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Backus, Functional Programming, and Ruby

 
 
Jesse Merriman
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      03-25-2007
Recently, John Backus, the father of Fortran, died. In 1977 he won the Turi=
ng=20
Award, and gave a lecture promoting functional programming. The lecture can=
=20
be accessed here:

=C2=A0 http://www.stanford.edu/class/cs242/readings/backus.pdf

and an interesting blog post about it is here:

=C2=A0=20
http://scienceblogs.com/goodmath/200...ional_pro_1.p=
hp

In section 5 of the lecture an example is given comparing calculation of th=
e=20
inner product of two vectors in an imperative style versus a functional=20
style. I thought it'd be fun to try to write it (functionally) in Ruby, and=
I=20
think it came out rather nifty, so I thought I'd post it here. The basic id=
ea=20
is to define an inner_product function as the compose of 3 other functions =
(a=20
sum insert, multiply mapping, and transpose), instead of doing it iterative=
ly=20
like this:

def inner_product(a, b)
sum =3D 0
(0...a.length).each do |i|
sum +=3D a[i] * b[i]
end
sum
end

Below is my more functional solution. Most of it is defining functions for=
=20
doing composes and such, and then almost at the end inner_product is define=
d=20
as Backus would've liked.

=2D--------------------------------------------------------------------
# Ruby implementation of John Backus' functional inner product example, from
# section 5.2 of his 1977 Turing Award Lecture, available here:
# http://www.stanford.edu/class/cs242/readings/backus.pdf

# Returns the transpose of a pair of vectors as a vector of pairs.
transpose =3D lambda do |pair_of_vec|
=C2=A0 pair_of_vec.first.zip(pair_of_vec.last)
end

# Returns a Proc that takes a vector of pairs and returns a vector of whate=
ver
# f returns.
apply_to_all =3D lambda do |f|
=C2=A0 lambda do |vec_of_pair|
=C2=A0 =C2=A0 vec_of_pair.map { |pair| f.call(pair.first, pair.last) }
=C2=A0 end
end

# Returns a Proc that takes a vector and returns a value of whatever f
# returns.
insert =3D lambda do |f|
=C2=A0 lambda do |vec|
=C2=A0 =C2=A0 # The reverse is taken so that, for example, when vec is [1,2=
,3,4], the
=C2=A0 =C2=A0 # result is: =C2=A01 <op> (2 <op> (3 <op> 4))
=C2=A0 =C2=A0 # instead of: ((1 <op> 2) <op> 3) <op> 4
=C2=A0 =C2=A0 # (Though its just a convention, and in many cases doesn't ma=
tter)
=C2=A0 =C2=A0 vec.reverse.inject { |memo, e| f.call(memo, e) }
=C2=A0 end
end

# Returns the composition of f and g as a Proc.
compose =3D lambda do |f,g|
=C2=A0 lambda { |*args| f.call(g.call(*args)) }
end

# Returns the composition of all given functions as a Proc.
compose_all =3D lambda do |*funcs|
=C2=A0 funcs.reverse.inject { |memo, f| compose.call(f, memo) }
end

# Convenience Procs.
add =C2=A0=3D lambda { |x,y| x+y }
mult =3D lambda { |x,y| x*y }

# Returns a value: the inner product of a pair of vectors.
inner_product =3D compose_all.call(
=C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 =C2=A0 insert.call(=
add), apply_to_all.call(mult), transpose)

# Test case from the lecture. Should print out 28.
puts inner_product.call([[1,2,3], [6,5,4]])
=2D--------------------------------------------------------------------

=2D-=20
Jesse Merriman
http://www.velocityreviews.com/forums/(E-Mail Removed)
http://www.jessemerriman.com/

 
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Joel VanderWerf
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      03-25-2007
Jesse Merriman wrote:
...
> # Test case from the lecture. Should print out 28.
> puts inner_product.call([[1,2,3], [6,5,4]])
> ---------------------------------------------------------------------


Ruby has its own way of doing things, not that there is anything wrong
with defining new ways:

[[1,2,3], [6,5,4]].transpose.map{|x,y|x*y}.inject{|s,x|s+x}
=> 28

--
vjoel : Joel VanderWerf : path berkeley edu : 510 665 3407

 
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Jesse Merriman
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      03-25-2007
On Sunday 25 March 2007 18:03, Joel VanderWerf wrote:
> Ruby has its own way of doing things, not that there is anything wrong
> with defining new ways:
>
> [[1,2,3], [6,5,4]].transpose.map{|x,y|x*y}.inject{|s,x|s+x}
> => 28


Yep, I was just seeing how close I could make Ruby code look like Backus'
code; the goal was to define inner_product as a compose of the three other
functions without any named variables.

--
Jesse Merriman
(E-Mail Removed)
http://www.jessemerriman.com/

 
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Devin Mullins
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      03-25-2007
Jesse Merriman wrote:
> On Sunday 25 March 2007 18:03, Joel VanderWerf wrote:
>
>>Ruby has its own way of doing things, not that there is anything wrong
>>with defining new ways:
>>
>>[[1,2,3], [6,5,4]].transpose.map{|x,y|x*y}.inject{|s,x|s+x}
>>=> 28

>
> Yep, I was just seeing how close I could make Ruby code look like Backus'
> code; the goal was to define inner_product as a compose of the three other
> functions without any named variables.


module Enumerable
def inner_product
transpose.map(&:*).inject(&:+)
end
end



(Not to slight Backus. Functional programming and BNF are both totally
cool.)

