Ultra-slow computer programs reveal fundamental mathematical limitations

Developers often seek to minimize the execution time of their code; But in 1962, Tibor Rado, Hungarian mathematician, posed the opposite problem. “How long does a simple computer program run before it stops?” He asked. Some programs are highly inefficient; But they still continue to work. Rado nicknamed the programs “Busy Beaver”.

Finding blue dog programs has been a misleading riddle for programmers since 1984, including math hobbies. However, in recent years, blue dog play has also become a research topic; Because it has been related to the basic concepts and problems of open mathematics. Scott Aranson“In mathematics, there is a very fine line between entertainment and important issues,” says a computer science theorist at the University of Texas at Austin.

New research shows that searching for computer programs with long execution times or slow programs can change the state of mathematical knowledge and reveal important points. Researchers say blue dog play is a coherent indicator for assessing the difficulty of specific issues, including Goldbach’s unresolved conjecture And Riemann hypothesis Is.

Uncountable computer game

In general, the blue dog game is about behavior Turing machines Is. Turing machines are the basic and ideal computers defined in 1936 by Alan Turing. The Turing machine performs operations on a string of finite bands divided into squares. This machine executes operations based on a list of rules. The first rule is as follows:

If the square contains the number 0, replace it with the number 1 and move the square to the right and check rule 2. If the square contains the number 1, leave 1 and go to the left of a square and check rule 3.

Each rule is like this: “Choose your adventure”. Some rules say refer to the previous rules, and eventually a law will include a “stop” instruction. Turing proved that this simple type of computer could perform possible calculations based on simple instructions and sufficient time.

According to Turing’s definition in 1936, the Turing machine is finally faced with the “stop” command to calculate anything and will not be caught in an infinite loop. Nevertheless, it proves that there is no safe and repetitive way to distinguish between machines that stop and machines that always run.

Busy blue dog issue

A visual representation of the slowest-running five-legged Turing machine. Each column of pixels represents a step in the calculation that moves from left to right. The black squares also indicate the positions in which the machine printed the number 1. The right column indicates the status of the calculations when the Turing machine stops.

The busy blue dog game begs the question: “Given a certain number of rules, what is the maximum number of steps a Turing machine can take before stopping?” For example, if you have only one rule and you want to make sure the Turing car stops, you should consider stopping immediately; As a result, the number of blue dogs busy for a monogamous car is equal to one (BB).

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By adding a few more rules, the number of machines also increases. Of the 6,561 possible cars with two rules, the car that runs the longest possible time before the stop operation (six stages) is the blue dog; But some other machines go on forever; As a result, none of them are blue dogs. How can these machines be removed? Turing proved that there is no way to automatically show that a machine that runs for a thousand or one million steps will not end up in the end; For this reason, it is very difficult to find busy blue dogs.

There is no general method for identifying Turing machines with the longest execution and the desired number of instructions. Better yet, the blue dog game is busy uncountable. It was difficult enough to prove that BB (2) = 6 and BB (3) = 107; As a Rado student, Shen Lane, With this project in 1965, succeeded in getting a doctorate. Rado (BB) called it quite disappointing, but the issue was finally resolved in 1983.

Threshold of ignorance

William Gasarch, A computer scientist at the University of Maryland, and other mathematicians are interested in using the game as a scale to measure the difficulty of important open problems in mathematics or to calculate mathematical knowledge. For example, Goldbach’s conjecture begs the question: “Is every even integer greater than two sums of the first two numbers?” Proving this conjecture is one of the most important events in number theory and allows mathematicians to gain a better understanding of the distribution of prime numbers.

In 2015, an anonymous GitHub user, alias Code Golf Addict, published a code for the Turing machine with 27 rules. This machine stops, if and only if Goldbach’s guess is wrong. This machine works by ascending counting of integers greater than four, and for each, examines all possible methods for obtaining an integer by adding the other two numbers to see if the obtained pair is prime. By finding the right pair of prime numbers, this machine deals with the next even integer and repeats the above process. If it finds an even number that is not the sum of even pairs of prime numbers, it stops.

Running the machine is not a practical way to solve Goldbach’s conjecture; Because you can not be sure of stopping it; But the busy blue dog game emphasizes this to some extent. If it were possible to calculate BB (27), an expectation ceiling would be created for the automatic solution of Goldbach’s conjecture; Because BB (27) corresponds to the maximum number of steps that a 27 Turing legal machine can perform. If we knew the exact number of steps, we could set the Turing machine to exactly the same number. If the Turing machine stops at that point, it means that Goldbach’s guess is wrong; But if it performs many steps, it can be said with certainty that the guesses are correct.

In 2016, Aranson collaborated with Yuri Matyasovich And Stephen O’Reilly Reached similar results. They found that the Turing machine would stop with 744 rules, if and only if Riemann’s hypothesis was wrong. Riemann’s hypothesis is about the distribution of prime numbers, and one of the problems of the Institute of General Mathematics is worth $ 1 million. The Aranson machine executes the automated solution in 744 steps (BB), which has a function similar to the Goldbach machine, and moves to higher numbers until a paradoxical pattern is found.

Of course, BB (744) is much larger than BB (27) in terms of number of rules, but working with simpler examples such as BB (5) can raise new questions in number theory that are interesting in their own right. Mathematics by name Pascal Michel In 1993, he proved that the five-dimensional Turing machine behaves similarly to the Kolatz conjecture function, which is a well-known open-ended problem in number theory. “Aranson says:

Much of the math can be encrypted with the question: Will this Turing machine stop? If you know all the numbers of busy blue dogs, you can answer all these questions.

Aranson recently used the Blue Dog Scale to measure the “threshold of ignorance” for all mathematical systems. Gلdel’s famous and incomplete theories from 1931 show that any set of basic rules that can serve as a logical and probabilistic basis for mathematics will suffer one of two fates: Rules can be inconsistent and lead to inconsistencies such as 0 = 1; ۲. Rules can be incomplete and can not prove some numerically correct theorems (for example, the fact that 2 + 2 = 4). The self-evident system of the whole of modern mathematics is known as the Zermelofrankel (ZF) set theory and has Gudley constraints. Now Aranson wants to use the busy blue dog game to identify the location of these systems.

In 2016, Aranson and his student, Adam یدیدیا, Defined a kind of Turing machine with 7,910 rules. This machine stops only when the ZF set theory is inconsistent; That is, BB (7910) is a calculation that ignores the rules of ZF theory. With these rules, one cannot prove the correctness of the result (BB (7910) and, for example, show that the result is 2 + 2 = 4, not 5.

Aurer designed a simpler machine with 748 rules. This machine stops only when the ZF is incompatible. In general, such thresholds of ignorance certainly exist, and, according to Aranson, this is a worldview that has existed since Godel, and the Blue Dog function is working on a way to implement them in real-world examples.

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