What do we call this (new?) higher-order function?
Question:
I am trying to name what I think is a new idea for a higher-order function. I want to call it a BinaryProjection, but my compatriot wants to call it the Plow. We tossed around the idea of a Snow Blower for a minute, too. Anyway, to the important part, here is the code in Python and Haskell to demonstrate the concept, which will be explained afterward.
Python:
>>> def plow(f, l):
return map(lambda t: f(*t), zip(l, l[1:]))
>>> plow(operator.add, [0, 1, 2, 3])
[1, 3, 5]
Haskell:
Prelude> let binaryProjection f xs = zipWith f xs (drop 1 xs)
Prelude> binaryProjection (+) [0,1,2,3]
[1,3,5]
As you may be able to infer, the sequence is being iterated through, utilizing adjacent elements as the parameters for the function you pass it, projecting the results into a new sequence. So, has anyone seen the functionality we’ve created? Is this familiar at all to those in the functional community? If not, what do we name it?
---- Update ----
There are three candidates for the name. meld, pinch, and pleat. In Haskell, they would be implemented as so (using "meld"):
Prelude> let meld xs = zip xs (drop 1 xs)
Prelude> meld [1..4]
[(1,2),(2,3),(3,4)]
Prelude> let meldWith f xs = zipWith f xs (drop 1 xs)
Prelude> meldWith (+) [1..4]
[3,5,7]
I think it’s time for a vote. I’m partial to pinch or pleat, myself.
Answers:
Here’s another implementation for Python which works if l is a generator too
import itertools as it
def apply_pairwise(f, l):
left, right = it.tee(l)
next(right)
return it.starmap(f, it.izip(left, right))
I think apply_pairwise is a better name
Since it’s similar to “fold” but doesn’t collapse the list into a single value, how about “crease”? If you keep “creasing”, you end up “folding” (sort of).
We could go with a cooking metaphor and call it “pinch”, like pinching the crust of a pie, though this might suggest a circular zipping, where the last element of the list is paired with the first.
def pinch(f, l):
return map(lambda t: f(*t), zip(l, l[1:]+l[:1]))
(If you only like one of “crease” or “pinch”, please note so as a comment. Should these be separate suggestions?)
So because there seems to be no name for this I suggest ‘merger’ or simple ‘merge’ because you are merging adjacent values together.
So merge is already taken so I now suggest ‘meld’ (or ‘merger’ still but that may be too close to ‘merge’)
For example:
meld :: (a -> a -> b) -> [a] -> [b]
meld _ [] = []
meld f xs = zipWith f (init xs) (tail xs)
Which can be used as:
> meld (+) [1..10]
[3,5,7,9,11,13,15,17,19]
> meld compare "hello world"
[GT,LT,EQ,LT,GT,LT,GT,LT,GT,GT]
Where the second example makes no real sense but makes a cool example.
I really can’t see any codified names for this anywhere in Python, that’s for sure. “Merge” is good but spoken for in a variety of other contexts. “Plow” tends to be unused and supplies a great visual of pushing steadily through a line of soil. Maybe I’ve just spent too much time gardening.
I also expanded the principle to allow functions that receive any number of parameters.
You might also consider: Pleat. It describes well the way you’re taking a list (like a long strand of fabric) and bunching segments of it together.
import operator
def stagger(l, w):
if len(l)>=w:
return [tuple(l[0:w])]+stagger(l[1:], w)
return []
def pleat(f, l, w=2):
return map(lambda p: f(*p), stagger(l, w))
print pleat(operator.add, range(10))
print pleat(lambda x, y, z: x*y/z, range(3, 13), 3)
print pleat(lambda x: "~%s~"%(x), range(10), 1)
print pleat(lambda a, b, x, y: a+b==x+y, [3, 2, 4, 1, 5, 0, 9, 9, 0], 4)
zipWithTail or adjacentPairs.
I vote for smearWith or smudgeWith because it’s like you are smearing/smudging the operation across the list.
BinaryOperate or BinaryMerge
In Python the meld equivalent is in the itertools receipes and called pairwise.
from itertools import starmap, izp, tee
def pairwise(iterable):
"s -> (s0,s1), (s1,s2), (s2, s3), ..."
a, b = tee(iterable)
next(b, None)
return izip(a, b)
So I would call it:
def pairwith(func, seq):
return starmap(func, pairwise(seq))
I think this makes sense because when you call it with the identity function, it simply returns pairs.
Hmm… a counterpoint.
(`ap` tail) . zipWith
doesn’t deserve a name.