 
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Jesse Merriman
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      03-25-2007
On Sunday 25 March 2007 18:47, Devin Mullins wrote:
> module Enumerable
> def inner_product
> transpose.map(&:*).inject(&:+)
> end
> end


That looks good, but is it right?

irb(main):001:0> module Enumerable
irb(main):002:1> def inner_product
irb(main):003:2> transpose.map(&:*).inject(&:+)
irb(main):004:2> end
irb(main):005:1> end
=> nil
irb(main):006:0> [[0,1,2], [6,5,4]].inner_product
TypeError: wrong argument type Symbol (expected Proc)
from (irb):3:in `inner_product'
from (irb):6

$ ruby --version
ruby 1.8.5 (2006-12-04 patchlevel 2) [i686-linux]

--
Jesse Merriman
(E-Mail Removed)
http://www.jessemerriman.com/

 
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M. Edward (Ed) Borasky
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      03-25-2007
Jesse Merriman wrote:
> Recently, John Backus, the father of Fortran, died. In 1977 he won the Turing
> Award, and gave a lecture promoting functional programming. The lecture can
> be accessed here:
>
> http://www.stanford.edu/class/cs242/readings/backus.pdf
>
> and an interesting blog post about it is here:
>
>
> http://scienceblogs.com/goodmath/200...onal_pro_1.php
>

Well ... let me throw a bit of historical perspective into the mix:

1. I read the article when it was published. At the time, I was working
as a mainly assembly language programmer in real-time operating systems.
I was quite familiar with functional programming styles, Lisp (1.5), the
lambda calculus and combinatory logic, and I was studying the formal
semantics of programming languages. The article resonated with me for
those reasons. However, I found it difficult to read in context with the
other computer science I was studying.

2. A few years later, functional languages, so-called single-assignment
languages and dataflow programming paradigms started to make an
appearance during the renaissance of high-performance scientific
computing that occurred during the Reagan years. Despite their
advantages, these, like the Backus paper, withered on the academic vine.

3. Maybe the time is *finally* ripe for the functional programming
paradigm to establish a firm commercial and business presence, given at
least a few useful applications written in Haskell, Erlang and OCaml.
After all, FORTRAN I came out in 1957 and LISP I only three years or so
later. I date the start of Lisp from McCarthy's 1960 paper, but I'm not
sure when the first actual LISP interpreter was put into operation.
Maybe ... but it's been about 50 years and the Von
Neumann/FORTRAN/imperative paradigm has continued to dominate the
computing world. The Church/McCarthy/LISP/Functional paradigm has held
its own -- once there were Lisp machines, after all -- but it is a
distinct second place runner.


--
M. Edward (Ed) Borasky, FBG, AB, PTA, PGS, MS, MNLP, NST, ACMC(P)
http://borasky-research.blogspot.com/

If God had meant for carrots to be eaten cooked, He would have given rabbits fire.


 
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Chris Carter
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      03-25-2007
On 3/25/07, Jesse Merriman <(E-Mail Removed)> wrote:
> On Sunday 25 March 2007 18:47, Devin Mullins wrote:
> > module Enumerable
> > def inner_product
> > transpose.map(&:*).inject(&:+)
> > end
> > end

>
> That looks good, but is it right?
>
> irb(main):001:0> module Enumerable
> irb(main):002:1> def inner_product
> irb(main):003:2> transpose.map(&:*).inject(&:+)
> irb(main):004:2> end
> irb(main):005:1> end
> => nil
> irb(main):006:0> [[0,1,2], [6,5,4]].inner_product
> TypeError: wrong argument type Symbol (expected Proc)
> from (irb):3:in `inner_product'
> from (irb):6
>
> $ ruby --version
> ruby 1.8.5 (2006-12-04 patchlevel 2) [i686-linux]
>
> --
> Jesse Merriman
> (E-Mail Removed)
> http://www.jessemerriman.com/
>
>


He is assuming you have the famed "Symbol#to_proc" method, you can
find it with a quick google.

--
Chris Carter
concentrationstudios.com
brynmawrcs.com

 
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Jesse Merriman
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      03-26-2007
On Sunday 25 March 2007 19:30, Chris Carter wrote:
> He is assuming you have the famed "Symbol#to_proc" method, you can
> find it with a quick google.


Ah, very nice. Very nice.

--
Jesse Merriman
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http://www.jessemerriman.com/

 
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Giles Bowkett
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      03-26-2007
> > He is assuming you have the famed "Symbol#to_proc" method, you can
> > find it with a quick google.

>
> Ah, very nice. Very nice.


I think this technique is massively underutilized. Imagine what it
would be like if overriding to_proc was a standard technique for all
kinds of objects. Things could get really weird.

(In a good way.)

--
Giles Bowkett
http://www.gilesgoatboy.org
http://gilesbowkett.blogspot.com
http://giles.tumblr.com/

 
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SonOfLilit
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      03-26-2007
oooh, shiny!

On 3/26/07, Giles Bowkett <(E-Mail Removed)> wrote:
> > > He is assuming you have the famed "Symbol#to_proc" method, you can
> > > find it with a quick google.

> >
> > Ah, very nice. Very nice.

>
> I think this technique is massively underutilized. Imagine what it
> would be like if overriding to_proc was a standard technique for all
> kinds of objects. Things could get really weird.
>
> (In a good way.)
>
> --
> Giles Bowkett
> http://www.gilesgoatboy.org
> http://gilesbowkett.blogspot.com
> http://giles.tumblr.com/
>
>


 
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