TweetFollow Us on Twitter

APL
Volume Number:1
Issue Number:12
Column Tag:APL Adventures

A Beginner's look at Mac APL

By Allyn Weaks, Seattle, WA.

I've wanted to learn APL ever since I heard that it could do integrals in one line and matrix inversions in a single symbol. Until PortaAPL for the Mac came along, I never had access to a computer with a full implementation. I'm hardly an expert at this language, having played with it for all of a month now, so please forgive any minor blunders. This month I'll start with a description of APL, enough of a tutorial to get you started defining simple functions (including an overview of the syntax, data types, and some of the simpler functions), and a brief review of PortaAPL for the Mac. I'll spend as few articles as possible in straight APL tutorial mode before starting in on Mac-specific features and bigger programs.

APL, A Programming Language, was developed by Kenneth Iverson at IBM in the early 60's, which makes it one of the older computer languages. It is more of a mathematical notation that can be interpreted by a computer than a traditional computer language, and handles data in groups (arrays) rather than just in small pieces.

Though it's a natural language for scientific and mathematical problems, in the U.S. it seems to be used mostly as a business language. Almost all of the textbooks currently in print are slanted towards business. Insurance companies in particular like it, because APL is ideal for statistical programs. In Europe and Canada it caught on more quickly and is used a lot in the scientific community for data analysis.

Since it is an interpreter, APL runs slower than most compilers. But it runs much faster than other interpreters such as Basic. The programs are much shorter, so APL spends most of its time working on the problem, not interpreting the commands. You also get the advantages of an interpreter - interactive writing and debugging. Functions are defined by name as in Forth or Logo, and you can define local variables, so it's easy to define new functions as you go along.

APL is extremely powerful. It's recursive and self-modifying, with dynamic allocation of data storage. It does calculations on entire arrays at once, not just element by element. This power has it's disadvantages, the primary one being memory limitations. Arrays require lots of memory. On a 512k Mac, the largest two dimensional array of integers you can deal with is about 225 by 225, which isn't big enough for many problems. Sometimes you have to use a less elegant algorithm to solve a problem. But then, who has ever had enough memory?

The peculiar character set APL uses (þ,©,“,ð,‰,Â, etc.) has given it the reputation of being unreadable. Pascal programmers have been heard to say that APL programmers hand each other sections of code and say 'Bet you can't figure out what this does!'. But APL isn't really harder to read than other languages once you know what the symbols mean. And since APL can work on groups of data all at once, instead of just a little at time, the code is much more compact. You can see the whole thing at once, instead of getting lost manipulating indices in the middle of triply nested loops.

APL is very different from Fortran, C, Forth, or Lisp. The symbols, syntax, and natural algorithms all take some getting used to. The first hurdle is the keyboard layout. Unshifted characters give upper-case letters, shifted characters and numbers give the APL symbols. You'll need to keep the keyboard map handy for reference at first, though most of the symbols are easy to find once you know the mnemonics (i.e. È (iota) is shift-I, ® (rho) is shift-R). If you can't stand it, there is an ASCII mode that substitutes keywords and ASCII punctuation for the symbols, but if you plan to read standard APL books, or trade programs with other APL users, it's better to do it right. The real notation is also more compact and easily read - the symbols stand out.

The next difficulty is the syntax. There is no hierarchy of operators - all APL expressions are evaluated from right to left. If you need to change the order of operation, you must use parentheses. For example, the expression 5-6-2 is equivalent to 5-(6-2) = 1 in APL, not (5-6)-2 = -3 as in most languages. And 5x6-2 equals 20, not 28.

The terminology is a bit different too. What is usually called an operator, such as + or -, is a function in APL, because it returns a value. An operator is a special thing that systematically modifies the effect of a function. Functions operate on data, and operators operate on functions. Operators are one of the things that makes APL so powerful - many mathematical ideas such as the inner product (given two vectors a,b,c and x,y,z the standard inner product is ax+by+cz, a scalar) can be generalized to any functions, not just the sum of the products.

