Turing machines based on unsharp quantum logic

TL;DR

This paper introduces unsharp quantum logic-based Turing machines, demonstrating their increased computational power over classical Turing machines and exploring their language recognition capabilities within a lattice-structured quantum logic framework.

Contribution

It defines E-valued non-deterministic and deterministic Turing machines based on unsharp quantum logic and compares their computational power and language recognition properties.

Findings

ENTMs are more powerful than EDTMs.

Width-first recognition is generally less or equal to depth-first recognition.

ENTMs with classical states have the same power as those with quantum states.

Abstract

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.