Turning a while loop into a recursive function

This is terribly simple, but it would appear I didn’t really know it for at least 20 years of my life, so it’s perhaps worth writing it down. (I suspect this is a consequence of working in languages which didn’t have tuple types):

Any while loop can be turned into a recursive function by doing little more than moving text around.

We can think of all while loops as doing the following:

state = init_state
constants = init_consts
while condition(state):
    state = f(state, constants)

Here state is shorthand for “everything that the while loop changes” whilst it is running, and f is shorthand for “the changes that the loop code makes”.

Given this generalized while loop we can rewrite this as:

def loop_func(state, constants):
    if condition(state):
         return loop_func(f(state, constants), constants)
    else:
        return state

state = loop_func(init_state, init_constants)

and life is once again incredibly dull.

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Going the other way, if you have a recursive function (not mutually recursive) like so

def f(state):
   if end_state(state):
       return f_known(state)
   new_state = change_state(state)
   return f(new_state)

then you can turn it into a while loop like this:

while not end_state(state):
      state = change_state(state)
return f_known(state)

Again, this is terribly simple once you start writing it down for yourself (though perhaps not if you are reading someone else write it)

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Post script
This also happens to let you write a “while function” which is analogous to the reduce function, though this seems to be noticeably lacking from haskell (I might just be mistaken about this). Perhaps, it is considered to produce code that is hard to read. This would look something like:

def while_reduce(condition, state_change, init_state):
         if condition(init_state):
                 return while_condition(condition, state_change, state_change(init_state)
         else:
                return init_state

Reduce (fold) for the imperatively minded

The reduce or fold function is a concept in functional programming languages which can be used to simulate looping behaviour without the need for a real loop, and in particularly without the need for any explicit variables. I believe it originated in the field of functional programming languages, where getting rid of any form of ‘state’ seems to be an overriding aim.

I suspect that, with suitable training, one may be able to grasp the meaning of a

Currying, state and higher-order functions

Take any higher order-function that returns a single function. It will always take the following form (upto some minor very minor changes):

def f(x):
     s = STATE(x)
     def g(y):
          return CODE(s,y)

g = f(arg)

This can immediately be converted to:

def f(x, y):
     s = STATE(x)
     return CODE(s,y)

g = lambda y: f(x, y)

This means that if we are working in a referentially transparent code with no side-effects, every higher-order function returning a function can be thought of as a function in several variables.

Its return value can be thought of as the function obtained by setting the values of some the variables in a multivariate function to constants. This means that, in a way, higher-order functions are just a mechanism for storing data together with a function. This is very much like what objects do in object-oriented programming, and in fact it is possible to replace any higher-order function with an equivalent class (I’ll talk about this in another post).

The only difference between the two function calls is that in the second STATE(arg) will get evaluated whenever g gets called, whereas in the first STATE(arg) will only ever get called once.

Second-order functions versus currying in python

Which code sample seems better:

Sample one:

def g(x):
    def f(y):
        return x + y
    return f

a = g(1)

Sample 2

def f(x,y):
    return x  + y

a = lambda x: f(x,1)

I’d argue that the second was simpler because:

• It is shorter
• Understanding functions of two arguments is easier than understanding functions that return functions
• The level of indirection to the code that actual does things is reduced by one.

Would you ever see the first example written? It probably depends how trendy the people you hang around with are. But if you are the kind of person who might feel inclined write this sort of thing, I think the second way is a better than the first.

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Afterthougts:
In case you want the title to make sense, I’d refer to the first example as using a second-order function and the first as using a type of currying. (Though the word currying tends to have a more precise meaning)

Also it is debatable whether such changes will make a large difference to the quality of code as compared to other changes.