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Turing Machine

Overview#

Turing Machine was invented in 1936 by Alan Turing, and is a mathematical model of a general computing machine.

Turing Machine is a theoretical device that manipulates symbols contained on a strip of tape. Turing Machines are not intended as a practical computing technology, but rather as a general model of a computing machine—anything from an advanced supercomputer to a mathematician with a pencil and paper. It is believed that if a problem can be solved by an algorithm, there exists a Turing Machine that solves the problem. Indeed, this is the statement of the Church–Turing thesis. Furthermore, it is known that everything that can be computed on other models of computation known to us today, such as a RAM machine, Conway's Game of Life, cellular automata or any programming language can be computed on a Turing Machine. Since Turing Machines are easy to analyze mathematically, and are believed to be as powerful as any other model of computation, the Turing Machine is the most commonly used model in complexity theory.

Turing Machine are used to define complexity classes, such as:

  • deterministic Turing machines
  • probabilistic Turing machines
  • non-deterministic Turing machines
  • quantum Turing machines
  • symmetric Turing machines
  • alternating Turing machines
They are all equally powerful in principle, but when you have a Bounded Context (such as time or space), some of these may be more powerful than others.

A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called Randomized algorithm.

A nondeterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. A nondeterministic Turing machine uses nondeterministic algorithms. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes.

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