Paraconsistent quantum logics

María Luisa Dalla Chiara, Roberto Giuntini

Newton C. A. da Costa, Christian DE RONDE

Norihiro Kamide

In this study, a new logic called symmetric paracon-sistent quantum logic (SPQL) is introduced as a dual monose-quent system, which is a restricted and indexed sequent calculus. The logic SPQL is regarded as an extension and generalization of Dalla Chiara and Giuntini's paraconsistent quantum logic (PQL). Theorems for syntactically and semantically embedding SPQL into PQL and vice versa are proved. Additionally, cut-elimination, symmetry-elimination, contraposition-elimination, and algebraic-completeness theorems are proved for SPOL.

Richard J. Greechie, Stanley Gudder

Juan C. Agudelo, Walter Alexandre Carnielli

We propose a method to define axiomatic theories for deterministic Turing machine computations. This method, when applied to axiomatizing computations in non-deterministic Turing machines, produces (in some cases) contradictory theories, therefore trivial theories (considering classical logic as the underlying logic). Substituting in such theories the underlying logic by the paraconsistent logic LFI1 ∗ permits us to define a new model of computation which we call paraconsistent Turing machine. We show that this initial model of computation allows the simulation of important quantum computing features. In particular, it allows to simulate the quantum solution of the well-known Deutsch’s and Deutsch-Jozsa problems. However, we show that this initial model of computation does not adequately represent the notion of entangled states, a key feature in quantum computing. In this way, the construction is refined by defining a paraconsistent logic with a connective expressing entangled states in a logical fashion, and this logic is used to define a more sharpened model of paraconsistent Turing machines, better approaching the quantum computing features. Finally, we define complexity classes for the models introduced and establish some surprising relationships with classical complexity classes.

Richard J. Greechie, Stanley Gudder