Chapter 11: Piercing the Phantom Sword

Youji clapped. His applause was one part perfunctory, three parts sincere. "But Xiang Shan, what does this have to do with what we call 'internal energy'..."

"Youji," Xiang Shan did not answer directly, but instead asked a question in return: "Do you think computers are powerful?"

Youji nodded. Controlling a prosthetic body—in truth, it was mostly the computer doing the heavy lifting. As long as there is a computer, even the most complex construction machinery can function. Youji also knew that computers could calculate things at unimaginable speeds, and the wealthy could use them for even more, like... like...

In any case, they were very powerful.

"Indeed. If we use symbolic operation as a benchmark, the performance of the human biological brain can be compared to a supercomputer—well, in this era, the parameters in my memory probably wouldn't count as a 'supercomputer' anymore. However, the software matched to this hardware is poorly optimized. If we are talking about pure calculation and performance, the biological brain cannot beat a computer."

This is normal. Biological evolution has always followed the principle of "good enough." Evolution does not pursue "ultimate performance." As long as a disadvantageous trait does not prevent an organism from "living long enough to reach reproductive age and bear offspring," it will not be eliminated by natural selection.

Natural humans are full of "idiotic" designs that lead to all sorts of ailments.

But there is one thing that computers cannot compare to the human brain.

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

Youji was a bit confused. "'Computational problems'..."

"Computational problems involve exploring whether all mathematical questions can be solved using the same calculation method. Based on this, a genius named Alan Turing designed a 'Turing machine,' and then... he refuted the human ideal of 'computability.' Not all mathematical problems can be solved by machines."

A Turing machine cannot understand many problems from the very beginning. It is not a matter of "insufficient computational resources to calculate," but rather that "the possibility of even starting the calculation does not exist."

For the simplest example, take parts of geometry—note, "parts of geometry," not "all." In mathematics, concepts like "numbers," "geometry," and "equations" can to some extent be converted into one another.

But from the start, there were many problems that computers could not calculate, or even identify.

In the early days of computer science, a professor sent one of his graduate students to solve the problem of "computer image recognition"—at the time, he optimistically believed that his student could completely conquer this problem in just two months.

But the truth was, it was impossible.

Even into the twenty-first century, "human-eye recognition of CAPTCHAs" remained the standard for certain network programs to determine "whether the user is human."

"Computer image recognition" is a terrifying field of study. Countless scholars around the world have struggled for it, yet programmers still use "human-eye CAPTCHAs" as a means to block malicious machine logins.

The reason why computer image recognition is so difficult is largely because...

"Problems that computers can understand are called 'Polynomial time problems'—or P problems for short. Computers can solve P problems quickly. And more difficult than P problems is Nondeterministic Polynomial time—that is, NP problems."

A large portion of geometric problems lie within NP. A Turing machine can quickly verify whether an answer is correct but cannot quickly provide the answer itself. And there are geometric problems even harder than NP.

"There was a time when mathematicians wanted to prove 'P=NP' to show that all NP problems could be converted into P problems, finding a path to make the computer a god. I still don't remember if it was ever successful..."

When the Turing machine was born, its limits were already set—because it disproved that "mathematics has absolute Turing computability."

The grand ideal of Mr. David Hilbert failed.

***

—If it weren't for the war, perhaps Anatoly might have been able to... what...

—Who is Anatoly again? Why do I know so many inexplicable, powerful figures?

A moment later, the man added with a sense of melancholy: "Probably not. There are things that computers are 'destined to be unable to do.' NP problems are destined to be things that electronic computers are incapable of solving. And NP problems aren't even the limit of complexity."

"Beyond NP problems, there are Polynomial Hierarchy [PH] problems. Beyond the Polynomial Hierarchy, there are Polynomial Space [PSPACE] problems. And beyond Polynomial Space, there exist Exponential Time [EXPTIME] problems."

"In this regard, quantum computers are one dimension stronger than electronic computers. However, the theoretical limit of a quantum computer's capability is called Bounded-error Quantum Polynomial time [BQP]. And the BQP category only includes a portion of PSPACE problems—even a quantum computer cannot reach EXPTIME. This is a realm bordering on the 'Dao'..."

Youji nodded with awe: "Although I don't understand it, it sounds very impressive. Then, Xiang Shan... what is EXPTIME? Can you give an example?"

"Let's use the simplest example." Xiang Shan nodded. "You are using an electronic program and feel that it is running a bit laggy. At this moment, you have to make a choice: do you decide to 'let it keep lagging like this, it will be fine in a moment,' or 'how much longer will I tolerate this before I reboot it'? This 'judgment' is an EXPTIME judgment."

