[How do programs run without a computer?] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
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How do programs run without a computer?
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Posted by: Roger Pingleton
I am posing this question because, to date, I have not seen a similar question posted.
Wolfram once stated something to the effect that he felt that if all of reality comes into existence from a very simple rule (or rules), that the program for, "life, the universe and everything" (as Douglas Adams might put it) might be only a few lines long.
My question is this, what would be "running" that "program"? And what runs any of the "programs," "rules," what-have-you?
Now some might accuse me of being too literal here, but I think this is an important question to ask. Not just, "can simple rules be used to explain complexity or the mechanics of the universe?" or "what are the rules?", but "what medium does the rule interact with?"
Wolfram talks about simple computer programs, but can a program run without a computer? No.
I can send a hex dump of a program to my printer and set the paper on my desk, but without the CPU, it's worthless.
This "meta" problem plagues me.
Now certainly I understand a model might be best limited to a narrow study, but even in the narrowest application, you need something to "run" the rules or "programs."
This is a sincere question.
Posted by: Jesse Nochella
Hello Roger.
I know the plague you're talking about, and I think there's a good reason for it being there.
I think that we only need to go so far as to say something to the effect of "...which implies that something such as ourselves would think that", where what we would be thinking is "what the hell!?, where do we come from?, what do we do now?"
And I believe the principle of computational equivalence does exactly this. For it implies that such undecidability is inevitable in systems as complex as ourselves.
The one thing I see happen with people who even may know this fact quite well is still the burden of traditional intuition that is in them – that makes the relentless attempt to define a seperation between the qualities of human behavior and the qualities of the external world. From that alone a crucial point that is so often missed is that not only does the PCE imply undecidability in us, but it also implies it in the whole universe as well.
I think it's sort of like the relationship between our conscious selves and the brains that are supposed to account for us. All the science leads to definite underlying structure, but we nevertheless have this constant curiosity of there being more there.
In a way the PCE seems to explain all of this quite well. For it's implications show us that simple rules really can make us in all our complexity, and that there really doesn't need to be something as complex as a god or a big computer behind it all for us to exist. It could be as simple as something like rule 30, but to the same level of satisfaction that would stop us from asking questions altogether I'm sure we would have to refine our models and polish our intuition to decide exactly what rule is for certain the fundamental mechanism at work.
Posted by: Vasily Shirin
Wolfram made a number of very strong statements,
which, IMO, decrease the value of his work.
One of these statements is that Universe is
a computer controlled by some simple program.
What he REALLY proves in NKS is:
there're cases when programs deliver a better
description of natural processes than
differential equations. He provides a number
of pretty convincing examples. There's also
some reasoning in NKS to the effect that
differential equations, except the simplest ones,
can be solved only by numeric methods,
so why to bother considering these equations
in the first place - we could be better off
starting from programs right away. There's certainly a grain of truth in this observation.
If wish he had just stopped here...
Posted by: Jesse Nochella
Vasily, there is no such statement in NKS saying that the universe is a giant computer. What he says is much stronger than that, and offers a kind of solution to the meta problem of which I talked about above.
Doesn't it? Am I missing something?
Posted by: Vasily Shirin
The idea that the whole Universe is a computer is what NKS
is all about. People are computers, too. This statement
was also repeated many times. I don't want to provide
any quotes here, because I'd have to quote the whole
book.
To verify that I was not alone with this interpretation
of NKS, please google "universe as computer", and
follow the very first link.
Posted by: Tony Smith
Roger Pingleton asked:what medium does the rule interact with?
The "medium" is often hypothesided to be a Planck-scale network whose elements are simple nodes (vertices in graph theory) and links (edges) which behave in a way that can be simulated by a simple program a general purpose computer, not because they are seriously purported to exist inside any such computer, but just because that is what they do. On a larger scale, gravitationally interacting bodies do not need some computer somewhere to calculate their conic section paths. Their interactions are intrinsic.
When you actually try to simulate even the simplest network rules on a computer, you face the problem of keeping track of which elementary item is which, a problem which the natural medium does not have to deal with because each electron, etc. carries its own identity with it even while behaving according to the same rules as every other electron.
... and, in the expectation that it was recursive:can a program run without a computer?
But it isn't recursive. Rather, this is all tied up in the idea that the "program" and the local data it considers are intrinsic to everything in the natural universe of space time energy matter. It is hypotheised that our universe will be found to emerge from such a simple enough local rule.
