## How Did Compute!’s Gazette’s MLX Program Work?

This is, at least, a retrocomputing-adjacent piece. I’m looking back at the logic of a common and useful tool from the early-to-mid-80s and why it’s built that way. I hope you enjoy. It has to deal with some of the fussier points about how Commodore 64 computers worked. If you find a paragraph is too much technical fussing for you, I ask you to not give up, just zip on to the next paragraph. It’s interesting to know why something was written that way, but it’s all right to accept that it was and move to the next point.

When the world and I were young, in the 1980s, we still had computers. There were two ways to get software, though. One was trading cassette tapes or floppy disks with cracked programs on them. (The cracking was taking off the copy-protection.) The other was typing. You could type in your own programs, certainly, just like you can make your own web page just by typing. Or you could type in a program. We had many magazines and books that had programs ready for entry. Some were serious programs, spreadsheets and word processors and such. Some were fun, like games or fractal-generators or such. Some were in-between, programs to draw or compose music or the such. Some added graphics or sound commands that the built-in BASIC programming language lacked. All this was available for the $2.95 cover price, or ten cents a page at the library photocopier. I had a Commodore 64 for most of this era, moving to a Commodore 128 (which also ran Commodore 64 programs) in 1989 or so. So my impressions, and this article, default to the Commodore 64 experience. These programs all had the same weakness. You had to type them in. You can expect to make errors. If the program was written in BASIC you had a hope of spotting errors. The BASIC programming language uses common English words for its commands. Their grammar is not English, but it’s also very formulaic, and not hard to pick up. One has a chance of spotting mistakes if it’s 250 PIRNT "SUM; " S one typed. But many programs were distributed as machine language. That is, the actual specific numbers that correspond to microchip instructions. For the Commodore 64, and most of the eight-bit home computers of the era, this was the 6502 microchip. (The 64 used a variation, the 6510. The differences between the 6502 and 6510 don’t matter for this essay.) Machine language had advantages, making the programs run faster, and usually able to do more things than BASIC could. But a string of numbers is only barely human-readable. Oh, you might in time learn to recognize the valid microchip instructions. But it is much harder to spot the mistakes on entering 32 255 120. That last would be a valid command on any eight-bit Commodore computer. It would have the computer print something, if it weren’t for the transposition errors. ## What Was MLX and How Did You Use It? The magazines came up with tools to handle this. In the 398-page(!) December 1983 issue of Compute!, my favorite line of magazines introduced MLX. This was a program, written in BASIC, which let you enter machine language programs. Charles Brannon has the credit for writing the article which introduced it. I assume he also wrote the program, but could be mistaken. I’m open to better information. Other magazines had other programs to do the same work; I knew them less well. MLX formatted machine language programs to look like this: 49152 :169,002,141,178,002,169,149 49158 :000,141,179,002,141,180,137 49164 :002,141,181,002,169,001,252 49170 :141,183,002,169,003,141,145 This was the first few lines of code for a game called Turnabout, by Mark Tuttle and Kevin Mykytyn. It ran in issue #28 of Commodore-computers-spinoff magazine Compute!’s Gazette. You can see the original at archive.org and imagine the poor wrists of people typing all this in. What did all this mean, though? These were lines you would enter in while running MLX. Before the colon was a location in memory. The numbers after the colon — the entries, I’ll call them — are six machine language instructions, one number to go into each memory cell. So, the number 169 was destined to go into memory location 49152. The number 002 would go into memory location 49153. The number 141 would go into memory location 49154. And so on; 000 would go into memory location 49158, 141 into 49159, 179 into 49160. 002 would go into memory location 49164; 141 would go into memory location 49170. And so on. MLX would prompt you with the line number, the 49152 or 49158 or 49164 or so on. Machine language programs could go into almost any memory location. You had to tell it where to start. 49152 was a popular location for Commodore 64 programs. It was the start of a nice block of memory not easily accessed except by machine language programs. Then you would type in the entries, the numbers that follow. This was a reasonably efficient way to key this stuff in. MLX automatically advanced the location in memory and would handle things like saving the program to tape or disk when you were done. The alert reader notices, though, that there are seven entries after the colon in each line. That seventh number is the checksum. It’s the guard that Compute! and Compute!’s Gazette put against typos. This seventh number was a checksum. MLX did a calculation based on the memory location and the first six numbers of the line. If it was not the seventh number on the line, then there was an error somewhere. You had to re-enter the line to get it right. The thing I’d wondered, and finally got curious enough to explore, was how it calculated this. ## What Was The Checksum And How Did It Work? Happily, Compute! and Compute!’s Gazette published MLX in almost every issue, so it’s easy to find. You can see it, for example, on page 123 of the October 1985 issue of Compute!’s Gazette. And MLX was itself a BASIC program. There are quirks of the language, and its representation in magazine print, that take time to get used to. But one can parse it without needing much expertise. One important thing is that most Commodore BASIC commands didn’t need spaces after them. For an often-used program like this they’d skip the spaces. And the : symbol denoted the end of one command and start of another. So, for example, PRINTCHR$(20):IFN=CKSUMTHEN530 one learns means PRINT CHR\$(20); IF N = CKSUM THEN 530.

