The Sword That Pierces Illusions

Youji applauded. There was three parts perfunctoryness and three parts sincerity in his applause. "But Xiang Shan, what does this have to do with the 'internal cultivation' we were talking about...?"

"Youji." Xiang Shan did not answer directly, instead posing a question. "Do you think computers are powerful?"

Youji nodded. The control prosthetics were, for the most part, powered by computers. As long as there was a computer, even the most complicated engineering machinery could operate.

Youji also knew that computers could calculate things at incredible speeds, and that wealthy people could use them for even more things, like... like...

Anyway, they were incredibly powerful.

"Indeed. If symbolic operations are the benchmark, the computing resources of a human biological brain can rival a supercomputer's—well, in this era, the specifications I remember probably no longer qualify as 'supercomputers.' But the software suited to this hardware is terribly optimized. If it is purely a contest of computation and performance, biological brains cannot compare to computers."

This was perfectly normal. Biological evolution had always followed the principle of "good enough." Evolution would never pursue the absolute limit of performance. As long as a detrimental trait did not prevent someone from "living to reproductive age and having children," natural selection would not eliminate it.

Natural humans were riddled with idiotic designs that caused all kinds of illnesses and injuries.

Yet in one respect alone, computers could not compare to the human brain.

"At least in the era I know, computers still could not escape the limitations of 'computability problems.'"

Youji was somewhat bewildered. "'Computability problems'...?"

"Computability problems explore whether all mathematical problems can be solved through a single method of computation. On that foundation, a genius named Alan Turing designed the 'Turing Machine,' and then... he disproved humanity's ideal of 'computability.' Not every mathematical problem can be solved by machines."

From the very beginning, Turing Machines could not understand many problems. It was not that "there were insufficient computational resources, so they could not calculate them," but that "the possibility of even beginning the calculation did not exist."

Take certain geometric problems, for example—note, "certain geometric problems," not "all of them." In mathematics, concepts such as "numbers," "geometry," and "equations" could, to a certain extent, be converted into one another.

Yet from the outset, there were many problems that computers could not calculate or even recognize.

In the early days of computers, a professor assigned one of his graduate students to solve the problem of "computer image recognition." At the time, he optimistically believed that the student would completely conquer the problem in just two months.

But in truth, that was impossible.

Even in the twenty-first century, "visual CAPTCHA recognition" was still used by certain online programs to determine whether a person logging in was human.

"Computer image recognition" was a terrifying field. Countless scholars across the world had struggled over it, yet programmers still used "visual CAPTCHA recognition" to stop malicious machine logins.

Computer image recognition was so difficult largely because...

"The problems computers can understand are called P problems, or Polynomial time problems. Computers can solve P problems quickly. More difficult than P problems are NP problems—Nondeterministic Polynomial time problems."

A great many geometric problems fell within NP. A Turing Machine could quickly verify whether an answer was correct, but it could not quickly produce that answer. Some geometric problems were even more difficult than NP.

"Mathematicians once wanted to prove that 'P = NP,' proving that all NP problems could be transformed into P problems, taking the first step toward allowing computers to enter the realm of gods. I still cannot remember whether they succeeded..."

The limits of the Turing Machine had been set the moment it was born—because it disproved the notion that "mathematics possesses absolute Turing computability."

Mr. David Hilbert's great ideal had failed.

—If it had not been for the war, perhaps Anatoly might have been able to accomplish... something...

—And who is Anatoly? How do I know so many ridiculously impressive people?

After a moment, the man added dejectedly, "Probably not. There are things computers are 'fated to be unable to do.' NP problems are things electronic computers are fated to be incapable of solving. And NP problems are not even the pinnacle of complexity."

"Beyond NP problems are problems in the polynomial hierarchy [PH]. Beyond the polynomial hierarchy are PSPACE problems. Beyond polynomial space are EXPTIME problems."

"In this respect, quantum computers are one dimension stronger than electronic computers. But the theoretical limit of quantum computers is called bounded-error quantum polynomial time [BQP]. And the BQP category includes only some PSPACE problems—even quantum computers cannot touch EXPTIME. That is a realm almost akin to the Dao..."

Youji nodded with an awestruck expression. "I do not understand any of it, but it sounds amazing. Then, Xiang Shan... what is EXPTIME? Could you give an example?"

"Let us use the simplest example." Xiang Shan nodded. "You are using an electronic program and feel that it is running a little sluggishly. At that moment, you must make a choice: decide whether 'I should let it keep lagging along, and it will be fine after a while,' or 'How much longer should I endure before restarting it?' That 'judgment' is an EXPTIME judgment."

Youji fell silent for a moment. "Huh?"

"That 'judgment' is an EXPTIME judgment."

"What?"

"That 'judgment' really is an EXPTIME judgment."

