Lambda in parameters

[Abdur-Rahmaan Janhangeer]

Greetings list,

Here is a famous question's solution

def cons(a, b):
def pair(f):
return f(a, b)
return pair

def car(c):
return c(lambda a, b: a)

print(cons(1, 2)(lambda a, b: a))
print(car(cons(1, 2)))

The aim of car is to return 1


but i don't understand how lambda achieves this

Kind Regards,


Abdur-Rahmaan Janhangeer

https://www.github.com/Abdur-RahmaanJ

Mauritius

sent from gmail client on Android, that's why the signature is so ugly.

[Cameron Simpson]

On 17Dec2020 23:52, Abdur-Rahmaan Janhangeer <arj.p...@gmail.com> wrote:
>Here is a famous question's solution

These look like implementations of Lisp operators, which I've never
found easy to remember. So I'll do this from first principles, but
looking at the (uncommented) code.

This is some gratuitiously tricky code. Maybe that's needed for these
operators. But there's a lot here, so let's unpick it:

>def cons(a, b):
> def pair(f):
> return f(a, b)
> return pair

Cons returns "pair", a function which itself accepts a function "f",
calls it with 2 values "a, b" and returns the result. For ultra
trickiness, those 2 values "a, b" are the ones you passed to the initial
call to cons().

This last bit is a closure: when you define a function, any nonlocal
variables (those you use but never assign to) have the defining scope
available for finding them. So the "pair" function gets "a" and "b" from
those you passed to "cons".

Anyway, cons() returns a "pair" function hooked to the "a" and "b" you
called it with.

>def car(c):
> return c(lambda a, b: a)

The function accepts a function, and calls that function with "lambda a,
b: a", which is itself a function which returns its first argument. You
could write car like this:

def car(c):
def first(a, b):
return a
return c(first)

The lambda is just a way to write a simple single expression function.

>print(cons(1, 2)(lambda a, b: a))

What is "cons(1,2)". That returns a "pair" function hooked up to the a=1
and b=2 values you supplied. And the pair function accepts a function of
2 variables.

What does this do?

cons(1, 2)(lambda a, b: a)

This takes the function returns by cons(1,2) and _calls_ that with a
simple function which accepts 2 values and returns the first of them.

So:

cons(1,2) => "pair function hooked to a=1 and b=2"

Then call:

pair(lambda a, b: a)

which sets "f" to the lambda function, can calls that with (a,b). So it
calls the lambda function with (1,2). Which returns 1.

>print(car(cons(1, 2)))

The "car" function pretty much just embodies the call-with-the-lambda.

>but i don't understand how lambda achieves this

If you rewite the lambda like this:

def a_from_ab(a,b):
return a

and then rewrite the first call to cons() like this:

cons(1,2)(a_from_ab)

does it make any more sense?

Frankly, I think this is a terrible way to solve this problem, whatever
the problem was supposed to be - that is not clear.

On the other hand, I presume it does implement the Lisp cons and car
functions. I truly have no idea, I just remember these names from my
brief brush with Lisp in the past.

Cheers,
Cameron Simpson <c...@cskk.id.au>

[Abdur-Rahmaan Janhangeer]

The Question:

# ---
This problem was asked by Jane Street.

cons(a, b) constructs a pair, and car(pair) and cdr(pair) returns the first
and last element of that pair. For example, car(cons(3, 4)) returns 3, and
cdr(cons(3, 4)) returns 4.

Given this implementation of cons:

def cons(a, b):
def pair(f):
return f(a, b)
return pair

Implement car and cdr.
# ---

Kind Regards,


Abdur-Rahmaan Janhangeer

https://www.github.com/Abdur-RahmaanJ

Mauritius

sent from gmail client on Android, that's why the signature is so ugly.

> --
> https://mail.python.org/mailman/listinfo/python-list
>

[Barry]

> On 18 Dec 2020, at 14:23, Abdur-Rahmaan Janhangeer <arj.p...@gmail.com> wrote:
>
> The Question:

>
> # ---
> This problem was asked by Jane Street.
>
> cons(a, b) constructs a pair, and car(pair) and cdr(pair) returns the first
> and last element of that pair. For example, car(cons(3, 4)) returns 3, and
> cdr(cons(3, 4)) returns 4.
>
> Given this implementation of cons:
>
> def cons(a, b):
> def pair(f):
> return f(a, b)
> return pair
>
> Implement car and cdr.

Why car and cdr?