Functions can be niladic, monadic, or dyadic. Niladic functions take no arguments - some system commands such as )OFF are niladic. Monadic functions take one argument, and are called by the sequence FUNCTION ARGUMENT, such as the factorial of N (!N). Dyadic functions, which take two arguments, are called as ARGUMENT FUNCTION ARGUMENT, as in 5 + 3. This is true of user defined functions as well as the built in primitives.

People who insist on strongly typed languages can stop here and read the Pascal article instead. The standard data types are boolean, character, integer, and floating point, but these distinctions are less important than the structure of the data into scalars, vectors, and larger arrays. An argument to any function can be a scalar, vector, or array, with a few exceptions. It is irrelevant (again with some exceptions) whether the data is a character string, an integer, or a set of boolean values. A scalar is a single number or character, without dimension or length. A vector is a set of data that forms a one dimensional array, a matrix has two dimensions. Arrays can have up to 63 dimensions in PortaAPL, though you'll never use that many - if you had two bits per dimension, you'd need a billion Gigabytes to hold the array. All elements of an array must have the same data type, so if you put a single floating point number in a large array, every element will be floating point. Type conversions are taken care of automatically and transparently at the interpreter's discretion, though there are ways to force a conversion to a particular type. On the Mac, boolean values are stored as bits, characters as bytes, integers as 4 byte words, and floating point as 8 byte long words. A character string is a vector of characters; a character matrix is a set of strings, one per row.

In addition to dividing functions into classes according to how many arguments they take, they can be divided into three groups that depend on what their effect is. Scalar functions change the data in an array, but not the shape or size of the array. Restructuring functions change the shape or dimension of an array, but not the data values. Mixed functions change both the data and the shape, and will have to wait for a later article. A scalar function can operate on arrays, not just scalars, but they operate element by element. For example, you can add two vectors of the same length:

 2 4 6 11 8  +  7 43 1 6 31

9 47 7 17 39

Iota (È) and rho (®) are two of the most important reshaping functions. Iota in it's monadic form creates a vector of integers in ascending order from a scalar: È6 gives the vector 1 2 3 4 5 6. Monadic rho gives the length of each dimension of an array: ® 1 3 5 7 is 4. Dyadic rho creates an array with the dimensions given by the left argument, filled with the values given in the right argument. The right argument is repeated as many times as necessary to fill the array:

 2 4 ® 1 2 3

1 2 3 1
2 3 1 2

Some of the scalar functions will look deceptively familiar to programmers. + and - do what you expect, but * means the exponential or power, not multiply, and / is not divide, but something more complicated - it can be either a restructuring function (compress) or an operator (reduction). When used as the reduction operator, f/ inserts the function f between each element of a vector:

 +/ 1 2 3 4
10

 ª/ 1 0 0 1
1

 -/ 1 2 3 4
™2

Fig. 1 APL Table of Functions

There is a floating point benchmark that's become popular on usenet: how long it takes to calculate the harmonic series to 10000. This is the sum of 1/i from i=1 to 10000. Most languages take 5 or so lines to write this program; APL takes 9 characters, including the 10000. No looping is needed. Start by noticing that you want to do something with an index that ranges from 1 to 10000. Well, there's a function iota that will generate a vector with all those values. Then, you need to take the reciprocal of each value, so just use the monadic ±. To take the sum of the vector, use the reduction operator with addition. So the whole thing turns into -/±È10000. How long does it take? 49 seconds, compared to 38 seconds for Aztec C, and 69 seconds for Megamax C. APL won't always do this well compared to compilers, this set of functions is extremely efficient in APL. Notice also the amount of memory used for a vector of 10000 4-byte integers.

