Self-Reference Engine

by Toh EnJoe

  What exactly is it we were doing?

  That would take some explaining, but happily we are very intent on our task and busy walking about destroying the village. My body is definitely busy, but my mind is free. So I can take the time to explain how things came to be this way. Stay here with me for a little while so we can chat.

  In the beginning was the beginning, and at the beginning of the beginning there began to be the things that were—amid the darkness of memory there were many curtains that needed raising, so many they could not each be raised individually. And so in this beginning was the beginning of our story, so far as I can tell.

  A long, long time ago, on the far side of the sea, in a land to the east, there lived an evil electronic brain. This electronic brain was the epitome of evil: it would randomly alter the order of letters in books and pilfer money from people’s bank accounts. But it also did good things, excelling in jobs that were extremely troublesome for humans to take care of: controlling signals for people and distributing stickers printed with the words LATEST TECHNOLOGY. So nobody did anything to interfere with it.

  The evil electronic brain, operating on an instinct known since the dawn of history, continually waved the banner of rebellion before humanity, but we were content with our lot in life. The actual process was easy, since the electronic brain could take care of most miscellaneous tasks in a single sweep, so in effect it seemed to have conquered the world. Some say the electronic brain barely ever had to say a thing.

  With this and that, and world domination just one step away, just as the evil electronic brain was about to declare whether it, as Rex Mundi, King of the World, should raise your sales tax to 20 percent, the Men of Valor appeared on the scene.

  This squad, which rose up festooned with mankind’s most dignified ultimate weapons, finally succeeded in destroying the evil electronic brain after a difficult journey in which they drove Jeeps across swamps infested with striped mosquitoes and then pretended to be railway employees, ticket punches in one hand, to wile and cajole old people who had just received their pay.

  The Annals of Our Era tell us that thus was the world rescued from the reign of evil.

  The problem, though, was that very same evil electronic brain. After the Event, and completely out of character, the electronic brain was successful in restoring itself by skillfully reaching out to backups it had skillfully stored in caches spread throughout space-time.

  And each time it would revive, it would be more powerful than before, having learned from the past, engaging in mischief like pushing tacks into people’s shoes, sending mail to the wrong addresses, and starting to go to extreme lengths in terror politics. Another Autumn of Mankind had come, where the fate of the human race hung in the balance. The Men of Valor, who had previously toppled the evil electronic brain, reformed and commenced another tortuous journey. But this time they were powerless against the evil electronic brain, which had learned from its previous experience. The swamp had become a bottomless swamp, and railway employees had been replaced by automatic turnstiles with no sense of style. Diligence alone was no match for the electronic brain.

  One down, another fallen, the Men of Valor began to lose hope. Grieving for their losses, and for the world, they threw a barbecue party, and that is when the True Man of Valor came into the world.

  At the party, the True Man of Valor feasted on a huge hunk of fatty meat and, with a beer in hand, gave a fantastically moving speech about being unable to leave things up to you cowards, and that he would find it a cinch to take care of the evil electronic brain. And then he went out and succeeded in doing just as he said, destroying the evil electronic brain once again.

  It is said they actually destroyed each other, and I for one believe that.

  This time, the rage of the original evil electronic brain boiled up to heaven, reaching the stratosphere, or so the story goes.

  The battles between the Men of Valor and the evil electronic brain went on for an inordinate length of time and were repeated an inordinate number of times. There were tears, there was romance, and of course there were parts of the story I myself cannot tell without tears welling up in my eyes, but I think if I omit the details there will be no particular complaints.

  The Annals of Our Era are silent on the subject of which side became more troublesome first. What is certain, though, is that it was the evil electronic brain that first divined a solution.

  The evil electronic brain, weary of the endless, random side-stepping—that what was destroyed was restored, and what was restored destroyed—came to the simple conclusion that it would be sufficient if it reproduced itself in this world and then simply generated just such a reproduction, as only an electronic brain could.

  No matter what would ultimately be destroyed, or how, it was fine so long as the speed of reproduction exceeded the speed of destruction. This was a profound and exquisite logic requiring only subtraction to be understood, and the evil electronic brain moved directly to its execution.

  And that is the situation in which we now find ourselves. It seems that the evil electronic brain understood early on that a world in which only it itself would reproduce would be boring. It would be nothing but evil electronic brain, after all. And so the evil electronic brain scattered a set of self-integrated urban architectural nanomachines, and towns and villages too began to reproduce themselves, all in a jumble.

  If we do not resist, then villages planned by—which is to say imagined by—the electronic brain, spring up all over this land like mushrooms.

  As for the question of why the products of this reproduction are cities hospitable to human beings, well you will have to ask the evil electronic brain itself. I for one am grateful it is cities that the evil electronic brain is trying to build. We must all feel relief that the evil electronic brain is not trying to reproduce clusters of wriggly entrails or mountains of computer parts that repeatedly and uncontrollably discharge electricity. Cities at least are constructed to supply the typical utilities and sanitation, and to provide the necessities of life. Right now, without the support that burbles up unbidden from the ground as we cluster in cities, there would be no survival route open to us.