Well obviously car is content of the address register and cdr is content of data register.
Apparently an artefact of a early implementation of lisp.

Barry

> --
> https://mail.python.org/mailman/listinfo/python-list
>

[Julio Di Egidio]

On Friday, 18 December 2020 at 15:20:59 UTC+1, Abdur-Rahmaan Janhangeer wrote:
> The Question:
>
> # ---
> This problem was asked by Jane Street.
>
> cons(a, b) constructs a pair, and car(pair) and cdr(pair) returns the first
> and last element of that pair. For example, car(cons(3, 4)) returns 3, and
> cdr(cons(3, 4)) returns 4.
>
> Given this implementation of cons:
> def cons(a, b):
> def pair(f):
> return f(a, b)
> return pair
> Implement car and cdr.
> # ---

Notice that you don't need (Python) lambdas to code it, plain function definitions are fine:

# ---

def cons(a, b):
def pair(f):
return f(a, b)
return pair

def car(pair):
def left(a, b):
return a
return pair(left)

pair = cons(1, 2)
assert car(pair) == 1
# ---

That said, few basic comments: In Python, that 'cons' does not construct a pair, it rather returns a function with values a and b in its closure that, given some function, applies it to those values. In fact, Python has tuples built-in, how to build them as well as how to access their members. I take it the point of the exercise is how to use a purely functional language, such as here a fragment of Python, to encode (i.e. formalize) pairs and their operations.

Julio

[Grant Edwards]

On 2020-12-18, Barry <ba...@barrys-emacs.org> wrote:
>> Implement car and cdr.
> Why car and cdr?
>
> Well obviously car is content of the address register and cdr is content of data register.
> Apparently an artefact of a early implementation of lisp.

While car and cdr are lisp operators, the "content of address
register" and "content of data register" etymology is apparently
apocryphal:

https://en.wikipedia.org/wiki/CAR_and_CDR#Etymology

--
Grant

[Philippe Meunier]

Abdur-Rahmaan Janhangeer wrote:
>The aim of car is to return 1

>but i don't understand how lambda achieves this

Cameron Simpson's explanation is very good. See also the example reduction
here: https://en.wikipedia.org/wiki/Church_encoding#Church_pairs

>This problem was asked by Jane Street.

Jane Street is well known for its love of functional programming in general
and of OCaml in particular. If you don't know OCaml yet, I highly
recommend it. You can think of it as Python with (static) types.

Best,

Philippe

[Cameron Simpson]

On 19Dec2020 07:39, Philippe Meunier <meu...@ccs.neu.edu> wrote:
>See also the example reduction
>here: https://en.wikipedia.org/wiki/Church_encoding#Church_pairs

Thank you for this reference. I've stuck it on my reading list.

Cheers,
Cameron Simpson <c...@cskk.id.au>

[Greg Ewing]

On 18/12/20 7:02 pm, Cameron Simpson wrote:
> Frankly, I think this is a terrible way to solve this problem, whatever
> the problem was supposed to be - that is not clear.

It demonstrates that a programming language doesn't strictly
need data structes -- you can do everything with nothing
but functions.

This is mainly of theoretical interest; it's not usually a
practical way to go about things. But it's a good exercise
in thinking about functions as objects to be manipulated.

--
Greg

[Abdur-Rahmaan Janhangeer]

Greetings,

Very clear explanations, rewriting lambdas as a function
was the key to understand it. Did not know was a lisp inspiration.

My mind still spiralling XD

Kind Regards,

Abdur-Rahmaan Janhangeer
about <https://compileralchemy.github.io/> | blog
<https://www.pythonkitchen.com>
github <https://github.com/Abdur-RahmaanJ>
Mauritius

[Abdur-Rahmaan Janhangeer]

Greetings list,

Why car and cdr?

Well obviously car is content of the address register and cdr is content of
data register.
Apparently an artefact of a early implementation of lisp.

Oh did not know that detail. More twists for sure. Thought lisp did not go
lowlevel.

[Abdur-Rahmaan Janhangeer]

Greetings list,

Jane Street is well known for its love of functional programming in general
and of OCaml in particular. If you don't know OCaml yet, I highly
recommend it. You can think of it as Python with (static) types.

Yes i know OCaml when i was exploring haskell and the like.
At that time i was looking for companies who use OCaml etc
Did not come across Jane Street (If you know someone from there
tell them the question writer has a really twisted mind ^^)/.
I don't understand why though you would prefer OCaml over Python, Go, Java
etc for a company ^^