Charming Python #19 (w-50039)

Even More Functional Programming in Python


David Mertz, Ph.D.
Applied Metaphysician, Gnosis Software, Inc.
June, 2001

Earlier installments of this columns touched on many basic concepts of functional programming (FP). This column continues the discussion by illustrating additional capabilities, especially those contained in Xoltar Toolkit: Currying, higher-order functions, and other specialized concepts.

What Is Python?

Python is a freely available, very-high-level, interpreted language developed by Guido van Rossum. It combines a clear syntax with powerful (but optional) object-oriented semantics. Python is available for almost every computer platform you might find yourself working on, and has strong portability between platforms.

Expression Bindings

Never content with partial solutions, one reader--Richard Davies--raised the issue of whether we might move bindings all the way into individual expressions. Let's take a quick look at why we might want to do that, and also show a remarkably elegant way of expressing this that a comp.lang.python contributor provided.

Let us first recall the Bindings class of the functional module. Using the attributes of that class, we were able to assure that a particular name means only one thing within a given block scope:

Python FP session with guarded rebinding

>>> from functional import * 
>>> let = Bindings() 
>>> let.car = lambda lst: lst[0] 
>>> let.car = lambda lst: lst[2] 
Traceback (innermost last): 
  File "<stdin>", line 1, in ? 
  File "d:\tools\functional.py", line 976, in __setattr__ 
    raise BindingError, "Binding '%s' cannot be modified." % name 
functional.BindingError:  Binding 'car' cannot be modified. 
>>> let.car(range(10)) 
0

The Bindings class does what we want within a module or function def scope, but there is no way to make it work within a single expression. In ML-family languages, however, it is natural to create bindings within a single expression:

Haskell expression-level name bindings

 
-- car (x:xs) = x  -- *could* create module-level binding 
list_of_list = [[1,2,3],[4,5,6],[7,8,9]] 
 
-- 'where' clause for expression-level binding 
firsts1 = [car x | x <- list_of_list] where car (x:xs) = x 
 
-- 'let' clause for expression-level binding 
firsts2 = let car (x:xs) = x in [car x | x <- list_of_list] 
 
-- more idiomatic higher-order 'map' technique 
firsts3 = map car list_of_list where car (x:xs) = x 
 
-- Result: firsts1 == firsts2 == firsts3 == [1,4,7] 
 

Greg Ewing observed that we can accomplish the same effect using Python's list-comprehensions; we can even do it in a way that is nearly as clean as Haskell's syntax:

Python 2.0+ expression-level name bindings

>>> list_of_list = [[1,2,3],[4,5,6],[7,8,9]] 
>>> [car_x for x in list_of_list for car_x in (x[0],)] 
[1, 4, 7]

This trick of putting an expession inside a single-item tuple in a list-comprehension does not provide any way of using expression-level bindings with higher-order functions. To use the higher-order functions, we still need to use block-level bindings, as with:

Python block-level bindings with 'map()'

>>> list_of_list = [[1,2,3],[4,5,6],[7,8,9]] 
>>> let = Bindings() 
>>> let.car = lambda l: l[0] 
>>> map(let.car,list_of_list) 
[1, 4, 7]

Not bad, but if we want to use map(), the scope of the binding remains a little broader than we might want. Nonetheless, it is possible to coax list comprehensions into doing our name bindings for us, even in cases where a list is not what we finally want:

"Stepping down" from Python list-comprehension

# Compare Haskell expression: 
# result = func car_car 
#          where 
#              car (x:xs) = x 
#              car_car = car (car list_of_list) 
#              func x = x + x^2 
>>> [func for x in list_of_list 
...       for car in (x[0],) 
...       for func in (car+car**2,)][0] 
2

What we have done is peform an arithmetic calculation on the first element of the first element of list_of_list while also naming the arithmetic calculation (but only in expression scope). As an "optimization" we might not bother to create a list longer than one element to start with, since we choose only the first element with the ending index 0:

Efficient stepping down from list-comprehension

 
>>> [func for x in list_of_list[:1] 
...       for car in (x[0],) 
...       for func in (car+car**2,)][0] 
2 
 
 

Higher-order Functions: Currying

Three of the most general higher-order functions are built into Python: map(), reduce() and filter(). What these functions do--and the reason we call them "higher-order"--is take other functions as (some of) their arguments. Other higher-order functions, but not these builtins, return function objects.