BTW, quicksilver says:
zip`ap`tail
The Aztec god of consecutive numbers
Using Mathematica
Plus @@@ Partition[{0, 1, 2, 3}, 2, 1]
or either of these more verbose alternatives
Apply[Plus, Partition[{0, 1, 2, 3}, 2, 1], {1}]
Map[Apply[Plus, #] &, Partition[{0, 1, 2, 3}, 2, 1]]
I have used and enjoyed this higher order function in many languages but I have enjoyed it the most in Mathematica; it seems succinct and flexible broken down into Partition and Apply with levelspec option.
this seems like ruby’s each_cons
ruby-1.9.2-p0 > (1..10).each_cons(2).to_a
=> [[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]]
This reminds me of convolution from image processing. But not sure if this is mathematically correct.
The generalized variant of the plain zip version of this is what I would think of as window. Not at a ghci terminal right now, but I think window n = take n . tails. Then your function is zipWith ([x,yj -> f x y) . window 2. This sort of style naturally works better when f is of type [a] -> b.
I’d be tempted to call it contour as I’ve used it for “contour” processing in music software – at the time I called it twomap or something silly like that.
There are also two specific named ‘contours’ in music processing one is gross contour – does the pitch go up or down. The other is refined contour where the the contour is either up, down, leap up or leap down, though I can’t seem to find a reference for how large the semitone difference has to be to make a leap.
in C++ Standard Template Library, it is called adjacent_difference (though the operator can be any operation, not just subtraction)
Nice idiom! I just needed to use this in Perl to determine the time between sequential events. Here’s what I ended up with.
sub pinch(&@) {
my ( $f, @list ) = @_;
no strict "refs";
use vars qw( $a $b );
my $caller = caller;
local( *{$caller . "::a"} ) = my $a;
local( *{$caller . "::b"} ) = my $b;
my @res;
for ( my $i = 0; $i < @list - 1; ++$i ) {
$a = $list[$i];
$b = $list[$i + 1];
push( @res, $f->() );
}
wantarray ? @res : @res;
}
print join( ",", pinch { $b - $a } qw( 1 2 3 4 5 6 7 ) ), $/;
# ==> 1,1,1,1,1,1
The implementation could probably be prettier if I’d made it dependent on List::Util, but… meh!
I am trying to name what I think is a new idea for a higher-order function. I want to call it a BinaryProjection, but my compatriot wants to call it the Plow. We tossed around the idea of a Snow Blower for a minute, too. Anyway, to the important part, here is the code in Python and Haskell to demonstrate the concept, which will be explained afterward.
Python:
>>> def plow(f, l):
return map(lambda t: f(*t), zip(l, l[1:]))
>>> plow(operator.add, [0, 1, 2, 3])
[1, 3, 5]
Haskell:
Prelude> let binaryProjection f xs = zipWith f xs (drop 1 xs)
Prelude> binaryProjection (+) [0,1,2,3]
[1,3,5]
As you may be able to infer, the sequence is being iterated through, utilizing adjacent elements as the parameters for the function you pass it, projecting the results into a new sequence. So, has anyone seen the functionality we’ve created? Is this familiar at all to those in the functional community? If not, what do we name it?
---- Update ----
There are three candidates for the name. meld, pinch, and pleat. In Haskell, they would be implemented as so (using "meld"):
Prelude> let meld xs = zip xs (drop 1 xs)
Prelude> meld [1..4]
[(1,2),(2,3),(3,4)]
Prelude> let meldWith f xs = zipWith f xs (drop 1 xs)
Prelude> meldWith (+) [1..4]
[3,5,7]
I think it’s time for a vote. I’m partial to pinch or pleat, myself.
Here’s another implementation for Python which works if l is a generator too
import itertools as it
def apply_pairwise(f, l):
left, right = it.tee(l)
next(right)
return it.starmap(f, it.izip(left, right))
I think apply_pairwise is a better name
Since it’s similar to “fold” but doesn’t collapse the list into a single value, how about “crease”? If you keep “creasing”, you end up “folding” (sort of).
We could go with a cooking metaphor and call it “pinch”, like pinching the crust of a pie, though this might suggest a circular zipping, where the last element of the list is paired with the first.
def pinch(f, l):
return map(lambda t: f(*t), zip(l, l[1:]+l[:1]))
(If you only like one of “crease” or “pinch”, please note so as a comment. Should these be separate suggestions?)
So because there seems to be no name for this I suggest ‘merger’ or simple ‘merge’ because you are merging adjacent values together.
