Friday, December 28, 2012

The Secret History of Lambda

Being a bit of an origins nut (I always want to know how something came to be or why it is a certain way), one of the things that always bothered me with regard to Lisp was that no one seemed to talking about the origin of lambda in the lambda calculus. I suppose if I wasn't lazy, I'd have gone to a library and spent some time looking it up. But since I was lazy, I used Wikipedia. Sadly, I never got what I wanted: no history of lambda. [1] Well, certainly some information about the history of the lambda calculus, but not the actual character or term in that context.

Why lambda? Why not gamma or delta? Or Siddham ṇḍha?

To my great relief, this question was finally answered when I was reading one of the best Lisp books I've ever read: Peter Norvig's Paradigms of Artificial Intelligence Programming: Case Studies in Common Lisp. I'll save my discussion of that book for later; right now I'm going to focus on the paragraph at location 821 of my Kindle edition of the book. [2]

The story goes something like this:
  • Between 1910 and 1913, Alfred Whitehead and Bertrand Russell published three volumes of their Principia Mathematica, a work whose purpose was to derive all of mathematics from basic principles in logic. In these tomes, they cover two types of functions: the familiar descriptive functions (defined using relations), and then propositional functions. [3]
  • Within the context of propositional functions, the authors make a typographical distinction between free variables and bound variables or functions that have an actual name: bound variables use circumflex notation, e.g. x̂(x+x). [4]
  • Around 1928, Church (and then later, with his grad students Stephen Kleene and J. B. Rosser) started attempting to improve upon Russell and Whitehead regarding a foundation for logic. [5]
  • Reportedly, Church stated that the use of  in the Principia was for class abstractions, and he needed to distinguish that from function abstractions, so he used x [6] or ^x [7] for the latter.
  • However, these proved to be awkward for different reasons, and an uppercase lambda was used: Λx. [8].
  • More awkwardness followed, as this was too easily confused with other symbols (perhaps uppercase delta? logical and?). Therefore, he substituted the lowercase λ. [9]
  • John McCarthy was a student of Alonzo Church and, as such, had inherited Church's notation for functions. When McCarthy invented Lisp in the late 1950s, he used the lambda notation for creating functions, though unlike Church, he spelled it out. [10] 
It seems that our beloved lambda [11], then, is an accident in typography more than anything else.

Somehow, this endears lambda to me even more ;-)

[1] As you can see from the rest of the footnotes, I've done some research since then and have found other references to this history of the lambda notation.

[2] Norvig, Peter (1991-10-15). Paradigms of Artificial Intelligence Programming: Case Studies in Common Lisp (Kindle Locations 821-829). Elsevier Science - A. Kindle Edition. The paragraph in question is quoted here:
The name lambda comes from the mathematician Alonzo Church’s notation for functions (Church 1941). Lisp usually prefers expressive names over terse Greek letters, but lambda is an exception. Abetter name would be make-function. Lambda derives from the notation in Russell and Whitehead’s Principia Mathematica, which used a caret over bound variables: x̂(x + x). Church wanted a one-dimensional string, so he moved the caret in front: ^x(x + x). The caret looked funny with nothing below it, so Church switched to the closest thing, an uppercase lambda, Λx(x + x). The Λ was easily confused with other symbols, so eventually the lowercase lambda was substituted: λx(x + x). John McCarthy was a student of Church’s at Princeton, so when McCarthy invented Lisp in 1958, he adopted the lambda notation. There were no Greek letters on the keypunches of that era, so McCarthy used (lambda (x) (+ x x)), and it has survived to this day.

[4] Norvig, 1991, Location 821.

[5] History of Lambda-calculus and Combinatory Logic, page 7.

[6] Ibid.

[7] Norvig, 1991, Location 821.

[8] Ibid.

[9] Looking at Church's works online, he uses lambda notation in his 1932 paper A Set of Postulates for the Foundation of Logic. His preceding papers upon which the seminal 1932 is based On the Law of Excluded Middle (1928) and Alternatives to Zermelo's Assumption (1927), make no reference to lambda notation. As such, A Set of Postulates for the Foundation of Logic seems to be his first paper that references lambda.

