David Mertz, Ph.D.
Applied Metaphysician, Gnosis Software, Inc.
Although users usually think of Python has a procedural and object oriented language, it actually contains everything one needs for a completely functional approach to programming. This article discusses general concepts of functional programming, and illustrates ways of implementing functional techniques in 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.
We better start with the hardest question: "What is functional programming (FP), anyway?" One approach would be to say that FP is what you do when you program in languages like Lisp, Scheme, Haskell, ML, OCAML, Clean, Mercury or Erlang (or a few others). That is a safe answer, but not one that clarifies very much. Unfortunately, it is hard to get a consistent opinion on just what FP is, even from functional programmers themselves. A story about elephants and blind men seems apropos here. It is also safe to contrast FP with "imperative programming" (what one does in languages like C, Pascal, C++, Java, Perl, Awk, TCL, and most others, at least for the most part).
While the author by all means welcomes the advice of those who know better, he would roughly characterize functional programming as having at least several of the following characteristics. Languages that get called functional make these things easy, and other things either hard or disallowed:
* Functions are first class (objects). That is, everything you can do with "data" can be done with functions themselves (such as passing a function to another function).
* Use of recursion as a primary control structure. In some languages, no other "loop" construct exists other than recursion.
* Focus on LISt Processing (e.g. the name
Lisp). Lists are often used with recursion on sub-lists as a substitute for loops.
* "Pure" functional languages eschew side-effects. This excludes the almost ubiquitous pattern in imperative languages of assigning first one, then another value, to the same variable to track program state.
* FP either discourages or outright disallows statements, and instead works with the evaluation of expressions (i.e. functions plus arguments). In the pure case, one program is one expression (plus supporting definitions).
* FP worries about what is to be computed rather than how it is to be computed.
* Much FP utilizes "higher order" functions (i.e. functions that operate on functions that operate on functions).
Advocates of functional programming argue that all these characteristic make for more rapidly developed, shorter, and less bug-prone code. Moreover, high theorists of computer science, logic, and math find it a lot easier to prove formal properties of functional languages and programs than of imperative languages and programs.
Python has had most of the characteristics of FP listed above since Python 1.0. But as with most Python features, they have been present in a very mixed language. Much as with Python's OOP features, you can use what you want and ignore the rest (until you need it later). With Python 2.0, a very nice bit of "syntactic sugar" was added with list comprehensions. While list comprehensions add no entirely new capability, they make a lot of the old capabilities look a lot nicer.
The basic elements of FP in Python are the functions
filter(), and the operator
Python 1.x, the
apply() function also comes in handy for
direct application of one function's list return value to
another function. Python 2.0 provides an improved syntax for
this purpose. Perhaps surprisingly, these very few functions
(and the basic operators), are almost sufficient to write any
Python program; specifically, the flow control statements
def) can all be handled
in a functional style using exclusively the FP functions and
operators. While actually eliminating all flow control
commands in a program is probably only useful for entering an
"obfuscated Python" contest (with code that will look a lot
like Lisp), it is worth understanding how FP expresses flow
control with functions and recursion.
The first thing to think about in our elimination exercise is
the fact that Python "short circuits" evaluation of boolean
expressions. This turns out to provide an expression version
else blocks (assuming each block calls one
function, which is always possible to arrange). Here is how:
# Normal statement-based flow control if <cond1>: func1() elif <cond2>: func2() else: func3() # Equivalent "short circuit" expression (<cond1> and func1()) or (<cond2> and func2()) or (func3()) # Example "short circuit" expression >>> x = 3 >>> def pr(s): return s >>> (x==1 and pr('one')) or (x==2 and pr('two')) or (pr('other')) 'other' >>> x = 2 >>> (x==1 and pr('one')) or (x==2 and pr('two')) or (pr('other')) 'two'
Our expression version of conditional calls might seem to be
nothing but a parlor trick; however, it is more interesting
when we notice that the
lambda operator must return an
expression. Since--as we have shown--expressions can contain
conditional blocks via short circuiting, a
is fully general in expressing conditional return values.
