Turing machine
If a Turing machine has a nondeterministic transition function there can be multiple transitions from one state to many others when reading a character. Again this does not enhance the power of Turing machines. However nondeterministic Turing machines (as they are called then) may possibly decrease the computation time by a strong amount. This question is covered in the P versus NP-discussion and is not solved yet. Most scientists assume however that nondeterministic machines can work much faster on certain problems.