"That cannot be right..." Youji deeply doubted whether his biological brain had malfunctioned. "Is that not... something even a child can understand?"

"From a mathematical perspective, the difficulty level of this problem is EXPTIME. A biological brain can make such a judgment easily, but computers cannot even 'understand' it. Among known computational models, there is only one that can reach this domain..." Xiang Shan pointed to his own head. "The human biological brain."

"Perhaps this is a limitation. Perhaps human brains cannot construct computational models more advanced than themselves on a mathematical level? Or perhaps they can, but humanity has not yet reached that domain? But I can be certain that humans today still have not solved this problem—because the martial artists I encountered today still retained biological brains."

Xiang Shan took out two strips of sheet metal and drilled holes into them by hand.

"This is the mystery of internal cultivation. Use your brain... use the wisdom you were born with, to command the vast resources brought by computers and form the 'internal force' that resists external evils."

"Because controlling prosthetics uses Turing Machines, recording intelligence uses Turing Machines, and even everything in this world belongs to Turing Machines. Therefore, when you can use internal force to command Turing Machines, you stand above all tools. The ruling class will be unable to stop you from obtaining information, nor can they steal anything from you. They can no longer control you."

"Practice martial arts without cultivating internal force, and all your efforts will amount to nothing. If you focus on training external martial arts while neglecting internal techniques, then any master of internal cultivation, any hacker, can freely read your memories, take away your martial arts algorithms and experiential convolutions, or even feed false signals into your sensory organs, making you live in illusions for your entire life. A martial artist without internal cultivation, no matter how strong he becomes, is nothing more than one of the oppressed. Only after mastering superior internal cultivation will you possess the strength to resist."

He inserted the metal strip into one of the machine's input slots.

This metal strip was storage. The presence or absence of holes represented the machine language of "yes" and "no." This was the computer's original soul.

Even without electronic components, it possessed every function a computer required.

"Now, I will teach you your first lesson. This may be the means by which you strike back when trapped in an illusion." Xiang Shan pointed at the crank. "Now, turn it."

Youji turned it as instructed. Driven by the gears, steel pins continuously fell. Some were blocked by the metal strip, while others passed through its holes and transferred kinetic energy to the gears below. As the gears moved, the metal strip was drawn inward.

Xiang Shan stared at the exposed mechanism, observing its operation.

With a ding, Youji looked incredulous.

"It is stuck, Xiang Shan..."

"Very good." Xiang Shan nodded. "The world has not vanished, and the machine has not vanished—this is a good sign, child. It means I am real, and you are real. We are not inside a simulator."

"Remember this. It is called Turing halting, the sword that cuts through illusions."

The Turing halting problem was the problem of determining whether any given program could finish running within a finite amount of time. It was equivalent to the following decision problem: whether there existed a program P that, for any input program w, could determine whether w would finish within a finite time or enter an infinite loop.

This problem could be simply stated as, "Can one Turing Machine easily determine whether another Turing Machine should halt?"

And Alan Turing had given his answer before computers were even born.

The Turing halting problem was logically unsolvable.

And so, it gave people a sword to cut through illusory realms.

If, within an illusory realm generated by a Turing Machine's computations, one could construct a structure mathematically isomorphic to that Turing Machine, then use it to determine the halting problem, that Turing Machine would gain the ability to exhaust the Matrix's resources and pierce through layer upon layer of Matrix.

So long as the higher-level Matrix was still a "Turing Machine."

The only way for one Turing Machine to understand another was to simulate that Turing Machine's operation within itself.

An instruction that could crash a virtual Turing Machine could also crash the Turing Machine simulating it.

—Of course, there is also the possibility that what simulates your world is not a Turing Machine, but a hyper-Turing machine. But honestly, any civilization capable of using a hyper-Turing machine to create a brain in a vat would already be far beyond human comprehension. In such a situation, there is fundamentally no method of escape within the scope of human cognition, so there is no need to think about it.

Turing Machines, the Turing halting problem, Hilbert's program... these were all the origin of internal cultivation.

That was precisely why Xiang Shan would rather temporarily put aside rebuilding his prosthetic body and complete this mysterious ritual first.

He could not be certain that he was not a "brain in a vat"—according to his understanding of technology, immersing a brain in nutrient fluid and making it mistake the surrounding illusion for "reality" was not difficult at all.

So he had to verify it.

Though a biological brain possessed an extraordinarily advanced mathematical structure, it was still something so easily deceived. In terms of "functionality," it could not compare to artificial creations.

How could something nature had blindly cobbled together according to the principle of "good enough" defeat methods intelligence had designed for the purpose of "deception"?

Only intelligence could contend with intelligence.

After confirming the "reality" around him, the world seemed to come alive in Xiang Shan's eyes.

He gently stroked Youji's head. "Very good. You are real, and so am I."