So how does it work? MLX is, as a program, convoluted. It’s well-described by the old term “spaghetti code”. But the actual calculation of the checksum is done in a single line of the program, albeit one with several instructions. I’ll print it, but with some spaces added in to make it easier to read.

FOR I = 1 TO 6:
CKSUM = (CKSUM + A(I))AND 255:
NEXT

Most of this you have a chance of understanding even if you don’t program. CKSUM is the checksum number. AD is the memory address for the start of the line. A is an array of six numbers, the six numbers of that line of machine language. I is an index, a number that ranges from 1 to 6 here. Each A(I) happens to be a number between 0 and 255 inclusive, because that’s the range of integers you can represent with eight bits.

## What Did This Code Mean?

So to decipher all this. Starting off. CKSUM = AD - INT(AD/256)*256. INT means “calculate the largest integer not greater than whatever’s inside”. So, like, INT(50/256) would be 0; INT(300/256) would be 1; INT(600/256) would be 2. What we start with, then, is the checksum is “the remainder after dividing the line’s starting address by 256”. We’re familiar with this, mathematically, as “address modulo 256”.

In any modern programming language, we’d write this as CKSUM = MOD(AD, 256) or CKSUM = AD % 256. But Commodore 64 BASIC didn’t have a modulo command. This structure was the familiar and comfortable enough workaround. But, read on.

The next bit was a for/next loop. This would do the steps inside for every integer value of I, starting at 1 and increasing to 6. CKSUM + A(I) has an obvious enough intention. What is the AND 255 part doing, though?

AND, here, is a logic operator. For the Commodore 64, it works on numbers represented as two-byte integers. These have a memory representation of 11111111 11111111 for ‘true’, and 00000000 00000000 for ‘false’. The very leftmost bit, for integers, is a plus-or-minus-sign. If that leftmost bit is a 1, the number is negative; if that leftmost bit is a 0, the number is positive. Did you notice me palming that card, there? We’ll come back to that.

Ordinary whole numbers can be represented in binary too. Like, the number 26 has a binary representation of 00000000 00011010. The number, say, 14 has a binary representation of 00000000 00001110. 26 AND 14 is the number 00000000 00001010, the binary digit being a 1 only when both the first and second numbers have a 1 in that column. This bitwise and operation is also sometimes referred to as masking, as in masking tape. The zeroes in the binary digits of one number mask out the binary digits of the other. (Which does the masking is a matter of taste; 26 AND 14 is the same number as 14 AND 26.)

The binary 00000000 0001010 is the decimal number 10. So you can see that generally these bitwise and operations give you weird results. Taking the bitwise and for 255 is more predictable, though. The number 255 has a bit representation of 00000000 11111111. So what (CKSUM + A(I)) AND 255 does is … give the remainder after dividing (CKSUM + A(I)) by 256. That is, it’s (CKSUM + A(I)) modulo 256.

The formula’s not complicated. To write it in mathematical terms, the calculation is:

$ck = \left(addr + \sum_{i = 1}^6 a_i\right) mod 256$

## Why Write It Like That?

So we have a question. Why are we calculating a number modulo 256 by two different processes? And in the same line of the program?

