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Open AccessArticle

Adapting Logic to Physics: The Quantum-Like Eigenlogic Program

1
Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190 Gif-sur-Yvette, France
2
Conservatoire National des Arts et Métiers, AFSCET, 75003 Paris, France
*
Author to whom correspondence should be addressed.
Entropy 2020, 22(2), 139; https://doi.org/10.3390/e22020139
Received: 15 December 2019 / Revised: 20 January 2020 / Accepted: 21 January 2020 / Published: 24 January 2020
(This article belongs to the Special Issue Quantum Information Revolution: Impact to Foundations)
Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet { 0 , 1 } using projection operators to the binary alphabet { + 1 , 1 } employing reversible involution operators. Also, many-valued logical operators are synthesized, for whatever alphabet, using operator methods based on Lagrange interpolation and on the Cayley–Hamilton theorem. Considering a superposition of logical input states one gets a fuzzy logic representation where the fuzzy membership function is the quantum probability given by the Born rule. Historical parallels from Boole, Post, Poincaré and Combinatory Logic are presented in relation to probability theory, non-commutative quaternion algebra and Turing machines. An extension to first order logic is proposed inspired by Grover’s algorithm. Eigenlogic is essentially a logic of operators and its truth-table logical semantics is provided by the eigenvalue structure which is shown to be related to the universality of logical quantum gates, a fundamental role being played by non-commutativity and entanglement. View Full-Text
Keywords: probabilistic logic; quantum computing gates; operator algebra probabilistic logic; quantum computing gates; operator algebra
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Toffano, Z.; Dubois, F. Adapting Logic to Physics: The Quantum-Like Eigenlogic Program. Entropy 2020, 22, 139.

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