The only place computers come in is in giving us the capability to simulate all manner of simple rules. Wolfram's hope is that by exploring what simple rules do we will gain understanding of the intrinsic natural workings of our universe.
Re the further discussion by Jesse and Vasily, as far as I can recall Wolfram has gone no further than to allow the recently fashionable notion that the universe could be a computer simulation. In a technical sense that could be hard to disprove, save for my keeping-track argumant above. But that is a long way short of claiming the universe is a computer.
Even more fashionably, every new theory gets to be scrutinised in terms of what may be read into it that might relate to the vastly overrated "consciousness problem". Wolfram's principle of computational equivalence insists that consciousness is no more sophisticated than any Class 4 or even Class 3 rule, though at least the Class 3 claim is admitted to be unproven.
All this is built on top of a belief that the universe is built of discrete components which behave deterministically through rules which are computationally irreducible, so the only way to find our where we are going is to go there. There is also an implicit claim that if this is true at the most micro level then it must also be true at all macro levels. Personally, I'm starting to have serious doubts as to whether that implicit claim necessarily holds across the vast range of scales we find in the natural universe, but I don't yet have a counterposition formulated.
Posted by: Vasily Shirin
If behaviour of Universe can be simulated, with 100%
accuracy, by a computer, as Wolfram claims, this is equivalent
to the statement that Universe IS a computer. Why?
BY DEFINITION. These systems are computationally
equivalent, and the whole class of these computationally
equivalent systems is CALLED (!) "computer".
IMO, NKS is the ultimate formulation of materialistic doctrine.
Wolfram just brought it to logical conclusion.
Posted by: Tommaso Bolognesi
The fact that we do need complex pieces of hardware (say a PC)
and of software (say Mathematica)
for running CA rules and simulating some natural behaviour
is indeed misleading, since it may
suggest that the real universe also needs some 'computer'
for running its 'program'; this idea leads to problematic questions,
such as speculating on the nature and origin of that 'computer',
on whether or not it is itself part of the universe...
My understanding of what Tony says (and I support his view)
is that we do not need to postulate the existence of a huge
'computer' animating a huge network of particles:
the only 'computer' is the individual particle itself,
which is intrinsically capable of interacting with its neighbors
according to some simple, universal rule.
All the rest is purely emergent, large scale behaviour,
that does not require any higher-level control.
While the huge computer would be very costly,
the tiny computing particle is very cheap.
Of course, setting up a huge network of cheap computing particles
is still costly.
But it would become cheap again if the whole construction were
carried out by one initial particle, in a big bang (self-replication
being one of the required emergent features) -- if, in other words,
the very creation of the physical substratum (space, matter, and so on)
could be regarded as an emergent feature.
Is it conceivable to look for some experimental evidence
in support of the last scenario by exploring rules
and computations on a (regular) computer?
The apparent contradiction here is that a regular computer
does already provide its own substratum (memory space and computing power),
while our experiment would try to simulate
(also) the very creation of some substratum, without making use
of any substratrum...
Posted by: Jason Cawley
I've moved this thread here because it strikes me as much more of an NKS way of thinking question than a pure investigation of the properties of simple programs question.
I think there is some confusion, or at least potential for it on the part of some readers, between the idea of a computer and the idea of a CPU. A CPU is part of a particular design for practical computers that we call the von Neumann architecture. (It is also, incidentally, not a physical object made out of carefully arranged sand, but anything that performs a certain functional role in such a set up). It is not a necessary part of the abstract nature of computation. It is an allowed and often convenient extra, a functional subsystem distinguished from other aspects of a computation.
Abstractly considered, a computation is a transformation operating on information. Information goes in, information comes out, differently. The difference can depend on the information or on any portion of it. But one set of rules must decide what happens to any variety of information given - that is the underlying regularity that makes it a computation.
(Note, information is not yet computation. Any number can be recorded on an abacus, but an abacus by itself is only a memory, not a computer. A memory is just an arrangement, what is arranged is a matter of indifference, electrons, beads, grains of sand... A set of rules for sliding beads back and forth on an abacus, on the other hand, may constitute a computer. Computers do not need to be made from electronics, that is just very convenient for handling large arrangements of delicate parts in a robust way, with minimal effort.)
