Turing machine


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How did Alan Turing came up with the idea of a Turing machine? Imagine you have a piece of paper with a grid where you solve math. You try to solve 2 + 3 = x. You write down every symbol in a square. Because 2D grid is too complicated we have only 1D grid paper where you can write one symbol in each square with unlimited length left and right. Today's computers store the algorithms in the same memory as the input data and result data. So we need to know the algorithm how to solve math. Math is just counting: II + III = IIIII. The algorithm looks like this: a + b = c, count(a) + count(b) = count(c). So we write down the algorithm on the same piece of paper, the 1D grid. We write down our input: 2 + 3 = x. An we see the algorithm: count(a) + count(b) = count(c). If the numbers were represented by matchsticks we could just count the match sticks. Every counting step is a state and the Turing machine moves to the next step by increasing the counter by one. It does it in a simple loop. Otherwise we need a lookup-table where all addition results are stored and just read the result from the table. This is also how monkeys calculate. We learn our facts, addition table and multiplication table, by heart. Than we use other algorithms to solve problems with bigger numbers. Intermediate results in addition with bigger numbers are written on the same piece of paper. This is like the working memory in a computer (RAM).

The Turing machine can also solve logic problems and deal with formal grammars. Formal grammars are a close concept to natural language. You can solve logical problems with resolution algorithm and you can determine if a string of characters is part of a language or not. The Chomsky hierarchy describes that there are languages which are less complex, and can be described with machines less powerful than Turing machines. A Turing machine can not calculate Pi with maximum precision. Because it is by design a machine in the field of discrete mathematics, which excludes Euclid geometry and calculus. It can not solve problems exactly where it needs to count to infinity. Symbols, words, sentences, stories and languages are part of discrete mathematics. Logic is also part of discrete mathematics, but not fuzzy logic. What is a fuzzy Turing machine? A geometric drawing machine? A human? Computers store and work with information in binary code. Binary code is part of discrete mathematics, for example integers.


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pseudocode:

string[] map = ["0", "I", "II", "III"];

string solvemath = "2+3";

solvemath = strreplace(1, map[1]);

solvemath = strreplace(2, map[2]);

solvemath = strreplace(3, map[3]);

solvemath = strreplace(+, '' ");

x = 0;

foreach(solvemath as character) {

x = x + 1;

};

print(x);







definition of an algorithm:


 * 1) The instruction for the algorithm needs to be describable in a finite amount of text. (finite)
 * 2) Every step of the algorithm must be executable. (executability)
 * 3) in each step only a finite amount of memory is allowed to be used. (dynamic finite, space complexity)
 * 4) There must be a finite amount of steps, so the algorithm can complete. (termination)
 * 5) The algorithm leads to the same results if the same input is used. (determined)
 * 6) In every step the next step must be known. (deterministic)

Turing machines do not understand and can not draw circles. The calculation of the number Pi with maximum precision will never halt. With this we can conclude that the number Pi does not really exist. The only thing that exists is an approximation of n-regular shapes like a pentagon, which we use to approximate the circumference of a circle. [https://en.wikipedia.org/wiki/Noli_turbare_circulos_meos! Noli turbare circulos meos!]

An analog computer with a compass can draw a circle faster than the calculation of the number Pi done by a Turing machine. For example a geometric drawing machine or some other machine with circles as parts, like electric generators and electric motors.

Since Pi is a convergentt infinite seires, you can not write it down. If you can not write down a number, the number does not exist. The number Pi does not really exist. Transcendent nuimbers do not really exist. We can only approximate them.