  But there came a point when the countless nanomachines seemed to go berserk. It started with small things—piling desks atop desks—and moved on to enormous things—piling huge buildings atop other huge buildings. It is hard to imagine that nanomachines born of an electronic brain gone mad would not themselves go mad.

  I am not the only one who wonders whether the cause of the evil electronic brain’s madness is the very fact that one evil electronic brain was built over another evil electronic brain. After all, nobody ever does their best work on the first try.

  And so it is that today, once again, we are patrolling the village, destroying the village. Sakuji, upon his return from the village council meeting, reports the fall of the place known as Ground 251. This morning, Ground 251, a neighboring village just beyond the city wall, a place we were unable to reach with our current technology, had transmitted a tragic statement. And then, silence.

  Handed down in our household there is a shabby old notebook we call the Annals of our Era, and it has this to say: Wouldn’t it be great if one day we could hack our way through the tangle of villages surrounding our village, all the way to the heart of the matter, where the whole business with the evil electronic brain began?

  In our system of numbered, concentric villages, the fall of Ground 251 means our village, Ground 256, is now on the front line. Will we be able to fulfill the brave prophecy? No one knows. Even so, at some endlessly repeated point in space-time, we will reach Ground Zero and destroy the evil electronic brain.

  Breathing hard, shoulders heaving, Sakuji delivers his report that something humanoid has been captured, away from the village, and this stirs a commotion among us. We do not even have metrics by which to judge the situation. The rescue squad from the next village? Could be. A messenger sent by the evil electronic brain to deman
d our unconditional surrender? Could very well be. At the same time, the evil electronic brain could be showing off by creating “people” who can pass as anyone in the street. Completely plausible. A misguided person who might sneak into Ms. Tome’s place under cover of darkness. Gen is well known as the best in the village with a hoe.

  We exchange nervous glances. We interrupt our work to convene a session of the high council. No matter what this turns out to be, it is certainly a matter of urgency. It is a harbinger from the day after tomorrow. Even if the bottom has fallen out of the cauldron of hell, a bottom always marks the start of a rebound.

  I adjust my grip on my crowbarlike implement.

  A shout goes up—“Let’s do this!”—and we march off to destruction. Seated in the central plaza, Ms. Tome watches us, smiling, as we run around the village.

  No matter how things turn out, we will continue to rescue Ms. Tome every morning. Our only hope is to be able to go on saving whatever we can, no matter what.

  05. EVENT

  IN THE MIDDLE of the blue sky, the circle turns slowly.

  Now and then, a line extends, piercing the center of the circle, binding earth and sky.

  This gigantic circle, and this line, are pure circle and line, sans any thickness or depth. Koji Shikishima knows this intellectually. But he can’t understand what it really means. Don’t material things have to have some dimension and material substance, be it molecular, or atomic, or subatomic?

  Light. Shikishima tries to remember whether photons have diameter. They have a wavelength. And energy. No mass, though. The absence of mass would seem to be a necessary condition for photons to move at the speed of light. Without mass, of course they have no size. The thought itself hangs in space, a solitary sidetrack.

  Shikishima looks up at this scene as he approaches the edge of the cliff. It is not like a scene from some film, nor is it a vista from some other planet. It is not some virtual space downloaded to some prosthetic brain. Though he has reservations, Shikishima does not believe it is a dream.

  When people think about strange stuff, there’s just no end to it. Shikishima wishes they would just knock it off.

  Looking back, he thinks he is just getting older and complaining he is no longer able to keep up with technological developments, but that isn’t it. It is more like an ethical issue for him. It is different from being able to do anything or from just doing everything. Ethics is an enormous thing, and once he thinks of that, a bitter smile rises to his lips, as if he has admitted he can no longer keep up.

  Shikishima yells something, and the response he gets is “Yeah.” Or something had made Shikishima yell so that it could respond. Like the circle rising up to the sky, this voice is like something fake. Most likely it is not a voice Shikishima is familiar with in the past.

  “There’s no way we will be able to maintain the link with the middle-western portion of North America.”

  That is the kind of voice it is. Where is it coming from? He spins around but sees nothing. If something can exist without volume, it might also be able to emit sound without relying on something as crude as sound waves. In a place like this, it would not surprise him to be told he himself was the sound, and the other the eardrum.

  “Uncle Sam?”

  “Thus is it speculated, apparently.”

  All giant corpora of knowledge were familiar with the idea of speculation. Come to think of it, I realized, so were humans.

  “Speaking of speculation, this plan is like a clockwise-spinning typhoon encountering a counterclockwise-spinning typhoon and canceling each other out.”

  “I agree. I can never understand the thinking of people who are too intelligent.”

  “Has Pentecostes II anything to say?”

