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David Mertz | Published June 1, 2001
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.
Let’s 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:
>>> from functional import ∗
>>> let = Bindings()
>>> let.car = lambda lst: lst0>>> let.car = lambda lst: lst2Traceback (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.
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:
‑‑ 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:
>>> 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:
>>> list_of_list = [1,2,3],[4,5,6],[7,8,9]>>> let = Bindings()
>>> let.car = lambda l: l0>>> 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:
#Compare Haskell expression:
#result = func car_car
# 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,)]02
We have performed 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:
>>> [func for x in list_of_list:1... for car in (x,)
... for func in (car+car∗∗2,)]02
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 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:
>>> def foo_factory():
"Foo function from factory"
>>> f = foo_factory()
Foo function from factory
The 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(). 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 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:
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:
>>> from functional import curry
>>> computation = lambda a,b,c,d: (a + b∗∗2 + c∗∗3 + d∗∗4)
>>> fillZero = curry(computation)
>>> fillOne = fillZero(1)
>>> fillTwo = fillOne(2)
>>> fillThree = fillTwo(3)
>>> answer = fillThree(5)
It is possible to further illustrate the parallel with closures by presenting the same simple tax-calculation program used in Part 2 (this time using curry()):
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 =", \
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). 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.
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(). 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 functional provides.
The functions sequential() 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:
>>> 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)
<functional.sequential instance at 0xb86ac>
5 10 15
x xx xxx
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 built-in map(), reduce(), filter() functions. But these particular higher-order functions from functional ask Boolean questions about collections of return values. For example:
>>> from functional import ∗
>>> isEven = lambda n: (n%2 == 0)
>>> any([1,3,5,8], isEven)
>>> any([1,3,5,7], isEven)
>>> none_of([1,3,5,7], isEven)
>>> all([2,4,6,8], isEven)
>>> all([2,4,6,7], isEven)
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:
>>> def minus7(n): return n‑7
>>> def times3(n): return n∗3
>>> minustimes = compose(times3,minus7)
>>> timesminus = compose(minus7,times3)
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.
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