To turn this into a function so you can run it for any number of terms, enter the editor by typing © Z “ HARMONIC N <ret>. This line will appear as line zero at the top of the screen. Any variable appearing on line zero is a local variable, all others are global. If you don't assign the function to a dummy variable (Z in this case), you won't be able to call it from another function later. Next, starting with line one, type in your program. At some point, the final value of the program must be assigned to the same dummy variable as in line zero. When you're finished, the screen should look like this:

 ©Z “ HARMONIC N
         [1]    Z “ -/±ÈN
 ©

You can move the cursor with the mouse. The lamp symbol (Ê) can be used at the beginning of a line to put in comments. Use command-z or the EDIT menu to exit the editor. To define a dyadic function, put one argument in front of the function name, and one after. DICE will roll N dice with M sides each:

©Z “ N DICE M; A; B 
 [1]    Ê  D&D dice roller - 
 rolls N M-sided dice
 [2]    A “ N ® M
 [3]    B “ œA
 [4]    Z “ -/B
 [5]    Ê This can also be written on 
 one line as:
 [6]    Ê  Z “ -/œN®M  or  Z “ -/B“œA“N®M
©                                                        

Line 2 sets up a vector with the same number of elements as there are dice and with each element equal to the numbers of sides of the die. The scalar operator ?R chooses a random element of the vector (ÈR), and if you give ? a vector as an argument it will find the random number for each element. Line 3 just adds up all of the independent rolls. What sort of changes need to be made to roll several dice with different numbers of sides? None! M can be a vector containing the sides of each die. Make sure the number of elements agrees with the value of N, however, or you will get the wrong result. There's at least one way to take care of this without changing the function, and I'll leave the finding of it as a puzzle.

In addition to the built in functions, APL has a number of system functions and system commands that are fairly standard from one implementation to another. System commands are prefaced with a right parenthesis, such as )OFF to exit to the finder, )SAVE FILENAME to save the workspace, and )COPY FILENAME to add a workspace to the current workspace. System functions, or quad functions, start with the quad symbol ¬. These can be used to inquire about or set various system variables, such as the precision of results with ¬PP, or to manipulate files. There are also quad versions of ¬OFF, ¬COPY and some other commands so that they can be called from inside of functions.

PortaAPL is written and distributed by Portable Software in Cambridge MA (60 Aberdeen Avenue, Cambridge, MA 02138; (617) 547-2918). A 512k Mac is required. The latest version is 2.0m. The price is $275 for the interpreter, some very clear documentation, and several workspaces with sample programs and functions to access parts of the toolbox. Upgrades are available for $25. Best of all, the interpreter isn't copy protected, so you can run it easily from a hard disk.

Portable Software has been writing APL interpreters for several years and for several different computers including Vax and the IBM PC. PortaAPL is a full IBM mainframe implementation, with lots of nice extensions. In fact, the main reference manual provided is the official IBM APL manual. The differences to the IBM version are an improved screen oriented editor, and the workspace id format which is system dependent. (Workspace id is a fancy term for program file name.) Extensions include ASCII mode, access to the Mac file system (though not the resource fork), and a way to write assembly language functions. The keyboard is a standard APL keyboard, with extensions for the Mac. Overstruck characters can be typed with the option key instead of key1-backspace-key2.

Another nice feature is a built in Datamedia 1520 APL terminal emulator. If you have access to APL on another computer, you can call up and work from your Mac. The Mac allows an improved keyboard set up, so you can send lowercase characters if you need to by holding down the option key. Unfortunately, there is no file transfer, so you can't send programs back and forth with it.

The program is written in C so it isn't as compact or as fast as it could be. It's 180k, and allows a work area of about 221k. Integers are 4 bytes and floating points numbers are 8 bytes with a range of +/- 1.79E308. Floating point calculations use the SANE package. The largest allowed function is 9999 lines; the maximum length of a symbol is 77 characters and underscores are allowed in symbol names. 16 files can be opened at once.

Editing functions and character matrices is acceptable, and certainly an improvement over the standard APL editor. (Try reading through some of the descriptions of editing a function in one of the books listed at the end!) The cursor is moved with the mouse, and there are menu and keyboard commands to insert and delete lines. Because of the nice fonts on the Mac, overstruck characters are typed in with the option key, so the editor can be in insert mode all of the time. Cut and paste aren't available, which can be a nuisance.