In the von Neumann architecture, a portion of the information in the input is considered "instructions" and another portion is considered "data". But they are instantiated in precisely the same way in memory. A definite instruction set operates on the instruction portion of the input to determine the sequence of operations to be performed on the memory portion of the input. (When this division is relaxed, and the results of operations on the memory portion can change the instruction portion of the input during run-time, we speak of “dynamic programming”).
A computer is general purpose if it can be made to implement any finite algorithm by changing what is fed into it. In the von Neumann architecture, by changing only the instruction portion of the input, we get the general purpose computer to perform any computable transformation on the rest of the data.
This exploits a finite version of the principle of universality. It is not necessary to have a different underlying rule for each sort of computation we want to perform. We can instead fix an instruction set (a BIOS for a contemporary computer e.g.), and then arrange sequences composed out of that fixed instruction set so as to mimic the behavior of the overall algorithm we want, for this particular run.
(Notice I say, "a finite version of...". The strict mathematical sense of universality depends on countably infinite sets, and no actual computer has infinite memory or has run for an infinite time. But as we see in practice, real world computation depends on the underlying flexibility of the system and not on its cardinality. Universality in the strict mathematical sense describes what e.g. a Pentium chip might be able to do with infinite memory or running for an infinite run time. When the answer is “any computable algorithm”, we find in practice that we have a general purpose computer. Even though the exact logical conditions of universality have not, sensu stricto, been met).
All that is necessary is for the limited instruction set to have sufficient internal richness to support emulation of an arbitrary computation. That is, the instruction set needs to be past the threshold of universality. Once it is, we do not in principle need to make it any more complicated, nor do we need to alter it to perform a different computation. We can instead leave the instruction set fixed, and modify the data passed to it, to perform a different sequence of those instructions (longer, looped, whatever).
When computation was being discovered, systems were deliberately designed to have universality as a property. But Turing is deservedly famous and so are his machines, because he proved that a system with a very simple underlying structure was already sufficient for this. Previous results in mathematics had already shown that any finite algorithm could be encoded as a problem of arithmetic. And Turing showed that precisely those problems that could be solved there, could be solved by one of his simple machines (idealizations, in fact, of the process whereby human beings did arithmetic).
In Turing’s day, “computer” did not mean “box on a desk that performs general algorithms”. It was the job description of a human being who added, subtracted, multiplied, and divided numbers. We can encode any arrangement as a number, both an input and an ouput. Any computation can then be specified as a certain transformation between numbers.
Now, after NKS, we know that systems that nobody ever designed to be able to perform universal computations can in fact perform universal computations, and that this is true even of remarkably simple systems. So simple they might readily arise physically in everyday systems, all around us. That, Wolfram conjectures, is the underlying cause of apparent complexity, the unifying aspect or marker that we notice as complexity. Some systems have sufficient internal richness that they are as programmable (capable of behaving in different ways when fed different inputs) as any artificial system that was meant to be.
The claim about complexity and the universe is then that its ongoing evolution in time corresponds to a computation of arbitrary sophistication. From prior state to next state there is some simple transformation. But a repeated sequence of simple transformations can do all sorts of complicated things.
Any given configuration of the universe then corresponds to a state of data or “memory”, if one wants to pursue the analogy. With no internal division between an instruction portion and a data portion, and no use made of the von Neumann scheme of functional differences imposed over memory, for the conceptual convenience of a human programmer. What corresponds to the underlying set of possible transformations, (which can then be concatenated into any finite algorithm)? The laws of physics, or rather, any transformation the laws of physics entail.
When Wolfram speaks of looking for the rule that might define the universe, he means finding a functional form that specifies those entailed transformations, in some appropriate, underlying representation of real states. Memory in is state of the universe at time t0. Memory out is state of the universe at time t1. Rule of the universe aka laws of physics are the transformation, r: M(t0) -> M(t1). The trick of course is to find a simple r that can still give rise to the great complexity we see all around us.
I hope this helps.
