Unambiguous Turing machine

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 every input ${\displaystyle w=a_{1},a_{2},...,a_{n}}$, there exists at most one sequence of configurations ${\displaystyle c_{0},c_{1},...,c_{m}}$ with the following conditions:

1. ${\displaystyle c_{0}}$ is the initial configuration with input ${\displaystyle w}$
2. ${\displaystyle c_{i+1}}$ is a successor of ${\displaystyle c_{i}}$ and
3. ${\displaystyle c_{m}}$ is an accepting configuration.

In other words, if ${\displaystyle w}$ is accepted by ${\displaystyle M}$, there is exactly one accepting computation.

Every deterministic Turing machine is an unambiguous Turing machine, as for each input, there is exactly one computation possible. Unambiguous Turing machines have the same expressivity as a Turing machines. 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 (potentially ambiguous) non-deterministic polynomial time.

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