David Mertz, Ph.D.
Applied Metaphysician, Gnosis Software, Inc.
This column continues an earlier installment's introduction to functional programming (FP) in Python. A number of intermediate or advanced FP concepts are demonstrated in here. Readers of this column will benefit from an introduction to different paradigms of program problem-solving.
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.
The previous Charming Python column on functional programming
introduced some basic concepts of FP. With the basics in mind,
this column will delve a little bit deeper into this quite rich
conceptual realm. For much of our delving, Bryn Keller's
"Xoltar Toolkit" will provide valuable assistance. Keller has
collected many of the strengths of FP into a nice little module
of pure Python implementations of the techniques. In addition
to the module
functional, Xoltar Toolkit includes the
module which supports structures that evaluate "only when
needed." Many traditionally functional languages also have
lazy evaluation, so between these components, the Xoltar
Toolkit lets you do much of what you might find in a functional
language like Haskell.
Attentive readers will remember a limitation I pointed out to the functional techniques described in the earlier column. Specifically, nothing in Python prevents the rebinding of names that are used to denote functional expressions. In FP, names are generally understood to be simply abbreviations of longer expressions, but the promise is implicit that "the same expression will always evaluate to the same result." If denotational names get rebound, the promise is broken. For example, let's say that we define some shorthand expressions that we'd like to use in our functional program, such as:
>>> car = lambda lst: lst >>> cdr = lambda lst: lst[1:] >>> sum2 = lambda lst: car(lst)+car(cdr(lst)) >>> sum2(range(10)) 1 >>> car = lambda lst: lst >>> sum2(range(10)) 5
Unfortunately, the very same expression
evaluates to two different things at two points in our program,
even though this expression itself does not use any mutable
variables in its arguments.
functional, fortunately, provides a class called
(proposed to Keller by your author) that prevents such
rebindings (at least accidentally, Python does not try to
prevent a determined programmer who wants to break things).
While use of
Bindings requires a little extra syntax, it
makes accidents hard to commit. In his examples within the
functional module, Keller names a
(I presume after the
let keyword in ML-family languages).
For example, we might do:
>>> from functional import * >>> let = Bindings() >>> let.car = lambda lst: lst >>> let.car = lambda lst: lst 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
Obviously, a real program would have to do something about catching these 'BindingError's, but the fact they are raised avoids a class of problems.
functional provides a
function to pull off a namespace (really, a dictionary) from a
Bindings instance. This comes in handy if you want to
compute an expression within a (immutable) namespace defined in
eval() is the Python function that allows
evaluation within a namespace. An example should clarify:
>>> let = Bindings() # "Real world" function names >>> let.r10 = range(10) >>> let.car = lambda lst: lst >>> let.cdr = lambda lst: lst[1:] >>> eval('car(r10)+car(cdr(r10))', namespace(let)) 1 >>> inv = Bindings() # "Inverted list" function names >>> inv.r10 = let.r10 >>> inv.car = lambda lst: lst[-1] >>> inv.cdr = lambda lst: lst[:-1] >>> eval('car(r10)+car(cdr(r10))', namespace(inv)) 17
One very interesting concept in FP is a closure. In fact, closures have proved sufficiently interesting to many developers that even generally non-functional languages like Perl and Ruby include closures as a feature. Moreover, Python 2.1 currently appears destined to add lexical scoping, which turns out to get one 99% of the way to closures.
So what is a closure, anyway? Steve Majewski has recently provided a nice characterization of the concept on the Python newsgroup:
An object is a piece of data with procedures attached to it... A closure is a procedure with a piece of data attached to it.
That is, a closure is something like FP's Jekyll to OOP's Hyde (or perhaps the roles are the other way around). A closure, like an object instance, is a way of carrying around a bundle of data and functionality, wrapped up together.
Let's step back just a bit to see what problem both objects and closures solve, and also to see how the problem can be solved without either. What a function returns is usually determined by a certain context around its calculation. The most common--and perhaps the most obvious--way of specifying this context is to pass some arguments to the function that tell it what values it should operate on. But sometimes also, there is a natural distinction between "background" and "foreground" arguments--between what the function is doing this particular time, and the way the function is "configured" for multiple potential calls.
There are a number of ways to handle background, while focussing on foreground. One way is to simply "bite the bullet" and pass every argument a function needs at every invocation. This often amounts to passing a number of values (or a structure with multiple slots) up and down a call chain, in the possiblity the values will be needed somewhere in the chain. A trivial example might look like:
In the cargo example, within
n has no purpose other
than being available to pass on to
c(). Another option is to
use global variables:
>>> N = 10 >>> def addN(i): ... global N ... return i+N ... >>> addN(7) # Add global N to argument 17 >>> N = 20 >>> addN(6) # Add global N to argument 26
N is simply available whenever you want to call
addN(), but there is no need to pass the global background
"context" explicitly. A somwhat more Pythonic technique is to
"freeze" a variable into a function using a default argument at
>>> N = 10 >>> def addN(i, n=N): ... return i+n ... >>> addN(5) # Add 10 15 >>> N = 20 >>> addN(6) # Add 10 (current N doesn't matter) 16
Our frozen variable is essentially a closure. Some data is
"attached" to the
addN() function. For a complete closure,
all the data present when
addN() was defined would be
available at invocation. However, in this example (and many
more robust ones), it is simple to make enough available with
default arguments. Variables that are never used by
thereby make no difference to its calculation.
