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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

What can quantum logic and fuzzy logic teach each other?

H. John Caulfield, Luis R. Lopez

Published Jul 1, 1992• 0 citations
Publisher

Abstract

This work began with our consideration of the question: `Does fuzzy logic help us understand quantum mechanics?' In arriving at a positive answer, we had to posit a nondeterministic defuzzifying rule. We suggest here that such a rule may be useful in fuzzy operations, especially fuzzy control. It is often possible to construct an optical quantum computer which directly implements nondeterministic defuzzification by quantum indeterminacy. Thus our attempt to use fuzzy logic to understand quantum mechanics leads to a system which uses quantum mechanics to implement fuzzy logic.

DOI
10.1117/12.140115

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This work began with our consideration of the question: `Does fuzzy logic help us understand quantum mechanics?' In arriving at a positive answer, we had to posit a nondeterministic defuzzifying rule. We suggest here that such a rule may be useful in fuzzy operations, especially fuzzy control. It is often possible to construct an optical quantum computer which directly implements nondeterministic defuzzification by quantum indeterminacy. Thus our attempt to use fuzzy logic to understand quantum mechanics leads to a system which uses quantum mechanics to implement fuzzy logic.
In this article we investigate a way in which quantum computing can be used to extend the class of fuzzy sets. The core idea is to see states of a quantum register as characteristic functions of quantum fuzzy subsets of a given set. As the real unit interval is embedded in the Bloch sphere, every fuzzy set is automatically a quantum fuzzy set. However, a generic quantum fuzzy set can be seen as a (possibly entangled) superposition of many fuzzy sets at once, offering new opportunities for modeling uncertainty. After introducing the main framework of quantum fuzzy set theory, we analyze the standard operations of fuzzification and defuzzification from our viewpoint. We conclude this preliminary paper with a list of possible applications of quantum fuzzy sets to pattern recognition, as well as future directions of pure research in quantum fuzzy set theory.
In this paper we present our proposal for a fuzzy logic model based on Quantum Computing. After analyzing how logical connectives are constructed and applied in fuzzy logic, we research how Quantum Computing can provide a new model to manage fuzzy variables, mainly using superposition of states for the implementation of fuzzy information. A comparison between our proposal and several classical connectives is carried out, including negation, conjunction and disjunction. We prove that quantum fuzzy connectives present a proper behavior, manifesting the peculiarities of the quantum paradigm, which is beneficial for the representation of uncertain knowledge. We conclude on how this work establishes a model that can be further expanded for the development of quantum fuzzy reasoning.
In this paper, the formulation of Quantum Mechanics in terms of fuzzy logic and fuzzy sets is explored. A result by Pykacz, which establishes a correspondence between (quantum) logics (lattices with certain properties) and certain families of fuzzy sets, is applied to the Birkhoff–von Neumann logic, the lattice of projectors of a Hilbert space. Three cases are considered: the qubit, two qubits entangled, and a qutrit ‘nested’ inside the two entangled qubits. The membership functions of the fuzzy sets are explicitly computed and all the connectives of the fuzzy sets are interpreted as operations with these particular membership functions. In this way, a complete picture of the standard quantum logic in terms of fuzzy sets is obtained for the systems considered.