Youji was silent for a moment: "Huh?"

"This 'judgment' is an EXPTIME judgment."

"What?"

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

"That's not right..." Youji doubted whether his biological brain had broken down: "Isn't this... something a child could understand?"

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

"Perhaps this is a limitation; perhaps the human brain cannot construct a computational model more advanced than itself on a mathematical level. Or perhaps it can, but humans just haven't reached that area yet. But I can be certain that humans today still haven't solved this problem—because the martial artist I met today still retained his biological brain."

Xiang Shan took out two strips of tin and used his hand to drill holes in them.

"This is the mystery of internal energy. Use your brain... use your innate wisdom to harness the vast resources brought by computers, forming an 'internal force' to resist external evils."

"Because controlling a prosthetic body uses a Turing machine, recording information uses a Turing machine, and even everything in this world is subordinate to the Turing machine. Therefore, when you can use internal energy to master the Turing machine, you transcend all tools. Those in power will be unable to prevent you from obtaining information, nor can they steal anything from you. They will no longer be able to control you."

"To practice martial arts without practicing internal energy is to end up with nothing. If you focus solely on external martial arts and ignore internal cultivation, then any internal energy expert or hacker can read your memory at will, steal your martial arts algorithms and experience convolutions, or even input false signals into your sensory organs, leaving you to live in illusions for the rest of your life. A martial artist without internal energy, no matter how strong they are physically, is merely an oppressed person. Only by mastering superior internal energy do you have the power to resist."

He inserted the tin strips into an input slot on the machine.

These tin strips were the storage medium. The presence or absence of holes represented the "yes" and "no" of machine language. This was the original soul of a computer.

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

"Now, I will teach you the first lesson. This might be the means for you to fight back after being trapped in an illusion." Xiang Shan pointed to the handle: "Now, turn it."

Youji turned it as instructed. Driven by the gears, the steel needles fell continuously. Some were blocked by the tin, while others passed through the holes, transferring kinetic energy to the gears below. As the gears moved, the tin strip shifted inward.

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

"Ding." Youji felt a sense of disbelief.

***

"It's stuck, Xiang Shan..."

"Very good." Xiang Shan nodded: "The world hasn't disappeared, the machine hasn't disappeared—this is a good sign, child. It shows that I am real, and you are real. We are not inside a simulator."

"Remember this well. This is called the 'Turing Halt,' a sword to shatter illusions."

The Turing halting problem is the question of determining whether any arbitrary program will finish running within a finite amount of time. This problem is equivalent to the following decision problem: Does there exist a program P that, for any input program w, can determine whether w will finish in finite time or enter an infinite loop?

This problem can be simply stated as: "Can one Turing machine simply determine whether another Turing machine should halt?"

And Alan Turing, before the birth of the computer, gave his judgment.

The Turing halting problem is logically unsolvable.

Thus, this provides people with a sword to shatter illusions.

If one can construct a structure mathematically isomorphic to a Turing machine within an illusion generated by a Turing machine's calculation, and then use it to perform a halting problem judgment, that Turing machine would have the capability to exhaust the resources of the matrix and pierce through layers of matrices.

As long as the superior matrix remains a "Turing machine."

The only way for one Turing machine to understand another is to simulate the work of that Turing machine within itself.

The command that can cause a virtual Turing machine to crash can also cause the Turing machine that is virtualizing it to crash.

—Of course, there is also the situation where "what is virtualizing your world is not a Turing machine, but a super-Turing machine." But honestly, a civilization capable of taking a super-Turing machine and turning it into a brain-in-a-vat has already far exceeded human understanding. In such a situation, there is no way to break out within the scope of human cognition, so there is no need to think about it.

Turing machines, the halting problem, the Hilbert program... these are the origins of internal energy.

This is exactly why Xiang Shan would rather temporarily put aside rebuilding his prosthetic body to complete this mysterious ritual.

He cannot be sure right now whether he is a "brain in a vat"—according to his understanding of technology, soaking a brain in nutrient solution and making it believe the surrounding illusions are "real" is not difficult.

So he must verify it.

Although the biological brain is extremely advanced in its mathematical structure, it is also something that is so easily deceived. It cannot compare to man-made objects in terms of "functionality."

How could something that nature blindly cobbled together with the principle of "as long as it works" compete against a means designed by intelligence with the explicit purpose of "deception"?

Only intelligence can counter intelligence.

Upon confirming the "reality" of his surroundings, the world seemed to come alive in Xiang Shan's eyes.

He gently stroked Youji's head: "Good. You are real, and I am real."