After Wang Qi appeared, Feng Luoyi and Turing took the initiative to give up a seat on the edge of the table and motioned for him to sit down. This means that they have recognized Wang Qi's ability as an arithmetic, and see him as a fellow practitioner of equal status rather than a junior. Turing also smiled at Wang Qi: "You can't drink tea with your manifested illusion, so I won't pour it for you." ”

Feng Luoyi first asked Wang Qi: "Is everything settled?"

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

"You don't have to tell me anything specific. 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. With the situation at that time, even if Gou Dabao could break away from that space station-type small Tianchen magic weapon, he would only be caught up by the Heavenly Sword and then completely evaporated.

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

In this universe, the concept of the von Neumann machine appeared earlier than the Turing machine. It's a pretty amazing thing. Feng Luoyi and Turing first relied on the imitation of the soul to standardize the Feng structure, and then further mechanized it to obtain a "Turing machine" that is not very strict.

Only this universe, which is particularly easy to form in the black box of technology, can support this phenomenon.

Now, the Shenzhou monks' understanding of the relationship between the abacus and the abacus is like this-first, the abacus is undoubtedly an advanced version of the abacus. But what about the fact that there is no excessive version between the calculator and the abacus?

Nobody knows.

And now, Turing real people have continued to make rapid progress in mathematical logic and fill this gap.

This research is not a fraudulent funding project that is called "filling in the gaps". It clarifies the foundation of the calculator and makes the development of the calculator "down-to-earth".

A lot of theories about calculators. With a more solid foundation. There is also more room for computer direction research.

And according to Turing, this is just a "by-product". He was thinking about how to bypass the "undecidable" and made improvements to the theory of calculators.

And the question that Turing really thinks about is what interests Wang Qi even more.

"You mean, you're trying to make a logical reasoning method to prove ...... Prove some problems?"

"I hope he will be able to bypass your incompleteness. But it seems to have failed. Turing spread out his manuscript and pointed: "First of all, we define the heavens, the earth, the earth, and the human elements in an unknown celestial change—just as we often do in the Tianyuan formula, using imaginary symbols to deal with the unknown changer. Then, let's assume that we have two 'buckets' and 'oxes' -- don't ask me what they are, suppose there is such a ......"

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

On the other side, Wang Qi has a big mouth.

"This ...... It's clearly ......"

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

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

Although Alan Turing finally proved that this lambda algorithm is equivalent to a Turing machine, both are proofs of undecidability. However, these two proofs have very different meanings in mathematics. In the subsequent development, Turing machine proofs and λ algorithm proofs have both walked out of a royal road.

Thinking of this, Wang Qi looked at Turing Zhenren's slightly delicate face, and felt a little embarrassed in his heart.

Unexpected, unexpected. Turing machine proofs and λ algorithm proofs are mutually reinforcing, with the same end but different paths. I used the "self-referential" Dongfeng to prove the undecidability with a Turing machine. Then the Turing real person completed Church's train of thought.

-- The overhaul of Shenzhou, although there are shadows of some people on the earth. But, after all, they have a life of their own.

Church was Alan Turing's mentor. But in this world, Turing Zhenren is self-taught, so he led the Thousand Machines Pavilion and suppressed the Fusang Ten Thousand Demons.

- However, Turing's thinking must not be quite the same as Church's, right? How did the Turing get this far?

Both Monks are quick thinkers. The communication is also very fast. In a moment when Wang Qi was distracted, Turing Real Man had already finished talking about his new idea. He smiled and said, "This algorithm is also quite good." I tried it myself, and he was able to prove almost all propositions, and express almost all arithmetic systems - it was a complete system. ”

This is a Turing man's reflection on undecidability. It is also his reflection on incompleteness.

Feng Luoyi said: "But Wang Qi has also proved before that completeness and compatibility will not exist at the same time. If this is a one-size-fits-all algorithm, then ......"

