NumPy another Iverson Ghost

During my recent SmugMug API and Python adventures I was haunted by an Iverson ghost: NumPy

An Iverson ghost is an embedding of APL like array programming features in nonAPL languages and tools.

You would be surprised at how often Iverson ghosts appear. Whenever programmers are challenged with processing large numeric arrays they rediscover bits of APL. Often they’re unaware of the rich heritage of array processing languages but in NumPy's case, they indirectly acknowledged the debt. In Numerical Python the authors wrote:

“The languages which were used to guide the development of NumPy include the infamous APL family of languages, Basis, MATLAB, FORTRAN, S and S+, and others.”

I consider “infamous” an upgrade from “a mistake carried through to perfection.”

Not only do developers frequently conjure up Iverson ghosts. They also invariably turn into little apostles of array programming that won’t shut up about how cutting down on all those goddamn loops clarifies and simplifies algorithms. How learning to think about operating on entire arrays, versus one dinky number at a time, frees the mind. Why it’s almost as if array programming is a tool of thought.

Where have I heard this before?

Ahh, I’ve got it, when I first encountered APL almost fifty years ago.

Yes, I am an old programmer, a fossil, a living relic. My brain is a putrid pool of punky programming languages. Python is just the latest in a longish line of languages. Some people collect stamps. I collect programming languages. And, just like stamp collectors have favorite stamps, I find some programming languages more attractive than others. For example, I recognize the undeniable utility of C/C++, for many tasks they are the only serious options, yet as useful and pervasive as C/C++ are they have never tickled my fancy. The notation is ugly! Yeah, I said it; suck on it C people. Similarly, the world’s most commonly used programming language JavaScript is equally ugly. Again, JavaScript is so damn useful that programmers put up with its many warts. Some have even made a few bucks writing books about its meager good parts.

I have similar inflammatory opinions about other widely used languages. The one that is making me miserable now is SQL, particularly Microsoft’s variant T-SQL. On purely aesthetic grounds I find well-formed SQL queries less appalling than your average C pointer fest. Core SQL is fairly elegant but the macro programming features that have grown up around it are depraved. I feel dirty when forced to use them which is just about every other day.

At the end of my programming day, I want to look on something that is beautiful. I don’t particularly care about how useful a chunk of code is or how much money it might make, or what silly little business problem it solves. If the damn code is ugly I don’t want to see it.

People keep rediscovering array programming, best described in Ken Iverson’s 1962 book A Programming Language, for two basic reasons:

  1. It’s an efficient way to handle an important class of problems.
  2. It’s a step away from the ugly and back towards the beautiful.

Both of these reasons manifest in NumPy‘s resounding success in the Python world.

As usual, efficiency led the way. The authors of Numerical Python note:

Why are these extensions needed? The core reason is a very prosaic one, and that is that manipulating a set of a million numbers in Python with the standard data structures such as lists, tuples or classes is much too slow and uses too much space.

Faced with a “does not compute” situation you can either try something else or fix what you have. The Python people fixed Python with NumPy. Pythonistas reluctantly embraced NumPy but quickly went apostolic! Now books like Elegant SciPy and the entire SciPy toolset that been built on NumPy take it for granted.

Is there anything in NumPy for programmers that have been drinking the array processing Kool-Aid for decades? The answer is yes! J programmers, in particular, are in for a treat with the new Python3 addon that’s been released with the latest J 8.07 beta. This addon directly supports NumPy arrays making it easy to swap data in and out of the J/Python environments. It’s one of those best of both worlds things.

The following NumPy examples are from the SciPy.org NumPy quick start tutorial. For each NumPy statement, I have provided a J equivalent. J is a descendant of APL. It was largely designed by the same man: Ken Iverson. A scumbag lawyer or greedy patent troll might consider suing NumPy‘s creators after looking at these examples. APL’s influence is obvious. Fortunately, Ken Iverson was more interested in promoting good ideas that profiting from them. I suspect he would be flattered that APL has mutated and colonized strange new worlds and I think even zealous Pythonistas will agree that Python is a delightfully strange world.

