Youji applauded, three parts perfunctory and three parts sincere. "But Xiang Shan, what does this have to do with the 'Inner Power' we were discussing...?"
"Youji," Xiang Shan did not answer directly, instead asking a question. "Do you think computers are powerful?"
Youji nodded. Controlling a prosthetic body was mostly the work of computers. As long as there was a computer, even highly complex engineering machinery could operate.
Youji also knew that computers could calculate things at unimaginable speeds. Rich people could use them for even more things, such as... such as...
Anyway, they were incredibly impressive.
"Indeed. If symbolic computation is the benchmark, the computing resources of the human Biological Brain can compare to a supercomputer—well, in this era, the figures I remember probably no longer qualify as 'supercomputing.' But the software matched to this hardware is terribly optimized. If it is purely a contest of computation and performance, the Biological Brain cannot compete with computers."
That was perfectly normal. Biological evolution had always followed the principle of "good enough." Evolution never pursued "ultimate performance." As long as a disadvantageous trait did not prevent one from living to reproductive age and having children, natural selection would not eliminate it.
Natural humans possessed a whole pile of idiotic designs that could cause all kinds of illnesses and injuries.
But there was one thing computers could not compare to the human brain in.
"At least in the era I knew, computers still could not escape the limits of 'computational problems.'"
Youji was somewhat bewildered. "'Computational problems'...?"
"Computational problems explore whether all mathematical problems can be solved through one and the same computational method. On that foundation, a genius named Alan Turing designed the 'Turing Machine,' and then... he denied humanity's ideal of Turing 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 "they lacked sufficient computing resources and therefore could not calculate them," but that "the possibility of even beginning the calculation did not exist."
The simplest example was certain geometric problems—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 start, there had been many problems computers could not calculate or even recognize.
In the early days of computers, a professor assigned one of his graduate students the problem of "computer image recognition." At the time, he optimistically believed that his student could completely conquer the problem in just two months.
But in truth, that was impossible.
Even in the twenty-first century, "visual CAPTCHA recognition" remained a standard by which some online programs determined whether a person logging in was human.
"Computer image recognition" was a terrifying field. Countless scholars around the world had struggled with 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 quickly solve P problems. More difficult than P problems are NP problems, or Nondeterministic Polynomial time problems."
A large portion of geometric problems fell within NP. A Turing Machine could quickly verify whether an answer was correct, but could not quickly produce the answer. Some geometric problems were even harder than NP.
"Mathematicians once wanted to prove 'P=NP,' proving that all NP problems could be transformed into P problems, taking the first step toward bringing computers into the realm of gods. I still do not remember whether they succeeded..."
The limits of the Turing Machine had been set the moment it was born—because it disproved the idea that "mathematics possessed absolute Turing computability."
Mr. David Hilbert's great ideal had failed.
—If not for the war, perhaps Anatoly could have accomplished... something...
—And who was Anatoly? How did I know so many inexplicably formidable people?
After a moment, the man added desolately, "Probably not. There are things computers are 'destined to be unable to do.' NP problems are things electronic computers are destined to be incapable of solving. And NP problems are not even the utmost limit of complexity."
"Beyond NP problems are polynomial hierarchy problems [PH]. Beyond polynomial hierarchy problems are PSPACE problems. Beyond polynomial space lie EXPTIME problems."
"In this regard, quantum computers are a whole 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 reach EXPTIME. That is a realm nearly approaching the Dao..."
Youji nodded with reverence. "I do not understand any of it, but it sounds amazing. Then, Xiang Shan... what is EXPTIME? Could you give me an example?"
"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 in 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' truly is an EXPTIME judgment."
"That cannot be right..." Youji suspected that his Biological Brain had malfunctioned. "Is this not... something even children can understand?"
"From a mathematical perspective, this problem belongs to the EXPTIME complexity class. A Biological Brain can make this judgment easily, but computers cannot even 'understand' it. Among known computational models, there is only one capable of reaching this domain..." Xiang Shan pointed at his own head. "The human Biological Brain."