Let's look next at an OOP approach to a slightly more realistic problem. The time of year has prompted my thoughts about those "interview" style tax program that collect various bits of data--not necessarily in a particular order--then eventually use them all for a caculation. Let's create a simplistic version of this:
TaxCalc class (or rather, in its instance), we can
collect some data--in whatever order we like--and once we have
all the elements needed, we can call a method of this object to
perform a calculation on the bundle of data. Everything stays
together within the instance, and further, a different instance
can carry a different bundle of data. The possibility of
creating multiple instances, differing only in their data is
something that was not possible in the "global variable" or
"frozen variable" approaches. The "cargo" approach can handle
this, but for the expanded example, we can see it might become
necessary to start passing around numerous values. While we
are here, it is interesting to note how a message-passing OOP
style might approach this (Smalltalk or Self are similar to
this, and so are several OOP xBase variants I have used):
class TaxCalc: def taxdue(self): return (self.income-self.deduct)*self.rate def setIncome(self,income): self.income = income return self def setDeduct(self,deduct): self.deduct = deduct return self def setRate(self,rate): self.rate = rate return self print "Smalltalk-style taxes due =", \ TaxCalc().setIncome(50000).setRate(0.30).setDeduct(10000).taxdue()
self with each "setter" allows us to treat the
"current" thing as a result of every method application. This
will have some interesting similarities to the FP closure
With the Xoltar toolkit, we can create full closures that have our desired property of combining data with a function, and also allowing multiple closures (nee objects) to contain different bundles:
from functional import * taxdue = lambda: (income-deduct)*rate incomeClosure = lambda income,taxdue: closure(taxdue) deductClosure = lambda deduct,taxdue: closure(taxdue) rateClosure = lambda rate,taxdue: closure(taxdue) taxFP = taxdue taxFP = incomeClosure(50000,taxFP) taxFP = rateClosure(0.30,taxFP) taxFP = deductClosure(10000,taxFP) print "Functional taxes due =",taxFP() print "Lisp-style taxes due =", \ incomeClosure(50000, rateClosure(0.30, deductClosure(10000, taxdue)))()
Each closure function we have defined takes any values defined within the function scope, and binds those values into the global scope of the function object. However, what appears as the function's global scope is not necessarily the same as the true module global scope, nor identical to a different closure's "global" scope. The closure simply "carries the data" with it.
In our example, we utilize a few particular functions to put
specific bindings within a closure's scope (income, deduct,
rate). It would be simple enough to modify the design to put
any arbitrary binding into scope. We also--just for the fun of
it--use two slightly different functional styles in the
example. The first successively binds additional values into
closure scope; by allowing
taxFP to be mutable, these "add to
closure" lines can appear in any order. However, if we were to
use immutable names like
tax_with_Income, we would have to
arrange the binding lines in a specific order, and pass the
earlier bindings to the next ones. In any case, once
everything necessary is bound into closure scope, we can call
the "seeded" function.
The second style looks a bit more like Lisp, to my eyes (the parentheses mostly). Beyond the aesthetic, two interesting things happen in the second style. The first is that name binding is avoided altogether. This second style is a single expression, with no statements used (see this column's predecessor for a discussion of why this matters).
The other interesting thing about the "Lisp-style" use of the
closures is how much it resembles the "Smalltalk-style"
message-passing methods given above. Both essentially
accumulate values along the way to calling the
function/method (both will raise errors in these crude versions
if the right data is not available). The "Smalltalk-style"
passes an object between each step, while the "Lisp-style"
passes a continuation. But deep down, functional and
object-oriented programming amount to much the same thing.
In this installment, we have knocked off a bit more of the
domain of functional programming. What remains is less (and
provably simpler?) than what did before (the title of the
section is a minor joke; unfortunately, its concept is not
explained herein). An excellent way to continue with a number
of FP concepts is by reading the
functional module's source.
The module is very well commented, and provides examples of
most of its functions/classes. Not covered in this column are
a number of simplifying meta-functions that make the
combinations and interaction of other functions simpler to
handle. These are definitely worth checking out for a Python
programmer seeking to continue the exploration of functional
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 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 email@example.com; his life pored over at http://gnosis.cx/publish/. Suggestions and recommendations on this, past, or future, columns are welcomed.