Enter the Immortal Cultivation

After Wang Qi appeared, Feng Luoyi and Turing took the initiative to give up a seat at the table and motioned for him to sit down. This means that they have recognized Wang Qi's ability as a mathematician and regard him as a colleague with equal status rather than a junior. Turing also smiled at Wang Qi: "Your manifested illusion can't drink tea, so I won't pour it for you."

Feng Luoyi first asked Wang Qi: "Has everything been resolved?"

"Yes, Gou Dabao's work has been completed, and there should be a report there. This time, over there..."

"You don't need to tell me the specific things. I just need to know that nothing happened." Feng Luoyi said calmly. When Wang Qi and Gou Dabao began to demonstrate, there were many heavenly swords facing that secret place. Based on the situation at that time, even if Gou Dabao could escape from the small space station-style Tianchen magic weapon, he would only be caught up by the Tianjian and then completely evaporated.

Feng Luoyi first briefly told Wang Qi the conversation he had just had with Master Turing. Wang Qi was first amazed by what Turing did, and even sighed at the magic of the physical laws of the universe.

In this universe, the concept of von Neumann machine appeared earlier than Turing machine. This is a very magical thing. Feng Luoyi and Turing first relied on imitation to standardize Feng's architecture, and then further mechanized it to come up with a less strict "Turing machine".

Only this universe, where technological black boxes are particularly easy to form, can support this phenomenon.

Now, the understanding between Chinese monks and calculators and abacus is this - first of all. The calculator is undoubtedly an advanced version of the abacus. But what if there is no over-the-top version between a calculator and an abacus?

no one knows.

Now, Turing has followed the rapid progress of mathematical logic and filled this gap.

This research is not a funding scam with the euphemistic name of “filling in the gaps”. It clarifies the foundation of calculators and makes the development of calculators "down to earth".

A lot of calculator theory. Have a more solid foundation. There is also more room for research in the direction of calculators.

According to Turing, this is just a "by-product". He made improvements to the calculator theory when he was thinking about how to bypass "undecidable".

The question that Master Turing was really thinking about was what interested Wang Qi even more.

"You mean, you were just trying to make a logical reasoning method to prove... prove some issues?"

"I hope he can get around your imperfections, but it seems to have failed." Master Turing spread out his manuscript. Pointing: "First of all. We define Tianyuan, Earthyuan and Renyuan in an unknown Heaven-changing style - just like what we often do in Tianyuan Style, we use fictitious symbols to deal with the unknown Heaven-changing style. Then, Let's assume that we have two changing types of 'Dou' and 'Niu' - don't ask me what they are, let's say there is such a..."

Feng Luoyi became a little interested. This kind of thinking that everything is a "hypothesis" is still very rare.

On the other side, Wang Qi opened his mouth.

"This...this is clearly..."

This is the lambda algorithm, another undecidable proof that has ever existed in the history of the earth.

That is, Alan Turing’s mentor. Mathematician Church's proof method.

Although Alan Turing finally proved that the lambda algorithm and the Turing machine are equivalent, they are both proofs of undecidability. However, these two proof methods have completely different meanings in mathematics. In the subsequent development, both Turing machine proof and lambda algorithm proof have taken a great road.

Thinking of this, Wang Qi looked at the slightly delicate face of Master Turing and felt a little sad.

Unexpected, unexpected. The Turing machine proof and the lambda algorithm proof are mutually exclusive, reaching the same goal but in different ways. I had previously relied on the "self-referential" Dongfeng to use the Turing machine to prove that it was undecidable, and then Turing completed Church's idea.

——The overhaul of China, although he has the shadow of some people on the earth. However, they have their own lives after all.

Church was Alan Turing's mentor. But in this world, Master Turing was self-taught, so he took charge of Qianji Pavilion and suppressed Fusang Ten Thousand Demons.

——However, Turing’s thinking probably doesn’t quite match Church’s, right? How did Turing take this step?

Both Xiaoyao monks have quick minds and communicate very quickly. Just when Wang Qi was distracted for a moment. Real Turing has finished talking about his new ideas. He smiled and said: "This algorithm is quite good. I have tried it myself, and it can prove almost all propositions and express almost any mathematical system - it is a complete system."

This is Turing's thinking on undecidability and his thinking on incompleteness.

Feng Luoyi said: "But Wang Qi has proven before that completeness and compatibility do not exist at the same time. If this is an all-powerful algorithm, then..."

