In this paper, we consider Turing machines based on unsharp quantum logic.
For a lattice-ordered quantum multiple-valued (MV) algebra E, we introduce
E-valued non-deterministic Turing machines (ENTMs) and E-valued deterministic
Turing machines (EDTMs). We discuss different E-valued recursively enumerable
languages from width-first and depth-first recognition. We find that
width-first recognition is equal to or less than depth-first recognition in
general. The equivalence requires an underlying E value lattice to degenerate
into an MV algebra. We also study variants of ENTMs. ENTMs with a classical
initial state and ENTMs with a classical final state have the same power as
ENTMs with quantum initial and final states. In particular, the latter can be
simulated by ENTMs with classical transitions under a certain condition. Using
these findings, we prove that ENTMs are not equivalent to EDTMs and that ENTMs
are more powerful than EDTMs. This is a notable difference from the classical
Turing machines.Comment: In Proceedings QPL 2011, arXiv:1210.029