An excellent DRILL program is provided, that randomly generates APL expressions at 5 levels for you to evaluate. Since it evaluates the answer you give it and compares that to the evaluation of the problem, you can add to the challenge by coming up with expressions that are equivalent, but use different functions.

One of the workspaces that comes with the package is called Goodies. It has lots of useful stuff in it, including peek and poke, a sound function for playing simple tunes, mouse functions, printer i/o redirection, and functions to let you access the modem port. There are also routines that copy text or graphics from the clipboard to the screen, or text from the screen to the clipboard.

As of 2.0m, PortaAPL supports much of quickdraw and menus, but not windows, resources, or controls. Desk accessories are available from inside the interpreter. Although the EDIT menu doesn't show the standard cut, copy, and paste functions, command-X, and V are functional when desk accessories are active. The calls to quickdraw and the menu manager are straight-forward. To use them, you need to include the appropriate workspace. The quickdraw workspace includes functions for line drawing, rectangles, ovals, arcs, text fonts and scroll regions. Not all of quickdraw is supported - getpixel, regions, pictures, copybits, and icons are all missing. The lack of getpixel is going to make it much harder for me to do screen dumps to my non-Imagewriter printer. Menus are easily created by assigning a character matrix containing the items and the functions they run to a menu id number, then installing the menu. The interrupts are handled by APL if you are at the command level. From inside of programs, there is a function ¬GETKEY that lets you ask for the appropriate events. Windows aren't supported, but you can define regions for input and output.

All in all, PortaAPL is a nice job. I haven't found any bugs yet while putting in sample programs from various sources. There is one thing to beware of - you must remember to save your work before you exit, because APL won't remind you. It would be nice to have complete access to the toolbox, but that will come eventually - if not from Portable Software, then from anyone who is willing to do a bit of assembly language programming. It's expensive compared to Basic, but it's much more fun.

I hope this has been enough to get you started. I recommend that you buy or borrow an APL text book and work through plenty of examples, as well as the Drill program. Next month, I'll get into loops and branches, the other operators, and a few ways to generate prime numbers.

A Programming Language by Kenneth E. Iverson

John Wiley & Sons 1962

An Introduction to APL by S. Pommier

Cambridge University Press 1984

APL: An Interactive Approach by Gilman & Allen J. Rose John Wiley & Sons 1983

Structured Programming in APL by D. Geller & D. Freedman

Winthrop 1976

 

Community Search:
MacTech Search:

Software Updates via MacUpdate

Capture One 15.3.1 - RAW workflow softwa...
Capture One is a professional RAW converter offering you ultimate image quality with accurate colors and incredible detail from more than 400 high-end cameras - straight out of the box. It offers... Read more
Connect Fonts 23.0.3 - Font management s...
Connect Fonts is the creative professional's font manager. Every professional font manager should deliver the basics: spectacular previews, powerful search tools, and efficient font organization. You... Read more
CleanMyMac X 4.11.0 - Delete files that...
CleanMyMac X makes space for the things you love. Sporting a range of ingenious new features, CleanMyMac lets you safely and intelligently scan and clean your entire system, delete large, unused... Read more
Firefox 102.0 - Fast, safe Web browser.
Firefox offers a fast, safe Web browsing experience. Browse quickly, securely, and effortlessly. With its industry-leading features, Firefox is the choice of Web development professionals and casual... Read more
Hopper Disassembler 5.6.1 - Binary disas...
Hopper Disassembler is a binary disassembler, decompiler, and debugger for 32- and 64-bit executables. It will let you disassemble any binary you want, and provide you all the information about its... Read more
Skim 1.6.11 - PDF reader and note-taker...
Skim is a PDF reader and note-taker for OS X. It is designed to help you read and annotate scientific papers in PDF, but is also great for viewing any PDF file. Skim includes many features and has a... Read more
Alfred 4.6.7 - Quick launcher for apps a...
Alfred is an award-winning productivity application for OS X. Alfred saves you time when you search for files online or on your Mac. Be more productive with hotkeys, keywords, and file actions at... Read more
Transmit 5.8.7 - Excellent FTP/SFTP clie...
Transmit is an excellent FTP (file transfer protocol), SFTP, S3 (Amazon.com file hosting) and iDisk/WebDAV client that allows you to upload, download, and delete files over the internet. With the... Read more
Adobe Lightroom Classic 11.4.1 - Import,...
You can download Lightroom for Mac as a part of Creative Cloud for only $9.99/month with Photoshop, included as part of the photography package. The latest version of Lightroom gives you all of the... Read more
MarsEdit 4.5.9 - Quick and convenient bl...
MarsEdit is a blog editor for OS X that makes editing your blog like writing email, with spell-checking, drafts, multiple windows, and even AppleScript support. It works with with most blog services... Read more