Posted by: Vasily Shirin
rule M(t0) -> M(t1) implies there's an absolute objective
time. One may argue (and Wolfram does exactly this) that the time measured by our clocks is not the same as real time of
universe, but still, the notion of time, in some or other
interpretation, is a basis for this theory. Which is not
surprising, because the notion of time is embedded in
our definition of computation (e.g., Turing machine employs
both space AND time). This theory
strikes me as being clearly antropocentric. From the fact that
we have an idea of time it eventually jumps to conclusion
that some variant of this idea is a basis of ULTIMATE
definition of Universe. On the same grounds, I can argue
that the notion of beauty (love, good, evil, etc.) is also ingrained in a fabric
of Universe -just because we happen to have these ideas in our
mind, too. Strangely, these notions never show up in Wolfram's
fundamental rules. I don't understand why the notion of
time is better than all those.
Posted by: Jason Cawley
Actually, strictly speaking it implies there is some underlying version of succession, but need not imply that this is invariant from place to place or corresponds to our sense of time in the emergent reality that results. A later time in the emergent sense in a certain frame of reference, may correspond to a whole series of update events having occurred in a whole subset of nodes in some underlying graph, such that locally considered, there are distinguishable predecessor and successor states of the local pattern of connections around whatever constitutes that frame of reference.
No globally synchronized absolute time is required for this. The minimal structure required for a rule-like mapping is succession, but how updating is sequenced is a free parameter to search on, not a prior assumption required by "rule-like-ness". There is deeper nesting of rule application here than there, in abstract data.
I should perhaps also point out that one can link chains of abstract states together with intervening events, or chains of abstract events together with intervening states. s-e-s-e-s-e sequences can be parsed in either "phase", starting from state and going to state s-e-s, or starting from event and going to event, e-s-e. You can have a space of mappings or functions as well as spaces they act on. That is formal tinkering, and just depends on the formal construct one thinks will produce a workable model (whether mathematical or program like). A local sequentially updating programmatic rule on an abstract network, is what Wolfram suggests.
Needless to say, any accurate model of reality must account for our emergent sense of time and the orderings it actually enables us to apply to real events. That is data; not corresponding to data implies any theory fails. But the manner in which the apparent ordering arises, may differ from theory to theory.
Posted by: Vasily Shirin
Jason, I'm sorry, but the idea of "succession" is not enough.
Theory has to predict not only WHAT will happen with the
object, but also WHEN, and you have no choice but using
time parameter to describe WHEN. Which is exactly what
you are doing in YOUR formula M(t0)->M(t1).
If we stick to cellular automation model, this translates
to the problem: which substitution is performed NEXT?
If we say that substitution rule is applied to cell X,
and then substitution rule is applied to cell Y, and
nothing happens in between, we already introduce
the notion of ABSOLUTE time. Succession is enough
only when you deal with a SINGLE cell, but if you consider
ALL cells, you need some notion of time that can be applied
GLOBALLY, and it's absolutely unclear how this global
notion may emerge from local successions. Obviously, the
order of substitutions is important. Wolfram certainly realizes
this problem, but his treatment of it is pretty vague.
I'm surprised that he ventured to take on
this problem at all; so far no one was able to say anything
meaningful about the nature of time, and saying just ANYTHING
about it exposes the person to a slew of very hard questions
for which he has no answers;
this could be easily avoided if NKS took a bit more modest
approach - e.g., just demonstrating that SOMETIMES programs
are as good (or even better) as diff.equations while
describing natural phenomena.
Posted by: Vasily Shirin
BTW, did anyone tried to play with CA driven by internal clock?
What I mean is: classic CA is driven by external clock: all the
rules ate applied simpltaneously to all cells, as if some external
clock generator was connected to each cell.
Instead, rule can be applied only if the state of neigbour changes.
E.g., if state of cell X is a function of states of cells Y and Z,
rule is applied only if either Y or Z (or both) changes its state.
If, as a result, state of X also changes, this recursively triggers
the changes in dependent cells, and so on (if state remains the same,
there's no further propagation). If change of state in
cell X affects several other cells, corresponding rules are applied
simulteneously. This is a natural definition of internal clock.
This process leads to natural definition of global (absolute) time,
as well as local time in each cell (the latter is defined simply
as a number of state changes within some interval of absolute time).
Posted by: JKR
Computers and the universe
Computers are artifacts just to transform input strings to output strings. The individual instructions of a CPU are like simple programs. Applying these instructions in certain sequences gives complex results.
Physics tries to find simple laws. Based on physics we know how atoms forms molecules and molecules forms even more complex structures like DNA. DNA created humans and humans created computers.
The thesis of Wolfram is that the metaphor of simple programs can be applied to the physical universe. The best way to proof that is to build programs that simulate the universe.