[10] Norvig indicates that this is simply due to the limitations of the keypunches in the 1950s that did not have keys for Greek letters.

[11] Alex Martelli is not a fan of lambda in the context of Python, and though a good friend of Peter Norvig, I've heard Alex refer to lambda as an abomination :-) So, perhaps not beloved for everyone. In fact, Peter Norvig himself wrote (see above) that a better name would have been make-function.

Tuesday, December 18, 2012

Seeking a Twisted Maintainer

Last week we posted on the Twisted Matrix blog about the maintainer position for the Twisted project being open. We are accepting applicants for a motivated and experienced release manager and core contributor. Our core maintainers are getting busier and busier with specialized Twisted work, and don't have the time that they used to be able to dedicate to maintaining Twisted.

The post on the Twisted Matrix blog gives a quick overview of the position; if you're interested, please check out the fellowship proposal for more details and email the address on that page (at the bottom).

Also, feel free to ping glyph, exarkun, or myself (oubiwann) on #twisted-dev on IRC to chat about it more.

Wednesday, December 12, 2012

Async in Python 3

Update: Guido has been working on PEP 3156; check on it regularly for the latest! (In the last two hours I've seen it updated with three big content changes.)

The buzz has died down a bit now, but the mellowing of the roaring flames has resulted in some nice embers in which an async for Python 3 is being forged. This is an exciting time for those of us who 1) love Python and 2) can't get us enough async.

I wanted to take the time to record some of the goodness here before I forgot or got too busy working on something else. So here goes:

The latest bout of Python async fever started in September of 2012 in this message when Christian M. Amsüss emailed the Python-ideas mail list about the state of async in Python and the hopes that a roadmap could be decided upon for Python 3. Note that this is the latest (re)incarnation of conversations that have been going on for some time and for which there is even a PEP (with related work on github).

After a few tens of messages were exchanged, Guido shared his thoughts, starting with:
This is an incredibly important discussion.
This seemed to really heat things up, eventually with core Twisted and Tornado folks chiming in. I learned a tremendous amount from the discussions that took place. There's probably a book deal in all that for a motivated archivist/interviewer...

After this went on for chunks of September and October, Guido stated that he'd like to break the discussion up into various sub-topics:
  • reactors
  • protocol implementations
  • Twisted (esp. Deferred)
  • Tornado
  • yield from vs. Futures

This was done in order to prevent the original thread from going over 100 messages and to better organize the discussion... but wow, things completely exploded after that (in good ways. mostly). It was async open season, and the ringing of shots in the air seemed continuous. If you scroll to about the half-way point of the October  archive page, you will see the first of these new threads ([Python-ideas] The async API of the future: Reactors). These messages essentially dominate the rest of the October archives. It's probably not unexpected that this continued into November. A related thread was started on Python-dev and it seemed to revive an old thread this month (on the same list).

All of this got mentioned on Reddit, too. It inspired at least two blog posts of which I am aware:  one post by Steve Dower, and another by Allen Short. Even better, though, Guido started an exploratory project called Tulip to test out some of these ideas in actual running code. As he mentions in the README, a tutorial by Greg Ewing was influential in the initial implementation of Tulip and initial design notes were made in the message [Python-ideas] Async API: some code to review.

Shortly after that, some of the Twisted devs local to Guido met with him at his former office in San Francisco. This went amazingly well and revolved mostly around the pros and cons of separating the protocol and transport functionality. Guido started experimenting with that in Tulip on December 6th. Yesterday, a followup meeting took place at the Rackspace office, this time with notes.

There's a long way to go still, but I find myself compulsively checking the commit log for Tulip now :-) It's exciting to imagine a future where Twisted and Tornado could easily interoperate with async support in Python 3 with a minimum of fuss. In fact, Glyph has already sketched out two classes which might be all that's needed for 2-way interoperation between Twisted and Python 3.

Here's to the future!