Some Numpy and J examples

Selected Examples from https://docs.scipy.org/doc/numpy-dev/user/quickstart.html Output has been suppressed here. For a more detailed look at these examples browse the Jupyter notebook:  NumPy and J Make Sweet Array Love.

Creating simple arrays

 
 # numpy
 a = np.arange(15).reshape(3, 5)
 
 NB. J
 a =. 3 5 $ i. 15

 # numpy
 a = np.array([2,3,4])
 
 NB. J
 a =. 2 3 4
 
 # numpy
 b = np.array([(1.5,2,3), (4,5,6)])
 
 NB. J
 b =. 1.5 2 3 ,: 4 5 6

 # numpy
 c = np.array( [ [1,2], [3,4] ], dtype=complex )
 
 NB. J
 j. 1 2 ,: 3 4
 
 # numpy
 np.zeros( (3,4) )
 
 NB. J
 3 4 $ 0
 
 # numpy - allocates array with whatever is in memory
 np.empty( (2,3) )
 
 NB. J - uses fill - safer but slower than numpy's trust memory method
 2 3 $ 0.0001 

Basic operations

 
 # numpy
 a = np.array( [20,30,40,50] )
 b = np.arange( 4 )
 c = a - b
 
 NB. J
 a =. 20 30 40 50
 b =. i. 4
 c =. a - b
 
 # numpy - uses previously defined (b)
 b ** 2
 
 NB. J
 b ^ 2

 # numpy - uses previously defined (a)
 10 * np.sin(a)
 
 NB. J
 10 * 1 o. a
 
 # numpy - booleans are True and False
 a < 35
 
 NB. J - booleans are 1 and 0
 a < 35

Array processing

 
 # numpy
 a = np.array( [[1,1], [0,1]] )
 b = np.array( [[2,0], [3,4]] )
 # elementwise product
 a * b

 NB. J
 a =. 1 1 ,: 0 1
 b =. 2 0 ,: 3 4
 a * b

 # numpy - matrix product
 np.dot(a, b)

 NB. J - matrix product
 a +/ . * b  
 
 # numpy - uniform pseudo random
 a = np.random.random( (2,3) )
 
 NB. J - uniform pseudo random
 a =. ? 2 3 $ 0
 
 # numpy - sum all array elements - implicit ravel
 a.sum(a)
 
 NB. J - sum all array elements - explicit ravel
 +/ , a
 
 # numpy
 b = np.arange(12).reshape(3,4)
 # sum of each column
 b.sum(axis=0)
 # min of each row
 b.min(axis=1)
 # cumulative sum along each row
 b.cumsum(axis=1)
 # transpose
 b.T     

 NB. J 
 b =. 3 4 $ i. 12
 NB. sum of each column
 +/ b
 NB. min of each row
 <./"1 b
 NB. cumulative sum along each row
 +/\"0 1 b
 NB. transpose
 |: b

Indexing and slicing

 
 # numpy 
 a = np.arange(10) ** 3 
 a[2]
 a[2:5]
 a[ : :-1]   # reversal

 NB. J
 a =. (i. 10) ^ 3
 2 { a
 (2 + i. 3) { a
 |. a

Typesetting UTF8 APL code with the LaTeX lstlisting package

UTF8 APL characters within a LaTeX lstlisting environment. Click for *.tex source code

Typesetting APL source code has always been a pain in the ass! In the dark ages, (the 1970’s), you had to fiddle with APL type-balls and live without luxuries like lower case letters. With the advent of general outline fonts it became technically possible to render APL glyphs on standard display devices provided you:

  1. Designed your own APL font.
  2. Mapped the atomic vector of your APL to whatever encoding your font demanded.
  3. Wrote WSFULL‘s of junk transliteration functions to dump your APL objects as font encoded text.

It’s a testament to either the talent, or pig headedness of APL programmers, that many actually did this. We all hated it! We still hate it! But, like an abused spouse, we kept going back for more.  It’s our fault; if we loved APL more it would stop hitting us!

When Unicode appeared APL’ers cheered — our long ASCII nightmare was ending. The more politically astute worked to include the APL characters in the Unicode standard. Hey if Klingon is there why not APL? Everyone thought it was just a matter of time until APL vendors abandoned their nonstandard atomic vectors and fully embraced Unicode. With a few notable exceptions we are still waiting. While we wait the problem of typesetting APL source code festers.