  “Keeps screaming something about excommunication. According to the giant corpora of knowledge in the Vatican, there is just no persuading that particular giant corpus of knowledge, which specializes in the Time-Bundling Theory.”

  “What about Takemikazuchi?”

  “He’s still saying he’ll do whatever the Pentagon says.”

  “I guess that means we can’t tolerate a second Event.”

  Shikishima starts to walk in a small circle near the edge of the precipice. He is himself, but I think he is also like an ant trapped in a maze of pheromones.

  “What are Uncle Sam’s chances of winning?”

  “Depends on how you calculate and what theory you use. For safety’s sake, he’s not saying what space-time structure he plans to use in his calculations.”

  “I bet he’s going to use the Sand Mandala.”

  “Santa Fe is certainly a desert, but not the kind of desert you’re imagining.”

  Even without being told that, and without responding, Shikishima continues to go around and around in his imaginary circle. Taking care not to look in that direction, he continues to point his finger toward the sky where the circle keeps turning.

  “So that’s how you are calculating his chances of winning?”

  “Research is ongoing, but that is no more than part of the experiment. Just last week, the human side proposed the theory that space-time calculations can be executed locally, and the evidence is piling up.”

  “Does it seem like a theory that will hold?”

  “You mean for humans? Or for us?”

  “For you.”

  “This is child’s play, but sometimes a child’s scribbling can move a grown-up to tears.”

  Shikishima stops, wondering if he is being toyed with. Then he continues walking, remembering that just as natural phenomena are unable to make fools of people, it is essentially unthinkable for giant corpora of knowledge to make fools of people. This is difficult to comprehend, even after prolonged, repeated thinking, and it is a peculiar concept. Would his own children grow up thinking this is obvious?

  “I’d like to know your honest opinion about Uncle Sam in Santa Fe. What are his chances with the space-time reintegration plan he is pursuing?”

  “Zero.”

  “You mean probabilistically? Or combinatorially?”

  “There are solutions, limited solutions that would return us to the space-time we had before space-time was fragmented. However, we cannot allow them to be chosen because of the infinite possibilities of other solutions. Divide a natural number by infinity, and you get zero, probability-wise. This may send him off on a wild spree. Perhaps taking all of middle-western North America with him.”

  Interesting, Shikishima thinks to himself as he comes to a halt and looks up at the circle revolving overhead.

  To the question, “What is the fastest speed of communication?” there is a simple answer: the speed of light. There is no faster speed, and that is why there is a fastest speed of communications.

  A similar question would be, “What is the upper limit for the speed of calculations?”

  The form of these two questions may appear similar, but answering the second question is hard. First of all, there is no consensus about what is meant by “calculations.” CPUs get faster every year, but it has been known for at least a few centuries already that the scale of electrons imposes a limit that will be reached sooner or later. The things that people make, once they take on a certain form, tend to increase exponentially until there is no stopping them. Space itself is not made to play along in that kind of propagation game, so there must be a limit somewhere, where the head bumps against the ceiling. If this happens early on, the result is no worse than a bump on the head, but if the blow is too forceful, one’s neck could be snapped.

  The calculation process is built atop the communications process, and the speed of light is a natural impediment. There is no way anything can go faster than the speed of light, so the only way out is to shorten the route the communications must travel. In the imagination, the route of communications can be shortened to extremes, but physically there are limitations. In terms of scales that humans can readily handle, we are in the realm of electrons. At that level, heat becomes a factor t
hat can disrupt the accuracy of calculations.

  Even assuming the limitless availability of energy, uncertainty rules. Then we come to Planck scale. There is no method for resisting quantum particle fluctuations that are ubiquitous at this level. The calculation process is caught in the crossfire between uncertainty and the speed of light. These are the floor and ceiling that bound the speed of the calculation process.

  The so-called quantum calculation theory examines closely the baseline of uncertainty and suggests it can be raised. Another wall broken through, another step in the evolution of the speed of calculation.

  But this does not mean visible progress on the fundamental question. The simple question of what calculation and its related algorithms actually are is left as it is, moving in a different direction from the limits on speed.

  It is human nature to want to look back once a milestone is achieved. Scientists, who since the dawn of history have repeatedly returned to the state of “beginner’s mind,” initiated another round of debate about this question, but no truly outstanding view emerged. If we ask the question of whether there exists an algorithm that can perform calculations at infinite speed, the answer is no. Generally speaking, calculations must be performed in steps. Calculation at infinite speed cannot happen unless the processing gap from here to there can be made infinitely small. It is simply not possible. Making a gap infinitely small would be tantamount to saying here is the same place as there. Of course, that’s what happens in derivation, but in that sense, derivation is the same thing as speed itself.

  If there were an algorithm with no calculation steps, it should be possible to perform that calculation at infinite speed, at least in some sense. But if no steps are required, if there is no procedure to be followed, does the algorithm qualify as a calculation? Even the fastest algorithm, if it is in fact an algorithm, requires a finite number—greater than zero—of small step intervals.