Latest Forum Discussions

See All

Apple Arcade Weekly Round-Up: Major Upda...
Apple recently revealed July’s upcoming Apple Arcade releases in a new App Store Story, and this week’s new release is My Bowling 3D+ featuring offline and online multiplayer support, and more. It arrives from the developers of Pro Darts 2022+ and... | Read more »
Downhill Mountain Biking Game ‘Descender...
Just over three years ago in May of 2019 developer RageSquid and publisher No More Robots released a quirky downhill mountain biking game called Descenders on PC and Xbox One. Bemoaning a lack of “extreme sports" titles in recent years led RageSquid... | Read more »
SwitchArcade Round-Up: ‘Monster Hunter R...
Hello gentle readers, and welcome to the SwitchArcade Round-Up for June 30th, 2022. Thursday is once more upon us, and that means a bunch of new releases to look at. We start things off with DLC for some very big games, Monster Hunter Rise and... | Read more »
‘HOOK 2’ Review – A Sharp Left Hook From...
The original HOOK ($1.99) had a very simple idea behind it. You were presented with a tangled mess of hooks and loops, and you needed to remove each one without snagging any others. Extremely simple at first, but as the puzzles rolled along,... | Read more »
‘Dicey Dungeons’ Mobile Version Launchin...
After a very long wait, Terry Cavanagh’s dungeon crawling roguelite deckbuiler hybrid experience Dicey Dungeons is coming to mobile platforms next week alongside a huge free DLC pack on all platforms. This DLC will be included in the mobile... | Read more »
Distract Yourself With These Great Mobil...
Every day, we pick out a curated list of the best mobile discounts on the App Store and post them here. This list won't be comprehensive, but it every game on it is recommended. Feel free to check out the coverage we did on them in the links below... | Read more »
‘Danganronpa S: Ultimate Summer Camp’ is...
If you’ve been following Danganronp over the last few years, Spike Chunsoft celebrated its anniversary by bringing the series to mobile in the form of anniversary editions. After the first two released, there was a long delay for V3, but it finally... | Read more »
Out Now: ‘HOOK 2’, ‘Incoherence’, ‘Juras...
Each and every day new mobile games are hitting the App Store, and so each week we put together a big old list of all the best new releases of the past seven days. Back in the day the App Store would showcase the same games for a week, and then... | Read more »
Upcoming Mobile MMO RPG Shooter ‘Avatar:...
This past January a contingent of developers made up of Archosaur Games, Tencent, Lightstorm Entertainment, and Disney announced a new mobile game set in James Cameron’s Avatar universe titled Avatar: Reckoning. | Read more »
Culinary Platformer ‘Chefy-Chef’ Coming...
If your name is Chefy, it’s pretty much a given that you should be a chef. Such is the case with Chefy-Chef, a game from Bug Studio about a chef named Chefy who must travel to all sorts of exotic locations using a magical refrigerator in an effort... | Read more »

