Legend of the Cyber Heroes
Chapter 747 - 002

Chapter 747: 002

A Turing Machine is not a specific machine or a class of machines, but a concept defined in the language of mathematics, a tool invented to answer the ultimate questions of mathematicians of the 20th century about mathematics. Its purpose is to abstract and simplify the complex mathematical activities conducted in the human brain.

Defining a Turing Machine requires seven sets, of which the two most important are "a finite non-empty set of states" and "a finite non-empty set of symbols." The former represents "the types of symbols that the machine can read and write on the tape" and is also called the "number of colors." The latter stands for "the states that can occur inside the machine," known as the "number of states"—this can be understood simply as the machine having "gears" like a car.

The remaining five include "the blank symbol," "the initial input symbols," "the initial state," "the set of transition functions," and the special "halt state," meaning "the machine must stop once it enters this state."

As long as these seven elements are present, a machine can be considered a Turing Machine.

In fact, the great mathematician Alan Turing did not design a machine that could actually operate when he created the concept. It was many years later that Von Neumann created a working model of the Turing Machine based on electricity.

For example, the abacus is divided into upper and lower sides by a horizontal beam. With two beads on top, by being close to or away from the beam, you can express 0 to 2, three types of symbols. With five beads on the bottom, you can express 0 to 5, six kinds of states. The abacus can be seen as a machine with three colors and six states.

At this time, if a mathematician writes down on paper all the transition functions corresponding to the three colors and six states and trains a person (or a monkey) to strictly manipulate the abacus according to the set of transition functions, then the abacus and its user could be considered "a Turing Machine."

Yawgmoth vaguely remembered that, many years ago, he seemed to have conducted similar research with Xiang Shan. Viewing DNA as the "tape" of a Turing Machine, the four nucleobases of DNA as "the set of symbols," and amino acids as "states," while ribosomes and mRNA possessed "the set of transition functions"...

The nuclei of Earth’s eukaryotic cells might be Turing Machines of four colors and N states.

Among many Turing Machines, there is a kind called the "Busy Beaver Machine" that is the most outrageous.

BB(n), where BB stands for "Busy Beaver", is a program that, given certain parameters, seeks the largest possible output of a terminating program. The simplest two-color two-state Busy Beaver Machine has only "on and off" states and can only output "0 or 1," stopping upon outputting a 1. A two-color two-state Busy Beaver Machine can produce four 1’s after six state transitions, so BB(2) = 6.

BB(3) = 21.

BB(4) = 107.

Then, humanity has not yet enumerated all the Turing Machines with five states. It is only known that BB(5) is either 47,176,870, or greater than 10 to the eleventh power.

BB(6) has been confirmed to be greater than 7.4*10^36537.

In 2016, a mathematician encoded the Goldbach conjecture into a Busy Beaver Machine of 4888 states. The encoding logic is straightforward, verifying each even number from small to large. If a certain even number can be expressed as the sum of two primes, then move to the next even number. If this Turing Machine completes BB(4888) steps without halting beforehand, humanity can assert that it will never stop, thus the Goldbach conjecture is true.

But the issue is that there are less than 10^80 atoms in the universe. This number is even far smaller than BB(6), let alone BB(4888).

Yawgmoth marveled at the outlandishness of this thought. All the matter and energy in the entire universe could not support calculation on such a scale. To record the precise value of this number, each atom in the universe would have to have more digits recorded on it than the number of atoms in the universe. It is an incalculable value.

"Even so, this is still a pursuit of ’infinity’ within ’finite’ steps," Xiang Shan introduced it as such, "If we were to exhaustively enumerate all even numbers in a more pedestrian way, the matter and energy we need would be ’infinite.’ A rational number is an infinite concept. This method can only guarantee use in the event the Goldbach conjecture is false—that is, within infinity, a counterexample to the Goldbach conjecture is found. But BB(4888), although larger than the universe, is a finite number. The Busy Beaver Machine finishes infinite exploration within a finite time through finite steps."

"But what’s even more wonderful is that humanity can find even more ingenious ways to prove the Goldbach conjecture—the final method used only a little bit of the light and heat spilling from the sun. You see, the effects of wisdom are terrifying."

The energy that nurtured all mathematicians from Pythagoras to Anatoly was just a tiny amount of energy released, fixed by plants, and converted by animals compared to the universe—insignificant.

But this insignificant amount of energy solved many answers that would have been hard to obtain, even burning through the entire universe.

Yawgmoth nodded, but then another question came to mind.

—Anatoly completed it in the year 20... whatever.

...

"Compared to ’exhausting infinity,’ the efficiency of BB(4888) is terrifying, isn’t it? But what’s more terrifying is that humans can find an even more efficient path to the answer," Xiang Shan raised his glass, "To our great minds."

Yawgmoth nodded. He still remembered that his friend’s long-cherished wish was to figure out "why the power of wisdom can be so efficient."

What, after all, is wisdom to this universe?

Yawgmoth felt his passion reignited. He was inspired by his friend’s words and once again remembered his dream.

In a trance, he had already finished his meal and was heading back to his study.

"Lately, I really am a bit..." Yawgmoth held his head. For him, "scientific research" was a comfort zone. It seemed like he had sealed himself in this interval for a long time, dealing with various biological studies every day, so busy that he forgot everything else. Such matters were a drain on his spirit.

Luckily, Xiang Shan was there.

But at the entrance to the lab, Ingrid stopped him.

"Hey, Yog," Ingrid said to him.

Yawgmoth felt a moment of daze. He vaguely remembered that recently, Inga seemed to have too much reverence for Xiang Shan... but another memory told him that there was a popular East Asian party game called "King Game" circulating in the company recently...

—Would Xiang Shan be the kind of person to play such a game?

—No, has Inga finished her meal already? That was fast?

"Do you remember? We, you, me, Xiang Shan, your ex-wife, the research we did on the Rama Project," Ingrid asked, "Viewing DNA as ’tape,’ ribosomes as ’read/write heads,’ mRNA as registers that store machine states. Amino acids were the set of states, and the DNA code table was treated as the command table... When doing research, the only time you are allowed to ’doubt’ is now, so remember the hint I gave you. Remember it well."