Turing machine, also known as Turing Computing, Turing Computer, is an abstract computing model proposed by mathematician Alan Maisson Turing (1912~1954), which abstracts the process of using paper and pen for mathematical operations. , by a virtual machine instead of people to perform mathematical operations.
The so-called Turing machine refers to an abstract machine, which has an infinitely long strip of paper. The paper strip is divided into small squares, each square has a different color. There is a machine head that moves around the tape. The machine head has a set of internal states and some fixed programs. At each moment, the machine head reads a square information from the current tape, and then uses its internal state to find the program table, output the information according to the program to the paper square, and convert its internal state, then Move.
As an abstract tool for studying the general nature of computing, it replaces people with mathematical operations and has the following effects:
1. As a language acceptor: the language accepted by M is recorded as L(M), which is a collection of such strings in Σ. When these strings are placed on the band of M, M is in the q0 state and M The leader is in the leftmost unit. These strings can cause M to enter a final state and stop. Given a Turing machine M that recognizes the language L, it is generally assumed that when the input is accepted, M is down, ie there is no next action. However, for unacceptable strings, M may never stop. The linguists accepted by the Turing machine are called recursive enumerable languages. A recursive collection is a subclass of a recursive enumerable collection that can always be accepted by a Turing machine that can stop all inputs.
2. As an integer function computer: The function calculated by the Turing machine is called a partial recursive function. In a sense, a partial recursive function is similar to a recursive enumerable language. Because the Turing machine that calculates it may not stop on a given input. The fully recursive function corresponds to recursive language education. Because it can be calculated by the Turing machine that always stops.
3. As a language generator: Let M be a multi-band Turing machine, which uses a band as an output band. On this band, the symbol can not be rewritten once it is written. The leader of the output tape cannot be moved to the left. Suppose that on the output tape, M writes some strings of a certain word table and separates them with separators. The set of strings finally printed on the output tape is called the language generated by M, which is denoted as G ( M), G(M) Hey. If L is a language generated by a Turing machine, then L is a recursive enumerable set and vice versa.
The birth of any scientific ideas and scientific concepts has its background, and there are often many fascinating stories in the background. The theory of computation dates back to 1900, when the famous great mathematician Hilbert presented the famous 23 mathematics problems to the international mathematics community at the conference of mathematicians at the turn of the century. The tenth question is this: Is there a finite, mechanical step that can determine whether the Diophantine equation has a solution? Here is the question of a limited, mechanical proof step, which in today's words is an algorithm. But at the time, people still didn't know what the "algorithm" was. In fact, many problems in the field of mathematics at the time were closely related to the "algorithm". Therefore, the definition of "algorithm" in science is coming to the fore. Then in the 1930s, two people finally proposed a method to accurately define the algorithm. One person is Turing, and the other is Church. The intuitive image of the Turing machine model proposed by Turing was quickly accepted by everyone.
I don't know if you have heard of the name Turing. Some people may know Newton, know Einstein, and even know von Neumann, but don't know Turing. However, Turing's contribution is no less than that of these masters of science. Turing's greatest contribution is to make the basic and profound concept of the algorithm clear with his Turing machine model. It is because of the theoretical foundation laid by Turing that it is possible to invent the greatest invention ever since the 20th century and even in human history: computers. Therefore, people call Turing: the father of computer theory.
The era of Turing's life experienced the Second World War. During World War II, he successfully deciphered the German password for the British government and thus made outstanding contributions to the UK. In fact, it was precisely because of World War II that the British government was willing to pay for Turing to make the most primitive computer. Of course, this kind of computer is specially used to decipher passwords, not the general-purpose computers we use now. (There is a film called "Password Love" English name "enigma" is based on the story of Turing's deciphering German password at the time, everyone is interested to find it.)
Turing is a very weird person who only likes to study with one person and does not like to communicate with others. And he is said to be a homosexual. To know that in the United Kingdom at the time, homosexual behavior was a big deal. Finally, he committed suicide when his career had just reached the top. To commemorate this great scholar, the computer industry has established the highest honor award: the ACM Turing Award.
The creation of the Turing machine laid the foundation for modern digital computers on the one hand (know that von Neumann designed the first computer according to Turing's vision). On the other hand, according to the basic simplicity of the Turing machine, we can also see what the limit is computable. In other words, in fact, the power of the computer is limited in principle. Please note that the limitation of the computer here is not to say that it can not eat, sweep the ground and other hardware limits, but from the perspective of information processing, the computer still has limits. This is the shutdown problem of the Turing machine. This problem is even more important in Turing's view. In his papers of the year, he actually proposed the Turing machine model to "speak with his hands".
Turing machine basic ideaTuring's basic idea is to use machines to simulate the process of people using paper and pencil for mathematical operations. He sees this process as two simple actions:
Write or erase a symbol on paper;
Move attention from one position of the paper to another;
At each stage, the person decides the next action, relying on the symbol of a certain position on the paper that the person is currently concerned about and the state of the person's current thinking.
To simulate this kind of computing process, Turing constructs a hypothetical machine consisting of the following parts:
1. An infinitely long tape TAPE. The tape is divided into small grids one after the other, each grid containing a symbol from a finite alphabet, and a special symbol in the alphabet indicates a blank. The grid on the tape is numbered 0, 1, 2, from left to right. .. , the right end of the tape can be extended infinitely.
2. A read/write head HEAD. The head can be moved left and right on the tape, it can read the symbols on the currently pointed grid and can change the symbols on the current grid.
3. A set of control rules TABLE. It determines the next action of the read/write head based on the current state of the machine and the symbol on the grid pointed by the current read/write head, and changes the value of the status register to bring the machine into a new state.
4. A status register. It is used to save the current state of the Turing machine. The number of all possible states of the Turing machine is limited and there is a special state called the shutdown state. See the shutdown issue.
Note that every part of the machine is limited, but it has a potentially infinitely long strip of paper, so this machine is just an ideal device. Turing believes that such a machine can simulate any calculation process that humans can perform.
In some models, the head moves along a fixed strip of paper. The instruction (q1) to be executed is displayed in the head. The "blank" tapes in this model are all zero. The shaded squares, including the white space scanned by the head, the squares labeled 1, 1, B, and the head and head symbols form the system state. (Drawn by Minsky (1967) p.121).
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