My preferred source code listing tool is the \LaTeX lstlisting package. lstlisting works well for standard ANSI source code.  I use it for J, C#, SQL, C, XML, Ocaml, Mathematica, F#, shell scripts and \LaTeX source code, i.e. everything except APL! lstlisting is an eight bit package; it will not handle arbitrary Unicode out of the box.  I didn’t know how to get around this so I handled APL by enclosing UTF8 APL text in plain \begin{verbatim} … \end{verbatim} environments. This works for XeLaTeX and LuaLaTeX but you lose all the lstlisting goodies. Then I saw an interesting tex.stackexchange.com posting about The ‘listings’ package and UTF-8. One solution to the post’s “French ligature problem” showed how to force Unicode down lstlisting‘s throat. I wondered if the same method would work for APL. It turns out that it does!

If you insert the following snippet of TeX code in your document preamble LuaLaTeX and XeLaTeX will properly process UTF8 APL text in lstlisting environments. You will need to download and install the APL385 Unicode font if it’s not on your system.  A test \LaTeX document illustrating this hack is available here. The compiled PDF is available here. As always these files can be accessed in the files sidebar.

% set lstlisting to accept UTF8 APL text
\makeatletter
\lst@InputCatcodes
\def\lst@DefEC{%
 \lst@CCECUse \lst@ProcessLetter
  ^^80^^81^^82^^83^^84^^85^^86^^87^^88^^89^^8a^^8b^^8c^^8d^^8e^^8f%
  ^^90^^91^^92^^93^^94^^95^^96^^97^^98^^99^^9a^^9b^^9c^^9d^^9e^^9f%
  ^^a0^^a1^^a2^^a3^^a4^^a5^^a6^^a7^^a8^^a9^^aa^^ab^^ac^^ad^^ae^^af%
  ^^b0^^b1^^b2^^b3^^b4^^b5^^b6^^b7^^b8^^b9^^ba^^bb^^bc^^bd^^be^^bf%
  ^^c0^^c1^^c2^^c3^^c4^^c5^^c6^^c7^^c8^^c9^^ca^^cb^^cc^^cd^^ce^^cf%
  ^^d0^^d1^^d2^^d3^^d4^^d5^^d6^^d7^^d8^^d9^^da^^db^^dc^^dd^^de^^df%
  ^^e0^^e1^^e2^^e3^^e4^^e5^^e6^^e7^^e8^^e9^^ea^^eb^^ec^^ed^^ee^^ef%
  ^^f0^^f1^^f2^^f3^^f4^^f5^^f6^^f7^^f8^^f9^^fa^^fb^^fc^^fd^^fe^^ff%
  ^^^^20ac^^^^0153^^^^0152%
  ^^^^20a7^^^^2190^^^^2191^^^^2192^^^^2193^^^^2206^^^^2207^^^^220a%
  ^^^^2218^^^^2228^^^^2229^^^^222a^^^^2235^^^^223c^^^^2260^^^^2261%
  ^^^^2262^^^^2264^^^^2265^^^^2282^^^^2283^^^^2296^^^^22a2^^^^22a3%
  ^^^^22a4^^^^22a5^^^^22c4^^^^2308^^^^230a^^^^2336^^^^2337^^^^2339%
  ^^^^233b^^^^233d^^^^233f^^^^2340^^^^2342^^^^2347^^^^2348^^^^2349%
  ^^^^234b^^^^234e^^^^2350^^^^2352^^^^2355^^^^2357^^^^2359^^^^235d%
  ^^^^235e^^^^235f^^^^2361^^^^2362^^^^2363^^^^2364^^^^2365^^^^2368%
  ^^^^236a^^^^236b^^^^236c^^^^2371^^^^2372^^^^2373^^^^2374^^^^2375%
  ^^^^2377^^^^2378^^^^237a^^^^2395^^^^25af^^^^25ca^^^^25cb%
  ^^00}
\lst@RestoreCatcodes
\makeatother

The Return of APL Fingers

APL typewriter ball (1970s)

APL typewriter ball (1970s)

I am programming in APL again after a six-year hiatus. My APL fingers are rusty but it’s amazing how deep muscle memory goes. I still know where all the beautiful APL glyphs’ hide on standard keyboards. I’ve programmed in almost a dozen programming languages but I maintain warm feelings for APL because I lost my coding virginity to her.

