Extending Classical Logic for Reasoning About Quantum Systems

A logic extending classical reasoning and supporting quantum reasoning is presented. The quantum logic is obtained by applying the exogenous semantics approach to propositional logic. The design is guided by the postulates of quantum mechanics and inspired by applications in quantum computation and information. The models of the quantum logic are superpositions of classical valuations. In order to achieve a recursive axiomatization, the superpositions are taken in inner product spaces over algebraic closures of arbitrary real closed fields.

[1]  David J. Foulis,et al.  A Half-Century of Quantum Logic What Have We Learned? , 1999 .

[2]  Martín Abadi,et al.  Decidability and Expressiveness for First-Order Logics of Probability , 1994, Inf. Comput..

[3]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[4]  Cristina Sernadas,et al.  Quantum Institutions , 2006, Essays Dedicated to Joseph A. Goguen.

[5]  Samson Abramsky,et al.  A categorical quantum logic , 2006, Math. Struct. Comput. Sci..

[6]  Amílcar Sernadas,et al.  Weakly complete axiomatization of exogenous quantum propositional logic , 2005, Inf. Comput..

[7]  Amílcar Sernadas,et al.  Reasoning About Quantum Systems , 2004, JELIA.

[8]  H. Dishkant.  Semantics of the minimal logic of quantum mechanics , 1972 .

[9]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[10]  Joseph Y. Halpern.  An Analysis of First-Order Logics of Probability , 1989, IJCAI.

[11]  Dov M. Gabbay,et al.  Fibred semantics and the weaving of logics. Part 1: Modal and intuitionistic logics , 1996, Journal of Symbolic Logic.

[12]  D. S. Arnon,et al.  Algorithms in real algebraic geometry , 1988 .

[13]  N. Bohr.  II - Can Quantum-Mechanical Description of Physical Reality be Considered Complete? , 1935 .

[14]  Ronald Fagin,et al.  A logic for reasoning about probabilities , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[15]  Manas K. Patra,et al.  A Logic for Probability in Quantum Systems , 2003, CSL.

[16]  Nils J. Nilsson,et al.  Probabilistic Logic * , 2022 .

[17]  Fahiem Bacchus,et al.  Representing and reasoning with probabilistic knowledge , 1988 .

[18]  Nils J. Nilsson,et al.  Probabilistic Logic Revisited , 1993, Artif. Intell..

[19]  Roberto Giuntini,et al.  Reasoning in quantum theory , 2004 .

[20]  Manas K. Patra,et al.  Knowledge in quantum systems , 2003, TARK '03.

[21]  Rajagopal Nagarajan,et al.  QMC: A Model Checker for Quantum Systems , 2007, CAV.

[22]  J. Linnett,et al.  Quantum mechanics , 1975, Nature.

[23]  Saul A. Kripke,et al.  Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi , 1963 .

[24]  M. K. Patra.  A LOGIC FOR QUANTUM COMPUTATION AND CLASSICAL SIMULATION OF QUANTUM ALGORITHMS , 2008 .

[25]  Rohit Chadha,et al.  Reasoning about probabilistic sequential programs , 2007, Theor. Comput. Sci..

[26]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[27]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004..

[28]  Dov M. Gabbay,et al.  Handbook of Quantum Logic and Quantum Structures: Quantum Logic , 2009 .

[29]  Samson Abramsky,et al.  A categorical semantics of quantum protocols , 2004, LICS 2004.

[30]  Fahiem Bacchus,et al.  On probability distributions over possible worlds , 2013, UAI.

[31]  P. Mateus,et al.  Exogenous Quantum Logic , 2004 .

[32]  P. Mateus,et al.  Exogenous Semantics Approach to Enriching Logics , 2005 .

[33]  Bengt Jonsson,et al.  A logic for reasoning about time and reliability , 1990, Formal Aspects of Computing.

[34]  H. Dishkant,et al.  Logic of Quantum Mechanics , 1976 .

[35]  Cristina Sernadas,et al.  Fibring Logics: Past, Present and Future , 2005, We Will Show Them!.