# Unambiguous Turing machine

In theoretical computer science, a Turing machine is a theoretical machine that is used in thought experiments to examine the abilities and limitations of computers. An unambiguous Turing machine is a special kind of non-deterministic Turing machine, which, in some sense, is similar to a deterministic Turing machine.

## Formal definition

A non-deterministic Turing machine is represented formally by a 6-tuple, ${\displaystyle M=(Q,\Sigma ,\iota ,\sqcup ,A,\delta )}$, as explained in the page non-deterministic Turing machine. An unambiguous Turing machine is a non-deterministic Turing machine such that, for all input w = a1a2 ... an, there exists at most one sequence of configurations c0,c1, ..., cm with the following conditions:

1. c0 is the initial configuration with input w
2. ci+1 is a successor of ci and
3. cm is an accepting configuration.

In other words, if w is accepted by M, there is exactly one accepting computation.

## Expressivity

A (deterministic) Turing machine is an unambiguous Turing machine. Indeed, for each input, there is exactly one computation possible.

On the one hand, unambiguous Turing machine have the same expressivity as a Turing machine. Indeed, they are a subset of non-deterministic Turing machines, which have the same expressivity as Turing machines.

On the other hand, unambiguous non-deterministic polynomial time is suspected to be strictly less expressive than non-deterministic polynomial time.

## References

Lane A. Hemaspaandra and Jorg Rothe, Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets, SIAM J. Comput., 26(3), 634–653