"Then it must be a self-contradictory algorithm." Wang Qi moved his fingers in the air, condensed the colored light, and wrote down a series of calculation formulas: "For a meaningful logical system, strong expression ability must be accompanied by indestructible limitations. We Now substitute some algorithms that are obviously wrong...you see, it makes sense."

If it is a theory, whether it is right or wrong, it can be explained from both sides, and it makes sense no matter how you say it, then it means that it makes no sense at all.

Seeing Wang Qi's quick reaction, Master Turing nodded appreciatively: "You are indeed amazing. I also realized this at the beginning. However, at that time I was still thinking that I could rely on imposing more restrictions. ' to avoid this contradiction."

"Impossible." Wang Qi shook his head and said, "Self-reference cannot be circumvented."

Self-reference is also a hurdle that the lambda algorithm cannot get around.

In this algorithm called λ, the prototype of λ is a function. When defining a lambda term, it is allowed to process any function into another lambda term.

Since it is "any function term", it must include the lambda term itself.

So, can it handle itself?

"Self-reference, what a magical concept." Turing praised: "I feel that the core part of our current work is closely related to it."

"The distinction between truth and falsehood of 'this statement is not true' has existed for tens of thousands of years, but before Wang Qi, no one realized its terrifying connotation." Feng Luoyi nodded and agreed with Turing's view.

It is also incomplete and undecidable. This "self-reference" seems to be a logical boundary. It is a conspicuous landmark on the boundary of arithmetic itself.

It seems that there is such a limit that firmly restricts the development of logic.

If you cross this boundary, you will either have to face an endless cycle of death, or you will have to contradict yourself and lose your footing.

How to get around this limit was once the biggest concern of mathematicians.

"This attempt is considered a failure." Master Turing said slowly: "This arithmetic, which I call the universal algorithm, has other meanings. I will sort it out later. What I will talk about next is my A little personal thought.”

"If - just if." Master Turing emphasized: "If we can really bypass that limit - here we don't care about the specific method of bypassing, just say that we can do this. Suddenly the way of heaven suddenly It has to be cheap, and the resonance between the world and the world, such as the "sound of the avenue" and "the chanting of Brahma saints", can be easily established. It is directly connected to the avenue and can give the answers to all specific problems. Answer……"

Wang Qi understood. What Turing said is called the "oracle machine" on earth.

The "oracle machine" is another Turing machine connected to "God" and a black box that contains the prophet. Suppose there is an oracle machine with a prime number determination oracle. In addition to doing everything that an ordinary Turing machine can do, it can also instantly determine whether a natural number written on a piece of paper is prime, without actually calculating it. .

So, can the oracle machine solve some problems that originally involve self-reference and contradiction?

For example, the contradiction caused by self-referentiality.

"This is another conjecture that has been made recently. Can this 'self-referential' be bypassed?" Turing smiled bitterly: "Can the fictitious Dao Shengyin solve this problem?"

Mathematics often has this method of proof, which is to first assume that something exists, and then make various logical deductions based on this premise.

The essence of Turing's deduction is to find an omnipotent existence in my imagination, assuming that it can solve the original undecidable problem, and then see what problem it can solve.

"Can the things we imagined be able to complete this undertaking?" Master Turing looked at Wang Qi and Feng Luoyi and asked calmly: "If it is feasible, it means there is still a glimmer of hope on our road. But even if The omnipotence in our illusion cannot be achieved..."

Feng Luoyi asked: "What is the result?"

Master Turing responded with a wry smile: "Not very good."

If there is a program A that determines the shutdown problem (the input required by A is a program), we then construct a new program B. This program calls A but is exactly the opposite of the output of A: if the input of B is judged to be shutdown by A, Then B will not stop; if B's ​​input is judged by A as not to stop, then B will stop.

Assume that there is an oracle machine with a shutdown problem oracle. If the "code" and input of an ordinary Turing machine are written on the paper tape of this machine, then it can know instantly without calculation that this ordinary Turing machine Will the machine stop when it encounters the indicated input?

However, if all the three words "Turing machine" in the proof are replaced by "oracle machine with a 'number theory problem' oracle", what if the other parts are not easy to say? The incomplete proof still holds!

Perhaps it is human beings' own thinking that limits their imagination. Even the omnipotent human imagination cannot cross this line.

Even the Holy Sound of the Great Dao cannot knock on this door. (To be continued ~^~)

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