"Perhaps this is a limitation: perhaps the human brain cannot construct a computational model mathematically more advanced than itself? Or perhaps it 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 retain their Biological Brains."
Xiang Shan took out two strips of sheet metal and drilled holes into them by hand.
"This is the mystery of Inner Power. Use your brain... use your innate wisdom to command the vast resources brought by computers and form Inner Power that resists external evils."
"Because prosthetic bodies are controlled through Turing Machines, information is recorded through Turing Machines, and even everything in this world belongs to Turing Machines. Therefore, when you can use Inner Power to command Turing Machines, you stand above all tools. Those in power 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 practicing Inner Power, and all your efforts will come to nothing. If you focus on cultivating Outer-School Martial Arts while neglecting Inner-School Techniques, then any master of Inner Power, any hacker, can freely read your memories, seize your Martial Arts Algorithm and accumulated experience, or even feed false signals into your sensory organs, making you live in hallucinations for your entire life. A martial artist without Inner Power, no matter how strong he becomes, is only one of the oppressed. Only by mastering superior Inner Power can you gain the strength to resist."
He inserted the strips of sheet metal into one of the machine's input slots.
The metal strips were storage devices. 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 handle. "Now, turn it."
Youji did as told and turned it. Driven by the gears, steel pins continuously fell. Some were blocked by the sheet metal, while others passed through its holes and transferred kinetic energy to the gears below. The sheet metal moved inward with the rotation of the gears.
Xiang Shan stared at the exposed machinery and watched it operate.
With a ding, Youji looked somewhat incredulous.
"It is stuck, Xiang Shan..."
"Very good." Xiang Shan nodded. "The world has not disappeared, and the machine has not disappeared—this is a good sign, child. It means I am real, and you are real. We are not inside a simulator."
"Remember this. This 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 terminate within a finite time or enter an infinite loop.
This problem could be simply expressed as: "Can one Turing Machine easily determine whether another Turing Machine should halt?"
And before computers were even born, Alan Turing had already given his answer.
The Turing Halting Problem was logically unsolvable.
And thus, it gave people a sword to cut through illusions.
If one could construct, within an illusory realm generated by a Turing Machine's calculations, a structure mathematically isomorphic to that Turing Machine, then use it to determine the Halting Problem, that Turing Machine would possess the ability to exhaust the Matrix's resources and break through layer upon layer of Matrix.
So long as the higher-level Matrix remained a "Turing Machine."
The only way for one Turing Machine to understand another Turing Machine was to simulate that Turing Machine's workings within itself.
An instruction that could crash a virtual Turing Machine could also crash the Turing Machine virtualizing it.
—Of course, there was also the possibility that "the thing virtualizing the world you exist in is not a Turing Machine, but a hyper-Turing machine." But honestly, a civilization capable of using a hyper-Turing machine to create a brain-in-a-vat scenario would already be far beyond human comprehension. Under those circumstances, there was simply no method of cracking it within the scope of human knowledge, so there was no need to think about it.
Turing Machines, the Turing Halting Problem, Hilbert's program... these were the origins of Inner Power.
That was precisely why Xiang Shan would rather temporarily set aside rebuilding his prosthetic body and complete this mysterious ritual first.
He could not yet be sure whether he was a "brain in a vat"—according to his understanding of technology, it was not difficult to soak a brain in nutrient solution and make him believe that the surrounding illusion was "real."
So he had to verify it.
Though the Biological Brain possessed an extraordinarily advanced mathematical structure, it was also something easily deceived. In terms of "functionality," it could not compare to artificial creations.
How could something nature had haphazardly fashioned according to the principle of "good enough" possibly contend with methods intelligence had designed for the purpose of "deception"?
Only intelligence could resist intelligence.
After confirming the "reality" around him, the world seemed to come alive in Xiang Shan's eyes.
He gently rubbed Youji's head. "Very good. You are real, and so am I."