So: NKS = Virtual Reality.
Posted by: Vasily Shirin
>> So: NKS = Virtual Reality
I agree with this. But note the irony: NKS is a manifesto of
20-th century materializm. However, this extreme form
of materializm immediately leads to negation of the very
idea of objective reality. I bet this wasn't Wolfram's intention.
Posted by: McQuinn
What about life processes and computation? The transcription of DNA to make proteins seems to me computational, with the “program” encoded in the geometry of chemical structures and their interaction. (Somewhere Wolfram undoubtedly says as much but I haven’t gotten far enough in NKS to cite it.)
For example, an enzyme needn’t consult any external instruction set; its shape and charge distribution constitute both software and hardware. From cell to organism to population, every component simply “knows” what to do. Natural selection over evolutionary time has filtered what works from what does not, serving more as quality assurance than as programmer. This seems to resemble NKS much more than it resembles Von Newman architecture.
While studying bacteriophage as an amateur, I started wondering whether phage and their bacterial hosts could constitute a computer, either engineered or discovered. If we ask the question, can phage and bacteria do computation, then what criteria for computation do we establish in order to design the investigation and evaluate the results? For a computer as we customarily conceive of the concept, we want it to add and subtract, etc. etc. In other words, the evaluation is defined in terms of input-output. At a more fundamental level, we could establish the essential Von Neumann markers for computation and ask, “Do phage demonstrate that they have those capacities?” If they do, then perhaps we could make a simple computer in a Petri dish. But this is trivial, for we would be imposing the performance standard on the system under examination, rather like training a dog to articulate simple one-syllable words.
A much more challenging question is this: how would we recognize computation already conducted by a phage and host population system? By studying NKS I’m hoping to gain some insight into how to ask such a question with more specificity. Bayesian network architecture may be relevant, also. (Not having much of the formal preparation required to readily understand such material, it seems absurd that I’m even interested.)
The quest is really for architectures where the consequences are implicit in the structure, that map well to the biology of phage-host interactions.
If none of this makes sense, my shortcoming. If any of it makes sense, my good luck.
SMcQ
Posted by: Roger Pingleton
Let me add aditional focus to this discussion.
There has been a lot of great discussion, and a lot of great thoughts to ponder, which I am pondering and will continue to ponder. Thank you to everyone who has posted.
But what I want to get back to is, that if we trace back the creation of everything (and this includes the creation of time itself which is part of the universe), we come to a point where everything is made out of a rule (which is what Wolfram postulates).
How does this work given that the rule has nothing to interact with? It is an idea. It is a design, but a design is just a blue print. It needs a medium to work with.
And what's more, as someone has also pointed out, since time is a feature of the universe, how can the rule follow it's course without time?
This is a very bizzarre area of thought. We start with a rule that interacts without a medium (perhaps with itself) and without time and this "rule" creates a world that then has a medium (the building blocks of the universe - sub atomic particals and the like), and it creates a universe that has time.
Does any of this make sense. I find it incredibly thought provoking and hard to fathom.
Perhaps there is a medium already for this rule to work with. Perhaps there is some bizarre notion of time apart from our own that simply alows a rule like this to work, but then, where did those two things come from? What explains them?
Could the universe be recursive? Could it depend on itself; it's own definitions to create a definition of itself?
Posted by: McQuinn
Roger,
There are two customary perspectives on your question, each very different in process and in outcome. There is also an ideosyncratic perspective, which I will get to at the end of this essay.
The psychological perspective (philosophical, if you will) is constrained by how the human mind/brain evolved to conceptualize the world so that we could survive within it. This skill set is fixed and hermetic, with learned responses serving as correction factors to keep our intuition from going astray. Within the psychological domain the questions you raise can only be “answered” through storytelling. Tools like NKS will always function in the psychological arena by analogy, allegory, anecdote and association. You can make up anything you like, whether it’s science fiction or religion or a philosophical code of conduct.
The analytical perspective creates a world apart where problems are tractable using mathematics. Analysis results in constructs that are tested for their “validity” in the “real” world by how well they account for data already observed or data yet to be discovered. The most elaborate analytical construct yet devised is probably string theory, though it has not produced any testable predictions. NKS seems a better candidate for the “theory of everything,” though that remains to be seen.
