Turingmachine

A Turing machine is an abstract mathematical model of computation introduced by Alan Turing in 1936. It is designed to formalize the notion of an algorithm and to study what problems can be computed in principle. A Turing machine consists of a tape that is conceptually infinite in both directions, divided into discrete cells that hold symbols from a finite alphabet. A read–write head can read the symbol in the current cell, write a new symbol, and move the head one cell to the left or right. The machine also has a finite control with a set of states and a transition function that determines the next action based on the current state and the symbol being read.

Operation proceeds by following the transition rules to write symbols, move the head, and switch states. The machine halts when it reaches a designated halting state or when no transition is defined for the current configuration. In the deterministic version, each state–symbol pair has a single defined action; non-deterministic variants allow multiple possible actions.

Significance lies in its role as a foundational model for computability theory. Through the Church–Turing thesis, it is posited that any function that can be computed by an algorithm can be computed by a Turing machine. A universal Turing machine can simulate any other Turing machine on any input, formalizing the concept of programmable computers.

Limitations include its abstract nature and focus on feasibility rather than efficiency. Not all problems are solvable by Turing machines; notably, the halting problem is undecidable. Variants such as multi-tape, non-deterministic, and probabilistic Turing machines explore different computational resources and capabilities. The model has profoundly influenced the theory of computation and the design of real-world computers.