Editing Complexity class (section)

====Deterministic Turing machines====
{{Main|Turing machine}}
[[File:Turing machine 2b.svg|thumb|right|An illustration of a Turing machine. The "B" indicates the blank symbol.]]
A '''Turing machine''' is a mathematical model of a general computing machine. It is the most commonly used model in complexity theory, owing in large part to the fact that it is believed to be as powerful as any other model of computation and is easy to analyze mathematically. Importantly, it is believed that if there exists an algorithm that solves a particular problem then there also exists a Turing machine that solves that same problem (this is known as the [[Church–Turing thesis]]); this means that it is believed that ''every'' algorithm can be represented as a Turing machine.

Mechanically, a Turing machine (TM) manipulates symbols (generally restricted to the bits 0 and 1 to provide an intuitive connection to real-life computers) contained on an infinitely long strip of tape. The TM can read and write, one at a time, using a tape head. Operation is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; if the symbol seen is 1, change into state 17; in state 17, if the symbol seen is 0, write a 1 and change to state 6". The Turing machine starts with only the input string on its tape and blanks everywhere else. The TM accepts the input if it enters a designated accept state and rejects the input if it enters a reject state. The deterministic Turing machine (DTM) is the most basic type of Turing machine. It uses a fixed set of rules to determine its future actions (which is why it is called "[[deterministic]]").

A computational problem can then be defined in terms of a Turing machine as the set of input strings that a particular Turing machine accepts. For example, the primality problem <math>\texttt{PRIME}</math> from above is the set of strings (representing natural numbers) that a Turing machine running an algorithm that correctly [[primality test|tests for primality]] accepts. A Turing machine is said to '''recognize''' a language (recall that "problem" and "language" are largely synonymous in computability and complexity theory) if it accepts all inputs that are in the language and is said to '''decide''' a language if it additionally rejects all inputs that are not in the language (certain inputs may cause a Turing machine to run forever, so [[Recursive set|decidability]] places the additional constraint over [[Recursively enumerable set|recognizability]] that the Turing machine must halt on all inputs). A Turing machine that "solves" a problem is generally meant to mean one that decides the language.

Turing machines enable intuitive notions of "time" and "space". The [[time complexity]] of a TM on a particular input is the number of elementary steps that the Turing machine takes to reach either an accept or reject state. The [[space complexity]] is the number of cells on its tape that it uses to reach either an accept or reject state.