We have this choice between psychological and analytical modes only if we constantly remind ourselves to keep them distinct. You asked, “How do programs run without a computer?” It seems framed as a psychological query, the concepts of program and computer defined by everyday association. An analytical response to your question can only be evaluated analytically. Nevertheless, we are all free to pick up analytical threads and weave them into stories, like spring robins building nests out of found objects.
Invariably, someone calls for the third way: the holistic view. Then they usually proceed to use narrative or analysis to explain what they mean. Wondering if the universe is recursive or self-defining is a holistic musing, the accepted answer depending on what mode of thought you prefer.
Now, if you don’t require explanations, but rather, you are seeking various expressions of perception, that third, holistic view can be found through Art.
SMcQ
Posted by: Philip Ronald Dutton
I do not think that a program (or more specifically, an algorithm) ever "runs." It just exists. Pretty simple. Computers do things specifically but they are not really executing programs. Computers utilize their own devices (atoms, electrons, maybe later quarks) to simulate the output of the algorithm for which it was programmed. Philosophically speaking, I really do not think my computer is even doing arithmetic when I run Mathematica (if i could afford it). It just is very good at simulating.
You can not prove to me that any computer executes mathematics. Computers just simulate. Why do I say this? Well, the basic arithmetic axioms (algebra or whatever system you prefer) do not even fully describe every thing... they take on a few assumptions.
The program exists and it already has an outcome no matter what the algorithm is. There exists an outcome or a state (infinite loop,etc.). If someone generates an algorithm with X amount of steps and the algorithm was generated in a random fashion such that no one in the history of the universe has ever "executed" it. Well I am sorry but the algorithm has an "answer" somewhere "out there" (in what I call the instant-answer-ether). As soon as you finish writing the last line of your algorithm on paper, the answer exists. You have to work to find it via simulation (arithmetic, computer program, logic, etc.). You have to work to extract it into our physical domain (brain/memory cells and cpu registers included in that domain).
As far as the algorithm is concerned, it never ever needs anything or anyone to perform it in order for the answer to first come into "being."
I would venture to say that a computer or any other "computation" device is only all about the business of simulation. We can't even express mathematics properly when writing out our lovely axioms so obviously these machines are just simulating computation. Mathematics is simulation by default since we have that little nasty axiom bug. A computer doing mathematics is just still doing simulation.
If you think of all the algorithms written by humans, we can easily say that they were not created by the humans but merely FOUND by the human. The algorithms exist and their answers exist without being run. But to bring the answers into our universe's domain we have to figure out how to simulate the algorithm. This discussion brings up lots of side items. My intent is just to provide another perspective on the subject.
Posted by: Val Smith
this surprising image was generated by a simple one line non-recursive function that you yourself could do on graph paper. There are a few very interesting anomalies about this:
1.None of the "Hieroglyphs" were designed by a man or a machine.
2.Restating 1: This is not a FONT.
3.This is a static map, a graph of f(x,y)
not a cellular automaton.
4.By coincidence the function has two constants, one of which is 23. 23 "appears" in the graph, but the other, 13, does not.
5. The appearance of "23" is as strange as if you dropped 23 marbles on the floor and they landed in
the form of the numerals 2 and 3.
6.The glyphs do look like a language don't they?
I see churches, houses, bridges, hearts, crosses,
and little people, as well as the numerals 2 and 3.
This and things like it can come from just a few logic gates, even directly connected to a TV!
NO Memory was required to generate this,
The "color" of any dot can be calculated
with one expression using it's coordinates.
however since it's on the internet, it had to get there through a computer.
This makes me wonder if there is a function to generate the bible, since this looks like
"information from no where", more interesting
to look at than digits of pi.
I don't remember the function except it used
Exclusive-OR, AND, and the numbers 13 and 23,
coordinates X,Y, and it was a "one-liner".
Posted by: Val Smith
Credibility is hard to come by so I had to find it!
Notes: This is a PC-DOS QBASIC notation.
The boolean functions operate on all bits of the numbers.
First (this makes it look nicer)
LET A=X+Y:B=ABS(X-Y):REM ROTATE 45'
Then
C=(((A XOR B)MOD 23)AND((A AND B)MOD 13)AND 8)/8
C is 1 if there is a dot in a glyph at X,Y otherwise it's 0
(So graph for all positive integers x and y)
MOD means the remainder of an integer long division
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