We get an answer by looking at the binary representation of 49152, which is 11000000 00000000. Remember that card I just palmed? I had warned that if the leftmost digit there were a 1, the number was understood to be negative. 49152 is many things, none of them negative.

So now we know the reason behind the odd programming choice to do the same thing two different ways. As with many odd programming choices it amounts to technical details of how Commodore hardware worked. The Commodore 64’s logical operators — AND, OR, and NOT — work on variables stored as two-byte integers. Two-byte integers can represent numbers from -32,768 up to +32,767. But memory addresses on the Commodore 64 are indexed from 0 up to 65,535. We can’t use bit masking to do the modulo operation, not on memory locations.

I have a second question, though. Look at the work inside the FOR loop. It takes the current value of the checksum, adds one of the entries to it, and takes the bitwise AND of that with 255. Why? The value would be the same if we waited until the loop was done to take the bitwise AND. At least, it would be unless the checksum grew to larger than 32,767. The checksum will be the sum of at most seven numbers, none of them larger than 255, though, so that can’t be the contraint. It’s usually faster to do as little inside a loop as possible, so, why this extravagance?

My first observation is that this FOR loop does the commands inside it six times. And logical operations like AND are very fast. The speed difference could not possibly be perceived. There is a point where optimizing your code is just making life harder for yourself.

My second observation goes back to the quirks of the Commodore 64. You entered commands, like the lines of a BASIC program, on a “logical line” that allowed up to eighty tokens. For typing in commands this is the same as the number of characters. Can this line be rewritten so there’s no redundant code inside the for loop, and so it’s all under 80 characters long?

Yes. This line would have the same effect and it’s only 78 characters:

Why not use that, then?

I don’t have a clear answer. I suspect it’s for the benefit of people typing in the MLX program. In typing that in I’d have trouble not putting in a space between FOR and I, or between CKSUM and AND. Also before and after the TO and before and after AND. This would make the line run over 80 characters and make it crash. The original line is 68 characters, short enough that anyone could add a space here and there and not mess up anything. In looking through MLX, and other programs, I find there are relatively few lines more than 70 characters long. I have found them as long as 76 characters, though. I can’t rule out there being 78- or 79-character lines. They would have to suppose anyone typing them in understands when the line is too long.

There’s an interesting bit of support for this. Compute! also published machine language programs for the Atari 400 and 800. A version of MLX came out for the Atari at the same time the Commodore 64’s came out. Atari BASIC allowed for 120 characters total. And the equivalent line in Atari MLX was:

This has a longer name for the address variable. It uses a different way to ensure that CKSUM stays a number between 0 and 255. But the whole line is only 98 characters.

We could save more spaces on the Commodore 64 version, though. Commodore BASIC “really” used only the first two characters of a variable name. To write CKSUM is for the convenience of the programmer. To the computer it would be the same if we wrote CK. We could even truncate it to CK for this one line of code. The only penalty would be confusing the reader who doesn’t remember that CK and CKSUM are the same variable.

And there’s no reason that this couldn’t have been two lines. One line could add up the checksum and a second could do the bitwise AND. Maybe this is all a matter of the programmer’s tastes.

In a modern language this is all quite zippy to code. To write it in Octave or Matlab is something like:

function [checksOut] = oldmlx(oneline)
entries = oneline(2:7);
keyedChecksum = oneline(8);

checksum = mod(address, 256);
checksum += sum(entries);
checksum = mod(checksum, 256);

checksOut = (keyedChecksum == checksum);
endfunction

This is a bit verbose. I want it to be easier to see what work is being done. We could make it this compact:

function [checksOut] = oldmlx(oneline)
checksOut = !(mod(sum(oneline(1:7))-oneline(8), 256));
endfunction

I don’t like compressing my thinking quite that much, though.

But that’s the checksum. Now the question: did it work?

