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this turing machine should run forever unless maths is wrong

by:QY Precision      2019-09-07
If a new computer program stops running, hundreds of Fifty Years of Mathematics by Jacob Aaron will prove to be wrong.
Thankfully, this is unlikely to happen, but the code behind it is testing the limits of the Math Field.
The program is an analog Turing machine and is a computational mathematical model created by code breaker Alan Turning.
In 1936 he showed that the actions of any computer algorithm can be imitated by a simple machine that reads and writes 0 and 1, or indicates, on an infinite length of tape by processing a set of states.
The more complex the algorithm is, the more states the machine needs.
Now, Scott Allen and Adam yadidia of MIT have created three Turing machines whose actions are intertwined with deep problems in mathematics.
This includes proof of 150-year-
The old Riemann hypothesis-the idea of controlling prime patterns.
Turing\'s machines have long been used to study these problems.
Their origins lie in a series of philosophical revelations that shook the world of mathematics in the 1930 th century.
First of all, Kurt Godel proves that some mathematical statements can never be proved to be true or false-they are not determinable.
He basically created the mathematical version of the phrase \"this sentence is false\": logical brain --
A tornado that contradicts itself.
Godel\'s assertion has a get-out clause.
If you change the basic assumption of establishing proof-axiom-you can make the problem determinable.
However, this will still leave other problems that cannot be decided.
This means that there is no axiom that will allow you to prove everything.
According to Godel\'s argument, Turing proved that there must be some Turing machine\'s behavior that cannot be predicted under the standard axiom-known as Zermelo --
Frenkel set theory with selection axiom (C)
What\'s more interesting is that ZFC supports most of the math.
But we don\'t know how complicated they will be.
Now, Yedidia and phononson have created a Turing machine for 7918 states with this property.
They named it \"Z \".
\"We\'re trying to make it concrete and say how many states it needs before you get into this unproven abyss? \"Alan Sun.
Terence Tao of the University of California, Los Angeles, said the couple simulated Z on a computer, but it was small enough to theoretically be built as a physical device.
\"If a person opened such a physical machine at that time, we believe what will happen is that it will run indefinitely,\" he said . \" Let\'s say you ignore the wear or energy needs of the body.
Z is designed to use its 7918 instructions in a loop forever, but it will prove that ZFC is inconsistent if it finally stops.
Mathematicians don\'t panic too much, though-they can simply turn to a slightly more powerful set of axiom.
Such an axiom already exists and can be used to prove the behavior of Z, but there is no benefit in doing so, because there will always be a Turing machine more than any axiom.
\"One can think that any given axiom system is like a computer with limited memory or processing power,\" Tao said . \".
\"People can switch to a computer that stores more, but no matter how much storage the computer has, there will still be some tasks that are out of their reach.
But ononson and Yedidia have created two more machines, which may give mathematicians more pause.
It is only when two well-known mathematical problems, which have long been considered true, are actually false, that these problems stop.
These are Goldbach\'s conjectures, which point out that each even number greater than 2 is a sum of two prime numbers, while Riemann assumes that all prime numbers follow a certain pattern.
The latter forms the basis of the modern number theory part, which, if unlikely, would be a major subversion.
In fact, to prove that these issues are wrong, the couple have no intention of running their Turing machine indefinitely.
Lance Fortnow of Georgia Tech in Atlanta says this is not a particularly effective way to solve this problem.
Expressing mathematical problems as Turing machines has different practical benefits: it helps to calculate how complex they are.
The Goldbach machine has 4888 states, the Riemann machine has 5372 states, and the Z machine has 7918 states, indicating that the ZFC problem is the most complex of these three states.
\"It will be in line with the intuition that most people have about such things,\" says dentonson . \".
Yedidia has put his code online and mathematicians may now be racing to narrow down the size of these Turing machines and push them to the limit.
There has already been a reviewer on the blog at phononson who claims to have created a 31-
While the couple has not verified this yet, the National Goldbach machine.
Fortnow says the actual size of the Turing machine does not matter.
\"The newspaper tells us that we can have a very compressed description that is beyond the ZFC\'s capabilities, but even if they are compressed more, he says: \"It won\'t give us too much pause on the basics of mathematics. \".
But ononson said that further narrowing down Z can say something interesting about the limitations of the mathematical foundation-things that Godel and Turing might want to know.
\"They might say \'Fine, but can you get 800 states? \'?
How about 80 states?
\"\" Said arrenson.
\"I wonder if there\'s 10-
The behavior is independent of the state machine of ZFC.
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