Wednesday, September 28, 2016

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Julia - Gödel 7

3. The Turing Machine

But on the ground, there is a problem, which is looming in the thoughts of the participants: they see problems everywhere, and there seems to be nothing to relate them. It is a fog that no one can see their way out of. A large part of the problem is that no one realizes that the mechanics of a new generation are fundamentally different from those in the past. This was the same with light and electricity, because no one knew what the difference was. And once it was found it took 50 years to discover the means by which it was enforced by nature. Now of course the Higgs will be known to all, but it was not that way for decades.

In the information space, the idea that it was a machine was not even in people's mind. Except one, who had been working on cleaning up the mathematics of Gödel, and he discovered a seemingly bizarre creation. Remember that no one had thought of it as a machine, even though the concept had turned up in Ancient Greece, in Renaissance France, and in Victorian England. But each time it had been dismissed, and the machines slumbered quietly until someone picked up the pieces.4

The man of course was Turing, who was at the right place with the right brain power. Because there were people who saw the signs and ignored them. As Winston Churchill said, they hit upon the truth and get up on their way as if nothing happened. This included Included Churchill, because Turing was a homosexual, which at the time was illegal, and immoral, and forced by people who were – many of them – homosexual. Such are the days the days are made of, and we can necessarily hope that we will go forward rather than backward (Because not only being homosexual is genetic, but it is likely that a revulsion against homosexuality is also genetic, But spreads out from there.)

But Turing was working on the machines, and saw a formula, which would work: an unlimited memory, a scanned symbol, and a simple set of instructions. And that is all that needs to be accomplished. All of the rest of the computer hardware that we possess is towards making it run better. He invented this in 1936, a few years after the GN was invented. Almost nobody else even thought of a machine for this problem.

But what this did is simply astonishing, partially because it solves the riddle of the Enigma – whose German scientists thought it to be unbreakable. Yet when the machine was perfected, that is a real machine, it was able to break things on and enigma code machine almost in real time. But it was a problem known to David Hilbert, and put on his list in 1900, called, appropriately enough, the “Hilbert's Problems”. Thus the uncrackable problem was indeed cracked by a simple machine in real time – though not in theory, because it was not possible. Remember that the mathematics behind this question was quite exact: first of all was the mathematics complete, second of all was the mathematics consistent, and third of all was the mathematics decidable?

Then on a day in Grantchester Turing had what he needed, because he knew that this problem was related to a universal machine. Once he had that concept, it was just working out what the machine had to do. And Turing had been the one man who was determined to see that a typewriter was far more than a mechanical device to produce letters.

All that was to be done is calculating the equivalent of Ohms. But again remember that the people at the time did not see this at all, because what they thought of as problems – rather than thinking of solutions – gripped their minds tightly. So we now move on to Nash, and one of the most brilliant papers of all time, and one of the most surprising.

So even as you play with Google, realize there is a long fight from 1900 when Hilbert enunciated the question, until you find what you are looking for by typing in a view words into a box.

Of course there are problems with the Turing machine – because of course it in his original form is not quite complete. But the problems are countable problems, and people throw themselves at solving them. Just recently, a man by the name of Manolis Kamvysselis granted his way to a complete schematic of the Universal Turing Machine – and leaves a copy for others to look at. The core of the machine is expressed in lines of code:

;; (#(_ _ _ _ _ _ _ 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s1 16)
;; (#(_ _ _ _ _ _ _ 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s1 7)
;; (#(_ _ _ _ _ _ _ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 8)
;; (#(_ _ _ _ _ _ _ 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 9)
;; (#(_ _ _ _ _ _ _ 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 10)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 11)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 12)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 13)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 14)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 15)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 16)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 17)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 -  -  _ _) s2 18)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 -  -  _ _) s2 19)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 -  -  _ _) s2 20)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 -  -  _ _) s2 21)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 -  -  _ _) s2 22)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 -  -  _ _) s2 23)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 -  -  _ _) s2 24)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -  -  _ _) s2 25)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -  -  _ _) s2 26)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -  -  _ _) s3 25)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -  -  _ _) s3 7)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -  -  _ _) s4 8)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -  -  _ _) h 7)
;; (#(_ _ _ _ _ _ _ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -  -  _ _) h 7)
Code responding to an idea of code. There is more written here then the author suspects.
So how was the Turing Machine set up? Since this is relatively one of the easiest answers, I do not need to do anything but explain what has already come to pass. Lots of people have described the Turing machine.

It Starts with the idea of a non-empty set of states. He then introduces the idea of a tape, like a recorded tape, which has a finite number of non-empty set of symbols on it. Remember when he did this this was a new concept, and for all practical purposes only the Germans were actually doing anything with them. But then that is all right because Gödel it was German. And touring was working on the problems that Gödel had set up.
Even needed a blank symbol, which was important because only the blank symbol could be unlimited, all of the rest of the symbols had to be limited. Even went on to say that the tape was the only source of input.
You then of course needs to translate the input symbols in two output symbols, and he does so by introducing the idea that the tape can move left, or right – though no motion at all can be represented as a shift zero. He then begins to work on what is called a busy beaver, which takes an input, and translates this into the output state. It has only 7 operations:


It can move left or right, or stop.
It can write 1 or 0.
it can read 1 or 0.
it can read the initial state.
It can halt
then it has the major trick, of computing the next operative in a table, which is the basis of the machine. It takes the input, which remember is only one or zero, and translates that in two a three digit state. Now , remember, most people want to get out of this current state as quickly as possible, they want to move back to the two rather than the three, and this table is the key.
A B C
Tape Write Move Next Write Move Next Write Move Next
0 1 R B 1 L A 1 L B
1 1 L C 1 R B 1 R HALT
You can see that want to get to 1 C – because that is the state where it halts. Now most of the time, programmers want to make it easier to build a machine, and you can look up the various topics that they consider on your own. But we are interested in the purely mathematical problems that this contains, if you want to go into computer science, there are many individuals who can take you from here. But there are few who wish to lead you into the realm of the purely mathematical, and most of them are interested in cleaning up the small details - which if you have not gotten it - these are large, small details.
Remember that recursion is already here, you can pick either the machine level, the translation to the operating system, or you can go up the ladder and work on the scripting language.

Machine i686
Operating System C, Go
User space C++,C#
Scripting language Ruby, Cassandra, Perl, Python

and so on...

But there is one point left to consider. If a Gödel is the number of states, and the Turning is the means to operate on these states, what confines the operating system? At which point we must reach the limiting factor, which brings us to Nash.


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