Legend of the Cyber Heroes

Yuki clapped her hands. The applause contained a mix of formality and sincerity, "But Xiang Shan, what does this have to do with the 'Inner Strength' we talked about..."

"Yuki," Xiang Shan didn't answer directly, instead, he asked a question, "Do you think, the computer, is a powerful thing?"

Yuki nodded. Controlling prosthetic bodies. Most of the effort is actually from the computer. As long as there's a computer, even complex engineering machinery can operate.

Yuki also knew computers could calculate things at an incredible speed, and rich people could use them to do even more things, such as... such as...

Anyway, they are really powerful.

"Indeed. The performance of the human biological brain, if benchmarked by symbolic operations, can compare with that of a supercomputer—well, in this era, the parameters in my memory probably can't be considered 'supercomputing' anymore. But the software that matches this hardware, the optimization is just too poor. If we simply compete in computing and seek performance, the biological brain is no match for computers."

This is quite normal. Biological evolution always follows the principle of "good enough is fine." Evolution is not about pursuing "extreme performance." A disadvantageous trait will not be eliminated by natural selection unless it affects the ability to "live to reproductive age and have children."

Natural Persons carry a multitude of designs that can lead to various diseases and intelligence deficits.

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

"At least in the era, I'm aware of, computers still couldn't break free from the confines of 'computational problems.'"

Yuki was a bit confused, "'Computational problems'...?"

"Computational problems refer to the exploration of whether all mathematical questions can be solved by one computational method. On this basis, a genius named Alan Turing designed the 'Turing Machine,' and then... he negated humanity's ideal of 'computability.' Not all mathematical problems can be solved by machines."

From the start, the Turing Machine couldn't understand many problems. It's not a matter of 'insufficient computational resources, unable to calculate' but of 'no possibility to even start calculating.'

The simplest examples are parts of geometry—notice, 'parts of geometry,' not 'all.' In mathematics, concepts like 'numbers,' 'geometry,' 'equations,' can be converted to some extent.

But from the start, there were many problems that computers couldn't calculate, or even recognize.

In the early days of computers, a professor sent one of his graduate students to solve the problem of 'computer image recognition'—he optimistically thought it would only take two months for his student to completely overcome this issue.

But the truth was, it was impossible.

Even into the twenty-first century, 'human eye recognition of CAPTCHAs' remained one of the standards for certain web programs to determine if a login attempt was from a human.

'Computer graphic recognition' is a terrifying subject. Scholars from all over the world struggle with it, yet programmers still use 'human eye recognition of CAPTCHAs' to deter malicious machine logins.

The difficulty of computer graphic recognition is largely due to...

"The problems computers can understand are called 'Polynomial time problems,' – P problems for short. Computers can quickly solve P problems. The problems that are more difficult than P problems are called non-deterministic polynomial time—Nondeterministic Polynomial time or NP problems."

A large part of geometry problems fall within NP. Turing Machines can quickly verify whether an answer is correct but cannot quickly provide an answer. And some geometry problems are even more difficult than NP.

"There was once a mathematician who tried to prove 'P=NP' to show that all NP problems could be turned into P problems, taking the first step toward elevating computers to the God Realm. I still can't remember if it was successful..."

From the moment the Turing Machine was born, limits were set—because it negated the 'absolute Turing computability' of mathematics.

Mr. David Hilbert's grand ideal had failed.

—If it weren't for the war, perhaps Anatoly could have possibly done... what...

—Who is Anatoly? How do I know so many bizarrely powerful characters?

After a moment, the man added dejectedly, "Probably not. Computers have 'things they are destined not to be able to do.' NP problems are just destined to be beyond the power of electronic computers to solve. And NP problems aren't even the epitome of complexity."

"Beyond NP problems, there are Polynomial Hierarchy problems—PH, beyond Polynomial Hierarchy problems, there are Polynomial Space problems—PSPACE problems, and beyond Polynomial Space, there are Exponential Time problems—EXPTIME problems."

"In this regard, quantum computers are a dimension stronger than electronic computers. But the theoretical limits of quantum computers' abilities are termed 'Bounded Error Quantum Polynomial Time problems'—BQP. And the BQP category includes only some PSPACE problems—even quantum computers cannot touch EXPTIME. It's a realm almost akin to Tao..."

Yuki nodded with awe, "Although I don't understand, it sounds like it's very powerful. So, Xiang Shan... what is EXPTIME? Can you give me an example?"

"Let's go with the simplest example," Xiang Shan nodded, "You are using an electronic program and feel that it's running a bit slow. At this moment, you need to make a choice—is it, 'let it run slowly, it'll be fine in a while,' or 'how much longer shall I endure before I restart it'? This 'judgment' is an EXPTIME decision."

Yuki fell silent for a moment, "Huh?"

"This 'judgment' is an EXPTIME decision."