Since Compute! and Compute!’s Gazette used it for years, the presumptive answer is that it did. The real question, then, is did it work well? “Well” means does it prevent the kinds of mistakes you’re likely to make without demanding too much extra work. We could, for example, eliminate nearly all errors by demanding every line be entered three times and accept only a number that’s entered the same at least two of three times. That’s an incredible typing load. Here? We have to enter one extra number for every six. Much lower load, but it allows more errors through. But the calculation is — effectively — simply “add together all the numbers we typed in, and see if that adds to the expected total”. If it stops the most likely errors, though, then it’s good. So let’s consider them.

The first and simplest error? Entering the wrong line. MLX advanced the memory location on its own. So if you intend to write the line for memory location 50268, and your eye slips and you start entering that for 50274 instead? Or even, reading left to right, going to line 50814 in the next column? Very easy to do. This checksum will detect that nicely, though. Entering one line too soon, or too late, will give a checksum that’s off by 6. If your eye skips two lines, the checksum will be off by 12. The only way to not have the checksum miss is to enter a line that’s some multiple of 256 memory locations away. And since each line is six memory locations, that means you have to jump 768 memory locations away. That is 128 lines away. You are not going to make that mistake. (Going from one column in the magazine to the next is a jump of 91 lines. The pages were 8½-by-11 pages, so were a bit easier to read than the image makes them look.)

How about other errors? You could mis-key, say, 169. But think of the plausible errors. Typing it in as 159 or 196 or 269 would be detected by the checksum. The only one that wouldn’t would be to enter a number that’s equal to 169, modulo 256. So, 425, say, or 681. There is nobody so careless as to read 169 and accidentally type 425, though. In any case, other code in MLX rejects any data that’s not between 0 and 255, so that’s caught before the checksum comes into play.

So it’s safe against the most obvious mistake. And against mis-keying a single entry. Yes, it’s possible that you typed in the whole line right but mis-keyed the checksum. If you did that you felt dumb but re-entered the line. If you even noticed and didn’t just accept the error report and start re-entering the line.

What about mis-keying double entries? And here we have trouble. Suppose that you’re supposed to enter 169, 062 and instead enter 159, 072. They’ll add to the same quantity, and the same checksum. All that’s protecting you is that it takes a bit of luck to make two errors that exactly balance each other. But, then, slipping and hitting an adjacent number on the keyboard is an easy mistake to make.

Worse is entry transposition. If you enter 062, 169 instead you have made no checksum errors. And you won’t even be typing any number “wrong”. At least with the mis-keying you might notice that 169 is a common number and 159 a rare one in machine language. (169 was the command “Load Accumulator”. That is, copy a number into the Central Processing Unit’s accumulator. This was one of three on-chip memory slots. 159 was no meaningful command. It would only appear as data.) Swapping two numbers is another easy error to make.

And they would happen. I can attest from experience. I’d had at least one program which, after typing, had one of these glitches. After all the time spent entering it, I ended up with a program that didn’t work. And I never had the heart to go back and track down the glitch or, more efficiently, retype the whole thing from scratch.

The irony is that the program with the critical typing errors was a machine language compiler. It’s something that would have let me write this sort of machine language code. Since I never reentered it, I never created anything but the most trivial of machine language programs for the 64.

So this MLX checksum was fair. It devoted one-seventh of the typing to error detection. It could catch line-swap errors, single-entry mis-keyings, and transpositions within one entry. It couldn’t catch transposing two entries. So that could have been better. I hope to address that soon.

## The Summer 2017 Mathematics A To Z: Functor

Gaurish gives me another topic for today. I’m now no longer sure whether Gaurish hopes me to become a topology blogger or a category theory blogger. I have the last laugh, though. I’ve wanted to get better-versed in both fields and there’s nothing like explaining something to learn about it.

# Functor.

So, category theory. It’s a foundational field. It talks about stuff that’s terribly abstract. This means it’s powerful, but it can be hard to think of interesting examples. I’ll try, though.

It starts with categories. These have three parts. The first part is a set of things. (There always is.) The second part is a collection of matches between pairs of things in the set. They’re called morphisms. The third part is a rule that lets us combine two morphisms into a new, third one. That is. Suppose ‘a’, ‘b’, and ‘c’ are things in the set. Then there’s a morphism that matches $a \rightarrow b$, and a morphism that matches $b \rightarrow c$. And we can combine them into another morphism that matches $a \rightarrow c$. So we have a set of things, and a set of things we can do with those things. And the set of things we can do is itself a group.