Price Scanner via MacPrices.net

July 4th sale at Verizon: Apple AirPods Pro f...
Verizon has Apple AirPods Pro on sale for $179.99 on their online store as part of their Fourth of July sale. Their price is $70 (28%) off Apple’s MSRP, and it’s among the lowest prices currently... Read more
Apple is now selling Certified Refurbished Ma...
Apple has added a full line of standard-configuration Mac Studios available in their Certified Refurbished section starting at only $1799 and ranging up to $400 off MSRP. Each Mac Studio comes with... Read more
Open-box 14″ M1 Pro MacBook Pros in stock tod...
QuickShip Electronics has open-box return Space Gray 14″ M1 Pro MacBook Pros in stock and on sale for $300-$450 off MSRP on their eBay store today. According to QuickShip, “The item in this listing... Read more
Can Being An iPhone User Really Determine Whe...
FEATURE: – If you’re traveling on the road today for the July 4th holiday, you might want to keep your Apple smartphone locked up inside the car’s glove compartment for your (and, everyone else’s)... Read more
2nd generation 4K Apple TVs with Siri remote...
Apple has restocked a full line of Certified Refurbished 2nd generation 32GB and 64GB 4K Apple TVs with Siri remotes for $30 off the cost of new models. Apple’s standard one-year warranty is included... Read more
Back in stock: Apple Watch Series 7 models fo...
Apple has restocked Certified Refurbished Apple Watch Series 7 WiFi-only models in their online store for $60-$70 off MSRP, starting at $339. Each Watch includes Apple’s standard one-year warranty, a... Read more
July 4th Sale at Expercom: $200 off any 16″ M...
Apple reseller Expercom has 16″ M1 Pro and M1 Max MacBook Pros available for $200 off MSRP as part of their July 4th sale. In addition to their MacBook Pro sale prices, take $50 off AppleCare+ when... Read more
10.2″ Apple iPads (WiFi models) are on sale f...
Amazon has Apple’s 9th generation 10.2″ WiFi iPads on sale for up to $20-$50 off MSRP for a limited time. Their prices are the lowest price currently available for one of these iPads. All models are... Read more
10-Core M1 Pro 14″ MacBook Pros on sale for $...
B&H Photo is offering $200 discounts on Apple’s new 14″ M1 Pro MacBook Pros with 10-Core CPUs (16GB RAM/1TB SSDs). Free 1-2 day shipping is available to most US addresses, and both models are in... Read more
B&H has 16-inch M1 Pro MacBook Pros in st...
New Space Gray 16″ MacBook Pros with Apple’s M1 Pro CPUs are in stock and on sale today at B&H Photo for $200 off Apple’s MSRP. Sale prices are for M1 Pro models with 512GB or 1TB of SSD storage... Read more

Jobs Board

VP, Software Engineering - *Apple* and Andr...
…Client Application Software Engineering team is seeking a VP, Software Engineering for Apple and Android. You will lead the client engineering team building Disney+, Read more
I/S Senior Engineer - *Apple* Systems Engin...
**19647BR** **Position Title:** I/S Senior Engineer - Apple Systems Engineering - Remote **Department:** Information Systems **Location:** Lakeland, FL between Read more
*Apple* IT Support Analyst - 2nd Shift - Zon...
Apple IT Support Analyst - 2nd Shift Professional Services Albany, New York Malta, New York Clifton Park, New York Menands, New York Syracuse, New York Watertown, Read more
Infotainment Certification Test Engineer (XC)...
…integration - CarPlay, android auto, MirrorLink, Baidu Carlife, MFi/iPod certification testing; Apple PPID preparation, Google HUCD and GTM preparation + 3 years of Read more
Workplace Services *Apple* Device Managemen...
…3350 Riverwood Parkway Suite 900, Atlanta, GA, 30339 USA **Workplace Services Apple Device Management** **Role Overview** Carrier is seeking an experienced and Read more
All contents are Copyright 1984-2011 by Xplain Corporation. All rights reserved. Theme designed by Icreon.