Python has always given users the ability to construct their own higher-order functions by virtue of the first-class status of function objects. A trivial case might look like:

Trivial Python function factory

>>> def foo_factory(): 
...     def foo(): 
...         print "Foo function from factory" 
...     return foo 
... 
>>> f = foo_factory() 
>>> f() 
Foo function from factory

The Xoltar Toolkit (see Resources), which I discussed also in early installments of this column, comes with a nice collection of higher-order functions. Most of the functions that Xoltar's functional module provides are ones that have been developed in various traditionally functional languages, and whose usefulness has proved itself over many years.

Possibly the most famous and most important higher-order function is traditionally called curry(). curry() is named after the logician Haskell Curry, whose first-name is also used to name the abovementioned programming language. The insight that underlies "currying" is that it is possible to treat (almost) every function as a partial function of just one argument. All that is necessary for currying to work is to allow the return value of functions to themselves be functions, but with the returned functions "narrowed" or "closer to completion." The way this works is quite similar to the closures that I wrote about in an earlier column--each successive call to a curried return function "fills in" more of the data involved in a final computation (data attached to a procedure).

Let's illustrate currying first with a very simple example in Haskell, then with the same example repeated in Python using the functional module:

Currying a Haskell computation

 
computation a b c d = (a + b^2+ c^3 + d^4) 
check = 1 + 2^2 + 3^3 + 5^4 
 
fillOne   = computation 1  -- specify "a" 
fillTwo   = fillOne 2      -- specify "b" 
fillThree = fillTwo 3      -- specify "c" 
answer    = fillThree 5    -- specify "d" 
 
-- Result: check == answer == 657 
 

Now in Python:

Currying a Python computation

>>> from functional import curry 
>>> computation = lambda a,b,c,d: (a + b**2 + c**3 + d**4) 
>>> computation(1,2,3,5) 
657 
>>> fillZero  = curry(computation) 
>>> fillOne   = fillZero(1)   # specify "a" 
>>> fillTwo   = fillOne(2)    # specify "b" 
>>> fillThree = fillTwo(3)    # specify "c" 
>>> answer    = fillThree(5)  # specify "d" 
>>> answer 
657

The parallel with closures can be illustrated further by presenting the same simple tax-calculation program used in the earlier installment (this time using curry()):

Python curried tax calculations

from functional import * 
 
taxcalc = lambda income,rate,deduct: (income-(deduct))*rate 
 
taxCurry = curry(taxcalc) 
taxCurry = taxCurry(50000) 
taxCurry = taxCurry(0.30) 
taxCurry = taxCurry(10000) 
print "Curried taxes due =",taxCurry 
 
print "Curried expression taxes due =", \ 
      curry(taxcalc)(50000)(0.30)(10000)

Unlike with closures, we need to curry the arguments in a specific order (left to right). But note that functional also contains an rcurry() class that will start at the other end (right to left).

The second print statement in the example at one level is a trivial spelling change from simply calling the normal taxcalc(50000,0.30,10000). But in a different level it makes rather clear the concept that every function can be a function of just one argument--a rather surprising idea to those new to it.

Miscellanous Higher-order Functions

Beyond the "fundamental" operation of currying, functional provides a grab-bag of interesting higher-order functions. Moreover, it is really not hard to write your own higher-order functions--either with or without functional. The ones in functional provide some interesting ideas, at the least.

For the most part, higher-order functions feel like "enhanced" versions of the standard map(), filter() and reduce(). A lot of the time the pattern in these functions is roughly "take a function or functions and some lists as arguments, then apply the function(s) to list arguments." There are a surprising number of interesting and useful ways to play on this theme. Another pattern is "take a collection of functions and create a function that combines their functionality." Again, numerous variations are possible. Let us look at some of what functional provides.