This describes a lot of stuff. Group theory fits seamlessly into this description. Most of what we do with numbers is a kind of group theory. Vector spaces do too. Most of what we do with analysis has vector spaces underneath it. Topology does too. Most of what we do with geometry is an expression of topology. So you see why category theory is so foundational.

Functors enter our picture when we have two categories. Or more. They’re about the ways we can match up categories. But let’s start with two categories. One of them I’ll name ‘C’, and the other, ‘D’. A functor has to match everything that’s in the set of ‘C’ to something that’s in the set of ‘D’.

And it does more. It has to match every morphism between things in ‘C’ to some other morphism, between corresponding things in ‘D’. It’s got to do it in a way that satisfies that combining, too. That is, suppose that ‘f’ and ‘g’ are morphisms for ‘C’. And that ‘f’ and ‘g’ combine to make ‘h’. Then, the functor has to match ‘f’ and ‘g’ and ‘h’ to some morphisms for ‘D’. The combination of whatever ‘f’ matches to and whatever ‘g’ matches to has to be whatever ‘h’ matches to.

This might sound to you like a homomorphism. If it does, I admire your memory or mathematical prowess. Functors are about matching one thing to another in a way that preserves structure. Structure is the way that sets of things can interact. We naturally look for stuff made up of different things that have the same structure. Yes, functors are themselves a category. That is, you can make a brand-new category whose set of things are the functors between two other categories. This is a good spot to pause while the dizziness passes.

There are two kingdoms of functor. You tell them apart by what they do with the morphisms. Here again I’m going to need my categories ‘C’ and ‘D’. I need a morphism for ‘C’. I’ll call that ‘f’. ‘f’ has to match something in the set of ‘C’ to something in the set of ‘C’. Let me call the first something ‘a’, and the second something ‘b’. That’s all right so far? Thank you.

Let me call my functor ‘F’. ‘F’ matches all the elements in ‘C’ to elements in ‘D’. And it matches all the morphisms on the elements in ‘C’ to morphisms on the elmenets in ‘D’. So if I write ‘F(a)’, what I mean is look at the element ‘a’ in the set for ‘C’. Then look at what element in the set for ‘D’ the functor matches with ‘a’. If I write ‘F(b)’, what I mean is look at the element ‘b’ in the set for ‘C’. Then pick out whatever element in the set for ‘D’ gets matched to ‘b’. If I write ‘F(f)’, what I mean is to look at the morphism ‘f’ between elements in ‘C’. Then pick out whatever morphism between elements in ‘D’ that that gets matched with.

Here’s where I’m going with this. Suppose my morphism ‘f’ matches ‘a’ to ‘b’. Does the functor of that morphism, ‘F(f)’, match ‘F(a)’ to ‘F(b)’? Of course, you say, what else could it do? And the answer is: why couldn’t it match ‘F(b)’ to ‘F(a)’?

No, it doesn’t break everything. Not if you’re consistent about swapping the order of the matchings. The normal everyday order, the one you’d thought couldn’t have an alternative, is a “covariant functor”. The crosswise order, this second thought, is a “contravariant functor”. Covariant and contravariant are distinctions that weave through much of mathematics. They particularly appear through tensors and the geometry they imply. In that introduction they tend to be difficult, even mean, creations, since in regular old Euclidean space they don’t mean anything different. They’re different for non-Euclidean spaces, and that’s important and valuable. The covariant versus contravariant difference is easier to grasp here.

Functors work their way into computer science. The avenue here is in functional programming. That’s a method of programming in which instead of the normal long list of commands, you write a single line of code that holds like fourteen “->” symbols that makes the computer stop and catch fire when it encounters a bug. The advantage is that when you have the code debugged it’s quite speedy and memory-efficient. The disadvantage is if you have to alter the function later, it’s easiest to throw everything out and start from scratch, beginning from vacuum-tube-based computing machines. But it works well while it does. You just have to get the hang of it.