A Theory of Computation Based on Quantum Logic (I)

TL;DR

This paper initiates a new theory of computation based on quantum logic, examining finite automata and revealing how quantum logic's non-distributivity affects automata properties and their potential recovery through commutativity.

Contribution

It introduces a foundational framework for quantum logic-based computation, highlighting the impact of non-distributivity and proposing conditions for property validity.

Findings

Many automata properties depend on distributivity of logic

Non-distributivity causes certain properties to fail in quantum logic

Imposing commutativity can recover some classical properties

Abstract

The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. It is found that the universal validity of many properties of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic.