In theoretical computer science, a nondeterministic Turing machine (NTM) is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations. That is, an NTM's next state is not completely determined by its action and the current symbol it sees, unlike a deterministic Turing machine.

NTMs are sometimes used in thought experiments to examine the abilities and limits of computers. One of the most important open problems in theoretical computer science is the P versus NP problem, which (among other equivalent formulations) concerns the question of how difficult it is to simulate nondeterministic computation with a deterministic computer.

Background

In essence, a Turing machine is imagined to be a simple computer that reads and writes symbols one at a time on an endless tape by strictly following a set of rules. It determines what action it should perform next according to its internal state and what symbol it currently sees. An example of one of a Turing Machine's rules might thus be: "If you are in state 2 and you see an 'A', then change it to 'B', move left, and change to state 3."

Intuition

Comparison of deterministic and nondeterministic computation

In contrast to a deterministic Turing machine, in a nondeterministic Turing machine (NTM) the set of rules may prescribe more than one action to be performed for any given situation. For example, an X on the tape in state 3 might allow the NTM to:

• Write a Y, move right, and switch to state 5

or

• Write an X, move left, and stay in state 3.

Definition

A nondeterministic Turing machine can be formally defined as a six-tuple ${\displaystyle M=(Q,\Sigma ,\iota ,\sqcup ,A,\delta )}$, where

• ${\displaystyle Q}$ is a finite set of states
• ${\displaystyle \Sigma }$ is a finite set of symbols (the tape alphabet)
• ${\displaystyle \iota \in Q}$ is the initial state
• ${\displaystyle \sqcup \in \Sigma }$ is the blank symbol
• ${\displaystyle A\subseteq Q}$ is the set of accepting (final) states
• ${\displaystyle \delta \subseteq \left(Q\backslash A\times \Sigma \right)\times \left(Q\times \Sigma \times \{L,S,R\}\right)}$ is a relation on states and symbols called the transition relation. ${\displaystyle L}$ is the movement to the left, ${\displaystyle S}$ is no movement, and ${\displaystyle R}$ is the movement to the right.

The difference with a standard (deterministic) Turing machine is that, for deterministic Turing machines, the transition relation is a function rather than just a relation.

Configurations and the yields relation on configurations, which describes the possible actions of the Turing machine given any possible contents of the tape, are as for standard Turing machines, except that the yields relation is no longer single-valued. (If the machine is deterministic, the possible computations are all prefixes of a single, possibly infinite, path.)

The input for an NTM is provided in the same manner as for a deterministic Turing machine: the machine is started in the configuration in which the tape head is on the first character of the string (if any), and the tape is all blank otherwise.

An NTM accepts an input string if and only if at least one of the possible computational paths starting from that string puts the machine into an accepting state. When simulating the many branching paths of an NTM on a deterministic machine, we can stop the entire simulation as soon as any branch reaches an accepting state.

Computational equivalence with DTMs

Any computational problem that can be solved by a DTM can also be solved by a NTM, and vice versa. However, it is believed that in general the time complexity may not be the same.

Bounded nondeterminism

An NTM has the property of bounded nondeterminism. That is, if an NTM always halts on a given input tape T then it halts in a bounded number of steps, and therefore can only have a bounded number of possible configurations.

Comparison with quantum computers

The suspected shape of the range of problems solvable by quantum computers in polynomial time (BQP). Note that the figure suggests ${\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}}$ and ${\displaystyle {\mathsf {NP}}\neq {\mathsf {PSPACE}}}$. If this is not true then the figure should look different.

Because quantum computers use quantum bits, which can be in superpositions of states, rather than conventional bits, there is sometimes a misconception that quantum computers are NTMs.^[4] However, it is believed by experts (but has not been proven) that the power of quantum computers is, in fact, incomparable to that of NTMs; that is, problems likely exist that an NTM could efficiently solve that a quantum computer cannot and vice versa.^[5] In particular, it is likely that NP-complete problems are solvable by NTMs but not by quantum computers in polynomial time.

Intuitively speaking, while a quantum computer can indeed be in a superposition state corresponding to all possible computational branches having been executed at the same time (similar to an NTM), the final measurement will collapse the quantum computer into a randomly selected branch. This branch then does not, in general, represent the sought-for solution, unlike the NTM, which is allowed to pick the right solution among the exponentially many branches.

See also

• Probabilistic Turing machine

References

External links

• C++ Simulator of a Nondeterministic Multitape Turing Machine (free software).
• C++ Simulator of a Nondeterministic Multitape Turing Machine download link from sourceforge.net