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 means of expression provided by a comp.lang.python contributor.
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:
Listing 1: Python FP session with guarded rebinding
>>> 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
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:
Listing 2: 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 it is possible to 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:
Listing 3: 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,)] [1, 4, 7]
This trick of putting an expression 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:
Listing 4: 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 >>> map(let.car,list_of_list) [1, 4, 7]
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:
Listing 5: "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,) ... for func in (car+car**2,)] 2
have performed an arithmetic calculation on the first element of the first
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
Listing 6: Efficient stepping down from list comprehension
>>> [func for x in list_of_list[:1] ... for car in (x,) ... for func in (car+car**2,)] 2
Three of the most general higher-order functions are
built into Python:
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 built-ins,
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 this:
Listing 7: Trivial Python function factory
>>> def foo_factory(): ... def foo(): ... print "Foo function from factory" ... return foo ... >>> f = foo_factory() >>> f() Foo function from factory
Xoltar Toolkit, which I discussed in Part
2 of this series, comes with a nice collection of higher-order
functions. Most of the functions that Xoltar's
functional module provides are ones developed
in various traditionally functional languages, and whose usefulness have
been proven over many years.
Possibly the most famous and most
important higher-order function is
curry() is named after the logician Haskell
Curry, whose first name is also used to name the above-mentioned
programming language. The underlying insight of "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." This works quite
similarly to the closures I wrote about in Part
2 -- each successive call to a curried return function "fills in"
more of the data involved in a final computation (data attached to a
Let's illustrate currying first with a very simple
example in Haskell, then with the same example repeated in Python using
Listing 8: 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"
Now in Python:
Listing 9: 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
is possible to further illustrate the parallel with closures by presenting
the same simple tax-calculation program used in Part
2 (this time using
Listing 10: 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)
with closures, we need to curry the arguments in a specific order (left to
right). But note that
functional also contains
rcurry() class that will start at the other
end (right to left).
taxcalc(50000,0.30,10000). In a different
level, however, 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.
"fundamental" operation of currying,
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
functional. The ones in
functional provide some interesting ideas, at
For the most part, higher-order functions feel like
"enhanced" versions of the standard
reduce(). Often, 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's look at
some of what
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
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:
Listing 11: 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'
conjoin() are similar to
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
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),
none_of() let you call the same function
against a list of arguments. In general structure, these are a bit like
functions. But these particular higher-order functions from
functional ask Boolean questions about
collections of return values. For example:
Listing 12: 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
particularly interesting higher-order function for those with a little bit
of mathematics background is
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:
Listing 13: 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
I hope this latest look at higher-order functions will arouse 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.
- Read all three parts in this series.
- Read more installments of Charming Python.
- Bryn Keller's "xoltar
toolkit", which includes the module
functional, adds a large number of useful FP extensions to Python. Since the
functionalmodule 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.
- Peter Norvig has written an interesting
Python for Lisp Programmers. While his focus 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.
- I've found it much easier to get a grasp
of functional programming in 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.
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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/dW/. Suggestions and recommendations on this, past, or future columns are welcome.