The functions seqential() and also() both create a function based on a sequence of component functions. The component functions can then be called with the same argument(s). The main difference between the two is simply that sequential() expects a single list as an argument, while also() takes a list of arguments. In most cases, these are useful for function side effects, but sequential() optionally lets you choose which function provides the combined return value:

Sequential calls to functions (with same args)

 
>>> def a(x): 
...     print x, 
...     return "a" 
... 
>>> def b(x): 
...     print x*2, 
...     return "b" 
... 
>>> def c(x): 
...     print x*3, 
...     return "c" 
... 
>>> r = also(a,b,c) 
>>> r 
<functional.sequential instance at 0xb86ac> 
>>> r(5) 
5 10 15 
'a' 
>>> sequential([a,b,c],main=c)('x') 
x xx xxx 
'c' 
 

The functions disjoin() and conjoin() are similar to sequential() and also() in terms of creating new functions that apply argument(s) to several component functions. But disjoin() asks whether any component functions return true (given the argument(s)), and conjoin() asks whether all components return true. Logical shortcutting is applied, where possible, so some side effects might not occur with disjoin(). joinfuncs() is similar to also(), but returns a tuple of the components' return values rather than selecting a main one.

Where the previous functions let you call multiple functions with the same argument(s), any(), all() and none_of() let you call the same function against a list of arguments. In general structure, these are a bit like the builtin map(), reduce(), filter() functions. But these particular higher-order functions from functional ask boolean questions about collections of return values. For example:

Ask about collections of return values

 
>>> from functional import * 
>>> isEven = lambda n: (n%2 == 0) 
>>> any([1,3,5,8], isEven) 
1 
>>> any([1,3,5,7], isEven) 
0 
>>> none_of([1,3,5,7], isEven) 
1 
>>> all([2,4,6,8], isEven) 
1 
>>> all([2,4,6,7], isEven) 
0 
 

A particularly interesting higher-order function for those with a little bit of mathematics background is compose(). The composition of several functions is a "chaining together" of the return value of one function to the input of the next function. The programmer who composes several functions is responsible for making sure the outputs and inputs match up--but then, that is true any time a programmer uses a return value. A simple example makes it clear:

Creating compositional functions

 
>>> def minus7(n): return n-7 
... 
>>> def times3(n): return n*3 
... 
>>> minus7(10) 
3 
>>> minustimes = compose(times3,minus7) 
>>> minustimes(10) 
9 
>>> times3(minus7(10)) 
9 
>>> timesminus = compose(minus7,times3) 
>>> timesminus(10) 
23 
>>> minus7(times3(10)) 
23 
 
 

Until Next Time

I hope this latest look at higher-order functions will picque readers' interest in a certain style of thinking. By all means, play with it. Try to create some of your own higher-order functions; some might well prove useful and powerful. Let me know how it goes, perhaps a later installment of this ad hoc series will discuss the novel and fascinating ideas that readers continue to provide.

Resources

The earlier installments of this series on functional programming in Python can be found at:

http://gnosis.cx/publish/programming/charming_python_13.html
http://gnosis.cx/publish/programming/charming_python_16.html

Bryn Keller's "xoltar toolkit" which includes the module functional adds a large number of useful FP extensions to Python. Since the functional module is itself written entirely in Python, what it does was already possible in Python itself. But Keller has figured out a very nicely integrated set of extensions, with a lot of power in compact definitions. The toolkit can be found at:

http://sourceforge.net/projects/xoltar-toolkit

A good starting point for functional programming is the Frequently Asked Questions for comp.lang.functional :

http://www.cs.nott.ac.uk/~gmh//faq.html#functional-languages

The author has found it much easier to get a grasp of functional programming via the language Haskell than in Lisp/Scheme (even though the latter is probably more widely used, if only in Emacs). Other Python programmers might similarly have an easier time without quite so many parentheses and prefix (Polish) operators.

http://www.haskell.org/

An excellent introductory book is:

Haskell: The Craft of Functional Programming (2nd Edition), Simon Thompson, Addison-Wesley (1999).

A book with a somewhat more applied feel which is an equally introduction to Haskell is:

The Haskell School of Expression: Learning Functional Programming Through Multimeia, Paul Hudak, Cambridge University Press (2000).

About The Author

Picture of Author Since conceptions without intuitions are empty, and intuitions without conceptions, blind, David Mertz wants a cast sculpture of Milton for his office. Start planning for his birthday. David may be reached at mertz@gnosis.cx; his life pored over at http://gnosis.cx/publish/. Suggestions and recommendations on this, past, or future, columns are welcomed.