Building on our example:
>>> pr = lambda s:s >>> namenum = lambda x: (x==1 and pr("one")) \ ... or (x==2 and pr("two")) \ ... or (pr("other")) >>> namenum(1) 'one' >>> namenum(2) 'two' >>> namenum(3) 'other'
The above examples have already witnessed the first class
status of functions in Python, but in a subtle way. When we
create a function object with the
lambda operation we have
something entirely general. As such, we were able to bind our
objects to the names "pr" and "namenum", in exactly the same
way we might have bound the number 23 or the string "spam" to
those names. But just like we can use the number 23 without
binding it to any name (i.e. as a function argument), we can
use the function object we created with
binding it to any name. A function is simply another value we
might do something with in Python.
The main thing we do with our first class objects, is pass them
to our FP builtin functions
Each of these functions accepts a function object as its first
map() performs the passed function on each
corresponding item in the specified list(s), and returns a list
reduce() performs the passed function on each
subsequent item and an internal accumulator of a final result;
reduce(lambda n,m:n*m, range(1,10)) means
"factorial of 10" (i.e. multiply each item by the product of
filter() uses the passed function
to "evaluate" each item in a list, and return a winnowed list
of the items that pass the function test. We also often pass
function objects to our own custom functions, but usually those
amount to combinations of the mentioned builtins.
By combining these three FP builtin functions, a surprising range of "flow" operations can be performed (all without statements, only expressions).
Replacing loops is as simple as was replacing conditional
for can be directly translated to
map(). As with
our conditional execution, we will need to simplify statement
blocks to single function calls (we are getting close to being
able to generally):
for e in lst: func(e) # statement-based loop map(func,lst) # map()-based loop
By the way, a similar technique is available for a functional
approach to sequential program flow. That is, imperative
programming mostly consists of statements that amount to "do
this, then do that, then do the other thing."
map() lets us
do just this:
# let's create an execution utility function do_it = lambda f: f() # let f1, f2, f3 (etc) be functions that perform actions map(do_it, [f1,f2,f3]) # map()-based action sequence
In general, the whole of our main program can be a
expression with a list of functions to execute to complete the
program. Another handy feature of first class functions is
that you can put them in a list.
while is slightly more complicated, but is still
# statement-based while loop while <cond>: <pre-suite> if <break_condition>: break else: <suite> # FP-style recursive while loop def while_block(): <pre-suite> if <break_condition>: return 1 else: <suite> return 0 while_FP = lambda: (<cond> and while_block()) or while_FP() while_FP()
Our translation of
while still requires a
function that may itself contain statements rather than just
expressions. But we might be able to apply further
eliminations to that function (such as short circuiting the
if/else in the template. Also, it is hard for <cond> to be
useful with the usual tests, such as
while myvar==7, since
the loop body (by design) cannot change any variable values
(well, globals could be modified in
while_block()). One way
to add a more useful condition is to let
a more interesting value, and compare that return for a
termination condition. It is worth looking at a concrete
example of eliminating statements:
# imperative version of "echo()" def echo_IMP(): while 1: x = raw_input("IMP -- ") if x == 'quit': break else: print x echo_IMP() # utility function for "identity with side-effect" def monadic_print(x): print x return x # FP version of "echo()" echo_FP = lambda: monadic_print(raw_input("FP -- "))=='quit' or echo_FP() echo_FP()
What we have accomplished is that we have managed to express a
little program that involves I/O, looping, and conditional
statments as a pure expression with recursion (in fact, as a
function object that can be passed elsewhere if desired). We
do still utilize the utility function
this function is completely general, and can be reused in every
functional program expression we might create later (it's a
one-time cost). Notice that any expression containing
monadic_print(x) evaluates to the same thing as if it had
x. FP (particularly Haskell) has the notion
of a "monad" for a function that "does nothing, and has a side
effect in the process."
After all this work in getting rid of perfectly sensible statements and substituting obscure nested expressions for them, a naturaly question is "Why?!" Reading over my descriptions of FP, we can see that all of them are achieved in Python. But the most important characteristic--and the one likely to be concretely useful--is the elimination of side-effects (or at least their containment to special areas like monads). A very large percentage of program errors--and the problem that drives programmers to debuggers--occur because variables obtain unexpected values during the course of program execution. Functional programs bypasses this particular issue by simply not assigning values to variables at all.