"What?"

"This 'judgment,' it really is an EXPTIME decision."

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"That's not right..." Yuki was doubtful whether his biological brain was malfunctioning: "Isn't this something that even children can understand?"

"From the mathematical perspective, the difficulty level of this problem is EXPTIME. Biological brains can make this judgment simply, but computers can't even 'understand' it. Among all known computational models, there's only one that can reach this domain..." Xiang Shan pointed to his head: "The human biological brain."

"Perhaps it is a limitation that the human brain can't construct a more advanced mathematical computational model than itself? Or possibly we can, it's just that humanity hasn't reached this domain yet? But I can assert that humans still haven't solved this problem—because the Martial Artists I encountered today still retained their biological brains."

Xiang Shan took out two strips of iron and drilled holes in them by hand.

"This is the intricacy of Inner Strength. Use your brain... use the wisdom of your talent to harness the vast resources brought by computers, forming an 'Inner Strength' that defends against external evils"

"Since controlling the prosthetic body, recording intelligence, and even everything in this world, is dependent on the Turing Machine, the moment you can use Inner Strength to drive the Turing Machine, you rise above all tools. The rulers will no longer be able to prevent you from obtaining information, nor steal anything from you. He will no longer control you."

"If you focus on practicing External Martial Arts without cultivating Inner Martial Arts, any stronger practitioner of Inner Strength, any hacker, can easily read your memories, take away your Martial Arts Algorithms, your Experience Accumulation, and even input false signals into your sensory organs, leaving you to live in illusion for a lifetime. A Martial Artist without Inner Strength, no matter how strong the practice, is just an Oppressor, and only by cultivating Superior Internal Power do you have the strength to resist."

He inserted the iron strips into an input slot of the machine.

These iron strips serve as the storage medium. The presence or absence of holes represents the machine language of "yes" and "no." It is the primal soul of the computer.

Even without an electronic structure, it possesses all the functionalities required by a computer.

"Now, I'll teach you the first lesson. This may be your means of a last stand after falling into an Illusion Technique." Xiang Shan pointed to the lever: "Now, turn it."

Yuki complied and turned the handle. Steel drills fell continually, driven by the gears. Some were blocked by the iron strips, while others passed through the holes in the iron, transferring kinetic energy to the gears below. The iron strips moved inward as the gears turned.

Xiang Shan stared at the exposed mechanism, watching it operate.

A "ding" sound. Yuki was somewhat incredulous.

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

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

"Savour this lesson. It's called the 'Turing Halting problem,' and it's the sword that cuts through illusion."

The Turing Halting problem is about determining whether any given program can finish running within a finite amount of time. This problem is equivalent to the decision problem: If there exists a program P that for any given input program w, can determine whether w will stop within finite time or fall into an infinite loop.

This problem could simply be phrased as "Can a Turing Machine simply determine whether another Turing Machine should halt."

Alan Turing, before the advent of the computer, had already made his judgment.

The Turing Halting problem is logically unsolvable.

Therefore, it also provides people with the sword to cut through the Illusory Realm.

If one could construct a structure mathematically isomorphic to the Turing Machine within an illusory realm created by the computations of a Turing Machine, and then use it to decide the halting problem, such a Turing Machine would have the capability to exhaust the resources of the matrix and break through the layers of matrix.

Provided that the higher-level matrix still constitutes a "Turing Machine."

The only way a Turing Machine can understand another Turing Machine is by simulating the workings of that Turing Machine within itself.

A command that can crash a virtual Turing Machine can also crash the Turing Machine that virtualizes it.

—Of course, there is also the situation where "the world you exist in is virtualized not by a Turing Machine but by a Super Turing Machine." But honestly, a civilization capable of using a Super Turing Machine to incarnate brains in vats has far surpassed human understanding. Under such circumstances, within the cognitive range of humans, there simply doesn't exist a method to break through, so there is no need to ponder over it.

The Turing Machine, the Turing Halting problem, the Hilbert's Program ... these are the origins of Inner Strength.

This is also exactly why Xiang Shan would rather temporarily set aside the reconstruction of his prosthetic body to first complete this mysterious ritual.

He is currently unable to confirm whether he is a "brain in a vat"—according to his understanding of technology, submerging a brain in nutrient fluid, making it believe that the surrounding illusions are "real," is not difficult.

Therefore, he must verify it.

The biological brain, although advanced in mathematical structure, is just so easily deceived. In terms of "functionality," it can't compete with artificial objects.

How could something haphazardly cobbled by nature with the principle "if it works, it's fine" stand against the deceptions designed with the purpose of "deceiving" by intelligence?

Only intelligence can counteract intelligence.

After confirming the "reality" of his surroundings, the world seemed to come alive for Xiang Shan.

He gently rubbed Yuki's head: "Good, you are real, and so am I."

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