Turing machine
Turing machine is a mathematical model of computation that defines an abstract machine which manipulates symbols on a strip of tape according to a table of rules. It was first described by Alan Turing in 1936 and remains a foundational concept in computer science, particularly in the theory of computation and computability. Despite its simplicity, the Turing machine is capable of simulating any computer algorithm, provided sufficient time and memory.
History
Alan Turing introduced the Turing machine in his 1936 paper "On Computable Numbers, with an Application to the Entscheidungsproblem". The paper addressed the decision problem posed by David Hilbert, proving that a general algorithm for determining the truth or falsity of mathematical statements within a formal system cannot exist. Turing's model provided a precise definition of what it means for a function to be computable, and his halting problem proof demonstrated fundamental limits of computation.
Components and operation
A Turing machine consists of:
- An infinitely long tape divided into cells, each containing a symbol from a finite alphabet.
- A head that reads and writes symbols on the tape and can move left or right.
- A state register that stores the current state of the machine, taken from a finite set of states.
- A transition function that, given the current state and the symbol under the head, specifies the next state, the symbol to write, and the direction to move the head.
The machine operates in discrete steps, starting from an initial state and an initial tape configuration. It halts when no transition is defined for the current state and symbol. The sequence of symbols left on the tape at halting is considered the output.
Significance
The Turing machine is central to the Church–Turing thesis, which asserts that any function computable by an effective procedure is computable by a Turing machine. The concept is used to define the class of computable functions and to analyze computational complexity. Variations such as the nondeterministic Turing machine and the universal Turing machine further extend its utility. The universal Turing machine, which can simulate any other Turing machine, prefigured the stored-program computer architecture.