Let's look at a fairly ordinary bit of imperative code. The goal here is to print out a list of pairs of numbers whose product is more than 25. The numbers that make up the pairs are themselves taken from two other lists. This sort of thing is moderately similar to things that programmers actually do in segments of their programs. An imperative approach to the goal might look like:
# Nested loop procedural style for finding big products xs = (1,2,3,4) ys = (10,15,3,22) bigmuls =  # ...more stuff... for x in xs: for y in ys: # ...more stuff... if x*y > 25: bigmuls.append((x,y)) # ...more stuff... # ...more stuff... print bigmuls
This project is small enough that nothing is likely to go
wrong. But perhaps our goal is embedded in code that
accomplishes a number of other goals at the same time. The
sections commented with "more stuff" are the places where
side-effects are likely to lead to bugs. At any of these
points, the variables
acquire unexpected values in the hypothetical abbreviated code.
Futhermore, after this bit of code is done, all the variables
have values that may or may not be expected and wanted by later
code. Obviously, encapsulation in functions/instances and care
as to scoping can be used to guard against this type of error.
And you can always
del your variables when you are done with
them. But in practice, the types of errors pointed to are
A functional approach to our goal eliminates these side-effect errors altogether. A possible bit of code is:
bigmuls = lambda xs,ys: filter(lambda (x,y):x*y > 25, combine(xs,ys)) combine = lambda xs,ys: map(None, xs*len(ys), dupelms(ys,len(xs))) dupelms = lambda lst,n: reduce(lambda s,t:s+t, map(lambda l,n=n: [l]*n, lst)) print bigmuls((1,2,3,4),(10,15,3,22))
We bind our anonymous (
lambda) function objects to names in
the example, but that is not strictly necessary. We could
instead simply nest the definitions. For readability we do it
this way; but also because
combine() is a nice utility
function to have anyway (produce a list of all pairs of
elements from two input lists).
dupelms() in turn is mostly
just a way of helping out
combine(). Even though this
functional example is more verbose than the imperative example,
once you consider the utility functions for reuse, the new code
bigmuls() itself is probably slightly less than in the
The real advantage of this functional example is that
absolutely no variables change any values within it. There are
no possible unanticipated side-effects on later code (or from
earlier code). Obviously, the lack of side-effects, in itself,
does not guarantee that the code is correct, but it is
nonetheless an advantage. Notice, however, that Python (unlike
many functional languages) does not prevent rebinding of the
starts meaning something different later in the program, all
bets are off. One could work up a Singleton class to contain
this type of immutable bindings (as, say,
s.bigmuls and so
on); but this column does not have room for that.
One thing distinctly worth noticing is that our particular goal is one tailor-made for a new feature of Python 2. Rather than either the imperative or functional examples given, the best (and functional) technique is:
print [(x,y) for x in (1,2,3,4) for y in (10,15,3,22) if x*y > 25]
This column has demonstrated ways to replace just about every Python flow-control construct with a functional equivalent (sparing side effects in the process). Translating a particular program efficiently takes some additional thinking, but we have seen that the functional built-ins are general and complete. In subsequent columns, we will look at more advanced techniques for functional programming; and hopefully we will be able to explore some more of the pros and cons of functional styles.
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:
Peter Norvig has written an interesting article, Python for Lisp Programmers. While the focus there is somewhat the reverse of my column, it provides very good general comparisons between Python and Lisp:
A good starting point for functional programming is the Frequently Asked Questions for comp.lang.functional :
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.
An excellent introductory book is:
Haskell: The Craft of Functional Programming (2nd Edition), Simon Thompson, Addison-Wesley (1999).
Since conceptions without intuitions are empty, and intutions without conceptions, blind, David Mertz wants a cast sculpture of Milton for his office. Start planning for his birthday. David may be reached at [email protected]; his life pored over at http://gnosis.cx/publish/. Suggestions and recommendations on this, past, or future, columns are welcomed.