = ->(lst) { LEFT.(RIGHT.(lst)) }
TAIL = ->(lst) { RIGHT.(RIGHT.(lst)) }

Ranges

We need an easy way to generate a list of numbers up to a maximum.
RANGE.(n) will return a list from 1 to n.

RANGE_BUILDER_IMPROVER = ->(partial) {
  ->(lst) {
    ->(n) {
      IS_ZERO.(n).(lst).( ->(x) { partial.(CONS.(n).(lst)).(DEC.(n)).(x) } )
    }
  }
}
RANGE = Y.(RANGE_BUILDER_IMPROVER).(EMPTY)

Fold and Map

In Ruby, we use each to build all the enumerable methods. Here we
will do the same, but starting with FOLD. FOLD is similar to
Ruby’s reduce. It is called like this:

        # Example
        FOLD.(list).(initial).(f)

Where f is a function of two arguments called like this:

        # Example
        f.(item).(previous_value)

The function f will be called agains each item in the list. The
second argument to f is the value of the previous call to f. The
first time f is called, the initial value will be passed in for the
previous value.

FOLD_IMPROVER = ->(partial) {
  ->(lst) {
    ->(initial) {
      ->(f) {
        IS_EMPTY.(lst).
        (initial).
        ( ->(x) { f.(HEAD.(lst)).(partial.(TAIL.(lst)).(initial).(f)).(x) } )
      }
    }
  }
}
FOLD = Y.(FOLD_IMPROVER)

We will build map from fold by using a function that conses
f(item) onto the previous return value. By initializing with the
empty list, the result will be a list of f applied to each item.

MAP = ->(lst) { ->(f) { FOLD.(lst).(EMPTY).( ->(item) { ->(lst) { CONS.(f.(item)).(lst) } } ) } }

Character Encodings and Strings

We can choose any character encoding we find convenient. Since we
only need the 10 digits and the letters ‘B’, ‘F’, ‘i’ ‘u’ and ‘z’,
we choose to let the numbers < 10 represent themselves and numbers
over 10 to represent the handful of letters we need. (This is the
same encoding that Tom Stuart used.)

Note that the to_char and to_string helper functions in
‘interface.rb’ are sensitive to this choice of encoding.

B = TEN
F = INC.(B)
I = INC.(F)
U = INC.(I)
Z = INC.(U)

Strings are just lists of characters. We need “Fizz”, “Buzz” and
“FizzBuzz”.

FIZZ = CONS.(F).(CONS.(I).(CONS.(Z).(CONS.(Z).(EMPTY))))
BUZZ = CONS.(B).(CONS.(U).(CONS.(Z).(CONS.(Z).(EMPTY))))
FIZZBUZZ = CONS.(F).(CONS.(I).(CONS.(Z).(CONS.(Z).(BUZZ))))

Division

A recursive definition for division. Keep subtracting the
denominator from the numerator until the numerator is less than the
numerator, then just return the number of times the subtraction was
done.

DIV_IMPROVER = ->(partial) {
  ->(num) {
    ->(denom){
      IS_LT.(num).(denom).
      (ZERO).
      ( ->(x) { INC.(partial.(SUB.(num).(denom)).(denom)).(x) } )
    }
  }
}
DIV = Y.(DIV_IMPROVER)

Converting numbers to strings

This is a standard number to string conversion method. The digits
are calculated starting with the least significant digit of the
result (n % 10). To prevent the digits from being built in
reverse order, we use a trick where we pass the “result so far” as
an argument. When the number is finished converting (n<10), we
just return the cons of n on the “result so far”. Initializing
the “result so far” with an empty list completes the trick.

Tom points out that choosing an encoding where a number < 10 is its
own character encoding greatly simplifies the overall conversion,
and I certainly agree.

TO_DIGITS_IMPROVER = ->(partial) {
  ->(result) {
    ->(n) {
      IS_LT.(n).(TEN).
      (CONS.(n).(result)).
      ( ->(x) { partial.(CONS.(MOD.(n).(TEN)).(result)).(DIV.(n).(TEN)).(x) } )
    }
  }
}
TO_DIGITS = Y.(TO_DIGITS_IMPROVER).(EMPTY)

The FizzBuzz Main Program

We now have everything in place to create the FizzBuzz program. The
program will create a list of FizzBuzz numbers (as strings) from 1
up to limit. We use RANGE to create the initial list of numbers
and then MAP to walk the list and transform it into either “Fizz”,
“Buzz”, “FizzBuzz”, or the string representation of the number.

PROGRAM = ->(limit){
  MAP.(RANGE.(limit)).(
    ->(n) {
      IS_ZERO.(MOD.(n).(FIFTEEN)).(
        FIZZBUZZ
      ).(
      IS_ZERO.(MOD.(n).(THREE)).(
        FIZZ
      ).(
      IS_ZERO.(MOD.(n).(FIVE)).(
        BUZZ
      ).(
        TO_DIGITS.(n)
      )))
    }
    )
}

Kick-start the FizzBuzz program and print its results.

Finally we run the program and collect the result in RESULT. We
then use the full Ruby-based interface functions to transcribe our
calculated result to something humans can read.

if $0 == __FILE__
  RESULT = PROGRAM.(HUNDRED)

  require "interface"
  puts to_strings(RESULT)
end
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