Cyber Heroes

Turing machine does not specifically refer to a certain machine or type of machine, but a concept defined in the language of mathematics. It is a tool invented to answer the ultimate questions of mathematics by mathematicians in the 20th century. Its purpose is to abstract and simplify the complex mathematical activities within the human mind.

Seven sets are required to define a Turing machine. The two most important ones are "a finite set of non-empty states (State)" and "a finite set of non-empty symbols (Symbol)". The former represents "the kinds of symbols this machine can read and write on paper tape" and is also known as the "color number". The latter represents "the states that can occur inside the machine" and is called the "state number" - which can be simply understood as "the machine can open first and second gears."

The remaining five include "blank symbol", "initial input symbol", "initial state", "transition function set", and the special "stop state", that is, "the machine must stop after entering this state."

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

In fact, when the great mathematician Alan Turing created this concept, he did not design a machine that could actually run it. It would be many years before von Neumann produced an engineering implementation of a Turing machine based on electricity.

For example, the abacus is divided into upper and lower sides by a beam. The two beads on the top can be used to express three symbols from 0 to 2 by leaning against the beam or not. Five beads can be used to express six states from 0 to 5 by leaning against the beam or not. The abacus can be viewed as a three-color, six-state machine.

At this time, if a mathematician writes down all the transfer functions corresponding to the three colors and six states on paper, and trains a person (or monkey) to move the abacus strictly according to the set of transfer functions, then the abacus and the user will It can be regarded as "a Turing machine".

Yawgmoth vaguely remembered that many years ago, he seemed to have done similar research with Xiang Shan. Think of DNA as the "paper tape" of the Turing machine, the four base pairs of DNA as a "set of symbols", and regard amino acids as a "state", and ribosomes and mRNA have a "set of transfer functions"...

The cell nucleus of eukaryotes on Earth may be a four-color N-state Turing machine.

Among the many Turing machines, one called the "Beaver Machine" is the most outrageous.

BB(n), BB means "Busy Beaver", busy beaver. is a terminateable program that finds the maximum output possible given arguments. The simplest two-color two-state beaver machine has only two states: "on and off", and can only output "0 or 1", and will stop when it outputs 1. The two-color two-state beaver machine can output 4 1's through 6 state transitions, so BB(2)=6.

BB(3)=21.

BB(4)=107.

However, humans have not yet enumerated all 5-state Beaver Turing machines. We only know that BB(5) is either 47, 176, 870, or it is greater than 10 raised to the eleventh power.

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

And in 2016, a mathematician encoded Goldbach's conjecture into a 43-state Beaver machine. The coding logic is very simple, which is to verify every even number from small to large. If you find that an even number can be expressed as the sum of two prime numbers, then examine the next even number. As long as this Turing machine completes BB(43) steps and does not stop before finishing, then humans can conclude that it will never stop, so Goldbach's conjecture is true.

But the problem is that there are less than 10^80 atoms in the universe. This number is even much smaller than BB(6), let alone BB(43).

Yawgmoth marveled at how outrageous the idea was. The entire universe of matter and energy cannot support this level of calculation. If you want to record the specific value of this number, you have to record more digits than the number of atoms in the universe on every atom in the universe. This is an uncalculable value.

"But even so, this is still the pursuit of 'infinity' in the 'finite' step." Xiang Shan introduced it this way, "If we use a more banal way to exhaust all even numbers, then The matter and energy we need are 'infinite'. Rational numbers are an infinite concept. This method can only be used when Goldbach's conjecture is false - that is, a Goldbach has been found in infinity. A counterexample to the conjecture. But BB(43), although it is larger than the universe, is a finite number. The Beaver machine completed the exploration of infinity through finite steps in a finite time."

"But what's even better is that humans can actually find better ways to prove Goldbach's conjecture - the final proof only consumes a little bit of the light and heat from the sun. You see, the effect of wisdom is How terrible."

The energy that nourished all the mathematicians from Pythagoras to Anatole was the tiny bit of energy released by the sun, solidified by plants, and transformed by animals. Insignificant compared to the universe.

But this tiny bit of energy has solved many answers that would be difficult to obtain even if the universe were burned.

Yawgmoth nodded. But then another question came to mind.

——Anatoly was completed in the 20's...

…

"Compared with 'exhaustive infinity', the efficiency of BB(43) is horribly high, isn't it? But what's even more horrifying is that humans can find more efficient paths leading to answers to problems." Xiang Shan raised the question Wine glass: "To our great minds."

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

What exactly is wisdom to this universe?

Yawgmoth felt his passion rekindled. He was inspired by the words of his best friend and once again thought of his dream.

In a daze, he had finished his meal and walked back to his research room.

"Recently, there has indeed been a bit..." Yawgmoth pressed his head. For him, “scientific research” is a comfort zone. He seems to have closed himself in this zone for a long time. He deals with various biological research every day and is so busy that he forgets everything. This kind of thing is a drain on his mind.

Fortunately, Xiangshan is here.

But at the door of the research room, Ingrid stopped him.

"Hey, Yog." Ingrid said to him.

Yawgmoth felt dazed. He vaguely remembered that Yingjia seemed to have too much respect for Xiangshan recently... But another memory told him that recently there was a popular East Asian party game called "King Game" in the company...

——Is Xiangshan someone who knows how to play this kind of game...

——No, Yingjia has finished her meal? So fast?

"Do you still remember. We, you, me, Xiangshan, your ex-wife, the research we did on the Rama Project." Ingrid asked, "Think of DNA as a 'paper tape' and ribosomes as a 'magnetic head' ', mRNA is regarded as a register that stores the state of the machine. Amino acids are a set of states, and the DNA code table is temporarily regarded as an instruction list... When conducting scientific research, it is the only time you are allowed to 'doubt', so remember my tips for you first. .Remember it well.”

Further research revealed that the difficulty of ZFC’s consistency proof has been reduced to BB(748)