"Then it must be a paradoxical algorithm. Wang Qi swiped his fingers in the air, condensed the colored light, and wrote down the Taoist arithmetic: "For a meaningful logical system, a strong ability to express itself must be accompanied by indestructible limitations. We now substitute some algorithms that are clearly wrong...... You see, it makes sense."

If it is a theory, whether it is right or wrong, the positive and negative sides can be explained, and no matter how you say it, it makes sense, then it is equivalent to no reason at all.

Turing real person saw Wang Qi's quick reaction and nodded appreciatively: "You are indeed amazing." I realized this at the beginning. However, at the time, I thought that I could circumvent this contradiction by imposing more 'limitations'. ”

"Impossible. Wang Qi shook his head and said: "Self-referential can't be bypassed. ”

Self-referential is also a hurdle that the λ algorithm cannot bypass.

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

Since it is "any function item", it must contain the λ term itself.

So, is it able to handle itself?

"Self-referential, what an amazing concept. Turing said, "I feel like the core of what we're doing now is all about 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 has realized the terrifying connotation in it. Feng Luoyi nodded, agreeing with Turing's opinion.

It is also incomplete, and so is it undecidable. This "self-referential" seems to be a logical boundary marker. It is a conspicuous boundary monument on the boundaries of arithmetic itself.

It seems that there is such a boundary, which 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 only be able to contradict yourself and be unstable.

How to bypass this limit was once the main concern of mathematicians.

"This attempt was a failure. Turing said slowly: "This arithmetic that I have called the Vientiane Algorithm has other meanings, and I will sort it out later." What I'm going to talk about next is a little idea of mine. ”

"What if—just what if. Turing really emphasized: "If we can really get around that line -- here we don't care about how we bypass it for the time being, let's just say we can do it." Just when the Heavenly Dao suddenly becomes cheap, the 'Sound of the Great Dao' and 'Brahmanic Meditation' and the like are filled with the heavens and the earth, and then we can easily build such a calculator, which directly connects to the Dao and can give the answer to all specific questions......"

Wang Qi knew it in his heart. What Turing said is called the "teleportation machine" on Earth.

The "Signal Machine" is another Turing machine connected to "God", and it is a black box that holds the prophet. For example, if there is an oracle machine with a prime number determination indication, it can do everything that a normal Turing function can do, and it can also instantly determine whether a natural number written on the paper tape is a prime number, without actually calculating it.

So, can some problems that originally involve self-referentiality and contradictions be solved by the Signal Machine?

For example, the contradiction caused by self-referentiality.

That's another conjecture that has been made recently. Can this 'self-referential' be bypassed?" Turing smiled bitterly: "Can the imaginary Dao Holy Sound solve this problem?"

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

The essence of the deduction made by the Turing real person is to find an omnipotent being in my imagination, assume that it can solve the otherwise undecidable problem, and then see what problem it can solve.

"Can we accomplish this undertaking with what we have imagined?" Turing looked at Wang Qi and Feng Luoyi, and asked calmly: "If it is feasible, it means that there is still a glimmer of hope on our path." But even the omnipotence in our illusions cannot be ......"

Feng Luoyi asked, "What is the result?"

Turing replied with a wry smile: "It's not very good." ”

If there is a program A that determines the shutdown problem (the input A needs is a program), we construct a new program B, which calls A but is the opposite of A's output: if B's input is judged to be down by A, B will not stop, and if B's input is judged by A to be down, then B will be stopped.

Suppose there is an indicator with a shutdown notice, and if the machine has the "code" and input of a normal Turing machine written on the tape, then it can instantly know whether the ordinary Turing machine will stop when it receives the input of the indication without calculation.

However, if we replace all the words "Turing machine" in the proof with "the word "indicating the 'number theory problem'", it will not be easy for the rest of the part to be word-for-word?

Perhaps it is the human mind itself that limits their imagination, and even the omnipotence of the human imagination cannot cross this line.

Even the Great Dao Shengyin couldn't knock on this door. (To be continued.) )

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