APL was gentle: abstract, clean, austere and so intoxicatingly elegant. I loved Iverson notation and how it manifested in what remains the most beautiful symbol set ever devised for a programming language. Blinded by passion I overlooked APL’s faults but as my ardor cooled I took notice of one glaring APL problem that persists to this day. Displaying, printing, emailing and now blogging with APL is a pain! Much has improved with the steady adoption of Unicode but even today handling APL imposes burdens. For example, without the APL385 Unicode font your browser will mangle the following APL characters.

!*+-/<=>?\^|~¨¯×÷←↑→↓∆∇∊∘∨∩∪∵∼≠≡≢≤≥⊂⊃⊖⊢⊣⊤⊥⋄⌈⌊

⌶⌷⌹⌻⌽⌿⍀⍂⍇⍈⍉⍋⍎⍐⍒⍕⍗⍙⍝⍞⍟⍡⍢⍣⍤⍥⍨⍪⍫⍬⍱⍲⍳⍴⍵⍷⍸⍺⎕◊○

By the way, if you have not installed APL385 Unicode I highly recommend downloading and installing it. You can get APL385 at VectorRegrettably installing a Unicode APL font will not fix all your APL character problems! In particular you cannot reliably:

  1. Copy and paste APL code between APL vendors or between APL and other languages.
  2. Typeset APL listings with ubiquitous \LaTeX packages like lstlisting.
  3. Post APL idioms on Twitter. Twitter’s 140 character limit is not such a big deal for APL.
  4. Print on arbitrary network printers! It’s the 21st century yet office printers are routinely limited by parochial IT policies to a small set of standard fonts.

APL’ers  almost relish these irritants after all problem solving is what APL’ers do! We don’t go crying to momma; we squash whatever is annoying us and get on with deeper problems.

Currently I am dealing with how to display APL functions on WordPress. WordPress supports a source code highlighting plug-in based on Alex Gorbatchev’s excellent Javascript SyntaxHighlighter. The WordPress plug-in produces wonderful results for well known languages like C# and by defining language specific Javascript classes you can highlight languages like APL. Eric Lescasse has done this for APL+WIN code so it is possible, (given Unicode APL fonts), to render highlighted web friendly APL code. Unfortunately WordPress does not support APL highlighting, (what a surprise), and has banned user Javascript classes on their freebie blogs. Apparently some programmers abuse JavaScript, (another surprise), and uncontrolled Javascript’ing might endanger the WordPress business model.

This leaves the users of peculiar programming languages with a problem. We can pester WordPress to support new languages or we can roll our own. I am starting a campaign to get APL and J on WordPress’s list of highlighted languages and while I am waiting for official support I will roll my own. Fortunately, highlighting code for blogs is not difficult. The much maligned MS Word (2007 and beyond) can crank out blog happy APL provided you have UTF-8 APL Unicode text to format. Getting UTF-8’ed APL is the tricky bit. Some APL systems like Dyalog directly support UTF-8 and others are planning to do so. APL+WIN cannot spit out UTF-8 but it’s not difficult to transform APL+WIN to UTF-8. The APL Wiki contains some slick APL+WIN functions to convert internal APL text to and from UTF-8.

To get a sense of why all this fuss is worthwhile consider the following APL function taken from Eugene McDonnell’s superb essay Life: Nasty, Brutish, and Short.

∇ z ← LifeKnuth w;v
v ← w
w ← w + (1 ⌽ w) + ¯1 ⌽ w
w ← w + (1 ⊖ w) + ¯1 ⊖ w
w ← w + w – v
z ← w ∊ 5 6 7

This little function is not the shortest APL Life function but in my opinion it’s the clearest and most concise description of Life’s generation rules out there. Tool of thought is not an empty APL marketing slogan. It’s the real deal!