TuringMaschine

The Turingmaschine, or Turing machine, is a theoretical model of computation that formalizes the concept of an algorithm. Introduced by Alan Turing in 1936, it serves as a foundational tool in computability theory to study what can be computed by a mechanical procedure.

A Turingmaschine consists of several basic components. It uses an infinite tape divided into discrete cells, each holding a symbol from a finite alphabet. A read–write head can read symbols on the tape, write new symbols, and move the tape one cell to the left or right. The machine also has a finite set of states, including a start state and one or more halting states. A transition function specifies, for a given current state and tape symbol, the symbol to write, the direction to move the head, and the next state.

Operationally, a Turingmaschine processes an input by following the transition rules. If the machine reaches a halting state, it stops; otherwise, it may continue indefinitely. In a deterministic Turingmaschine, each state–symbol pair has a unique transition. Non-deterministic variants allow multiple possible transitions, though any such machine can be simulated by a deterministic one with a suitable encoding.

A universal Turingmaschine is a single machine capable of simulating any other Turingmaschine when provided with a description of the latter and its input. This concept underpins the idea of universal computation and the possibility of a general-purpose computer.

The Turingmaschine is central to the formalization of the Church–Turing thesis, which posits that any function that can be effectively calculated can be computed by a Turingmaschine. It also helps define complexity and decidability, including the halting problem, which is provably unsolvable by any Turingmaschine. Variants include multi-tape, probabilistic, and quantum extensions, all exploring different computational resources.