So merge is already taken so I now suggest ‘meld’ (or ‘merger’ still but that may be too close to ‘merge’)
For example:
meld :: (a -> a -> b) -> [a] -> [b]
meld _ [] = []
meld f xs = zipWith f (init xs) (tail xs)
Which can be used as:
> meld (+) [1..10]
[3,5,7,9,11,13,15,17,19]
> meld compare "hello world"
[GT,LT,EQ,LT,GT,LT,GT,LT,GT,GT]
Where the second example makes no real sense but makes a cool example.
I really can’t see any codified names for this anywhere in Python, that’s for sure. “Merge” is good but spoken for in a variety of other contexts. “Plow” tends to be unused and supplies a great visual of pushing steadily through a line of soil. Maybe I’ve just spent too much time gardening.
I also expanded the principle to allow functions that receive any number of parameters.
You might also consider: Pleat. It describes well the way you’re taking a list (like a long strand of fabric) and bunching segments of it together.
import operator
def stagger(l, w):
if len(l)>=w:
return [tuple(l[0:w])]+stagger(l[1:], w)
return []
def pleat(f, l, w=2):
return map(lambda p: f(*p), stagger(l, w))
print pleat(operator.add, range(10))
print pleat(lambda x, y, z: x*y/z, range(3, 13), 3)
print pleat(lambda x: "~%s~"%(x), range(10), 1)
print pleat(lambda a, b, x, y: a+b==x+y, [3, 2, 4, 1, 5, 0, 9, 9, 0], 4)
zipWithTail or adjacentPairs.
I vote for smearWith or smudgeWith because it’s like you are smearing/smudging the operation across the list.
BinaryOperate or BinaryMerge
In Python the meld equivalent is in the itertools receipes and called pairwise.
from itertools import starmap, izp, tee
def pairwise(iterable):
"s -> (s0,s1), (s1,s2), (s2, s3), ..."
a, b = tee(iterable)
next(b, None)
return izip(a, b)
So I would call it:
def pairwith(func, seq):
return starmap(func, pairwise(seq))
I think this makes sense because when you call it with the identity function, it simply returns pairs.
Hmm… a counterpoint.
(`ap` tail) . zipWith
doesn’t deserve a name.
BTW, quicksilver says:
zip`ap`tail
The Aztec god of consecutive numbers
Using Mathematica
Plus @@@ Partition[{0, 1, 2, 3}, 2, 1]
or either of these more verbose alternatives
Apply[Plus, Partition[{0, 1, 2, 3}, 2, 1], {1}]
Map[Apply[Plus, #] &, Partition[{0, 1, 2, 3}, 2, 1]]
I have used and enjoyed this higher order function in many languages but I have enjoyed it the most in Mathematica; it seems succinct and flexible broken down into Partition and Apply with levelspec option.
this seems like ruby’s each_cons
ruby-1.9.2-p0 > (1..10).each_cons(2).to_a
=> [[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]]
This reminds me of convolution from image processing. But not sure if this is mathematically correct.
The generalized variant of the plain zip version of this is what I would think of as window. Not at a ghci terminal right now, but I think window n = take n . tails. Then your function is zipWith ([x,yj -> f x y) . window 2. This sort of style naturally works better when f is of type [a] -> b.
I’d be tempted to call it contour as I’ve used it for “contour” processing in music software – at the time I called it twomap or something silly like that.
There are also two specific named ‘contours’ in music processing one is gross contour – does the pitch go up or down. The other is refined contour where the the contour is either up, down, leap up or leap down, though I can’t seem to find a reference for how large the semitone difference has to be to make a leap.
in C++ Standard Template Library, it is called adjacent_difference (though the operator can be any operation, not just subtraction)
Nice idiom! I just needed to use this in Perl to determine the time between sequential events. Here’s what I ended up with.
sub pinch(&@) {
my ( $f, @list ) = @_;
no strict "refs";
use vars qw( $a $b );
my $caller = caller;
local( *{$caller . "::a"} ) = my $a;
local( *{$caller . "::b"} ) = my $b;
my @res;
for ( my $i = 0; $i < @list - 1; ++$i ) {
$a = $list[$i];
$b = $list[$i + 1];
push( @res, $f->() );
}
wantarray ? @res : @res;
}
print join( ",", pinch { $b - $a } qw( 1 2 3 4 5 6 7 ) ), $/;
# ==> 1,1,1,1,1,1
The implementation could probably be prettier if I’d made it dependent on List::Util, but… meh!