Advertisement

Quantum Institutions

  • Carlos Caleiro
  • Paulo Mateus
  • Amilcar Sernadas
  • Cristina Sernadas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4060)

Abstract

The exogenous approach to enriching any given base logic for probabilistic and quantum reasoning is brought into the realm of institutions. The theory of institutions helps in capturing the precise relationships between the logics that are obtained, and, furthermore, helps in analyzing some of the key design decisions and opens the way to make the approach more useful and, at the same time, more abstract.

Keywords

Hilbert Space Quantum Logic Satisfaction Condition Conservative Extension Classical Propositional Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Mateus, P., Sernadas, A.: Exogenous quantum logic. In: Carnielli, W.A., Dionísio, F.M., Mateus, P. (eds.) Proceedings of CombLog’04, Workshop on Combination of Logics: Theory and Applications, 1049-001 Lisboa, Portugal, Departamento de Matemática, Instituto Superior Técnico (2004), pp. 141–149 (2004), Extended abstractGoogle Scholar
  2. 2.
    Mateus, P., Sernadas, A.: Reasoning about quantum systems. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 239–251. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  3. 3.
    Mateus, P., Sernadas, A.: Weakly complete axiomatization of exogenous quantum propositional logic. Information and Computation (in print) ArXiv math.LO/0503453Google Scholar
  4. 4.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  5. 5.
    Mateus, P., Sernadas, A., Sernadas, C.: Exogenous semantics approach to enriching logics. In: Sica, G. (ed.) Essays on the Foundations of Mathematics and Logic, Polimetrica. Advanced Studies in Mathematics and Logic, vol. 1, pp. 165–194 (2005)Google Scholar
  6. 6.
    Foulis, D.J.: A half-century of quantum logic. What have we learned? In: Quantum Structures and the Nature of Reality. Einstein Meets Magritte, vol. 7, pp. 1–36. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  7. 7.
    Chiara, M.L.D., Giuntini, R., Greechie, R.: Reasoning in Quantum Theory. Kluwer Academic Publishers, Dordrecht (2004)MATHGoogle Scholar
  8. 8.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Annals of Mathematics 37, 823–843 (1936)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Nilsson, N.J.: Probabilistic logic. Artificial Intelligence 28, 71–87 (1986)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Nilsson, N.J.: Probabilistic logic revisited. Artificial Intelligence 59, 39–42 (1993)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Bacchus, F.: Representing and Reasoning with Probabilistic Knowledge. In: MIT Press Series in Artificial Intelligence, MIT Press, Cambridge (1990)Google Scholar
  12. 12.
    Bacchus, F.: On probability distributions over possible worlds. In: Uncertainty in Artificial Intelligence. Machine Intelligence and Pattern Recognition, vol. 4(9), pp. 217–226. North-Holland, Amsterdam (1990)Google Scholar
  13. 13.
    Fagin, R., Halpern, J.Y., Megiddo, N.: A logic for reasoning about probabilities. Information and Computation 87, 78–128 (1990)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dishkant, H.: Semantics of the minimal logic of quantum mechanics. Studia Logica 30, 23–32 (1972)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Goguen, J., Burstall, R.: A study in the foundations of programming methodology: specifications, institutions, charters and parchments. In: Poigné, A., Pitt, D.H., Rydeheard, D.E., Abramsky, S. (eds.) Category Theory and Computer Programming. LNCS, vol. 240, pp. 313–333. Springer, Heidelberg (1986)Google Scholar
  16. 16.
    Goguen, J., Burstall, R.: Institutions: abstract model theory for specification and programming. Journal of the ACM 39, 95–146 (1992)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Goguen, J., Roşu, G.: Institution morphisms. Formal Aspects of Computing 13, 274–307 (2002)MATHCrossRefGoogle Scholar
  18. 18.
    Meseguer, J.: General logics. In: Proceedings of the Logic Colloquium 1987, pp. 275–329. North- Holland, Amsterdam (1989)CrossRefGoogle Scholar
  19. 19.
    Tarlecki, A.: Moving between logical systems. In: Haveraaen, M., Dahl, O.-J., Owe, O. (eds.) Abstract Data Types 1995 and COMPASS 1995. LNCS, vol. 1130, pp. 478–502. Springer, Heidelberg (1996)Google Scholar
  20. 20.
    Mossakowski, T.: Different types of arrow between logical frameworks. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 158–169. Springer, Heidelberg (1996)Google Scholar
  21. 21.
    Cerioli, M., Meseguer, J.: May I borrow your logic (Transporting logical structures along maps). Theoretical Computer Science 173, 311–347 (1997)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Bridges, D.S.: Computability. Graduate Texts in Mathematics, vol. 146. Springer, Heidelberg (1994)MATHGoogle Scholar
  23. 23.
    Caleiro, C., Carnielli, W.A., Coniglio, M.E., Marcos, J.: Two’s company: “The humbug of many logical values”. In: Béziau, J.Y. (ed.) Logica Universalis, pp. 169–189. Birkhäuser, Basel (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Carlos Caleiro
    • 1
  • Paulo Mateus
    • 1
  • Amilcar Sernadas
    • 1
  • Cristina Sernadas
    • 1
  1. 1.CLC, Department of MathematicsISTLisbonPortugal

Personalised recommendations