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Title
Abstract
Introduction
Conclusion
References

Algebraic Logic and Computer Science

Profile image of Manuel MartinsManuel Martins

2011

Abstract

Algebraic Logic and Computer Science Jacinta Poças, Manuel Martins, and Carlos Caleiro 1 Dep. Mathematics, IST TU Lisbon, Portugal 2 Dep. Mathematics, U Aveiro, Portugal 3 SQIG Instituto de Telecomunicações Abstract. Abstract algebraic logic (AAL) is a branch of logic that Abstract algebraic logic (AAL) is a branch of logic that uses universal algebra to study the properties of logical systems by associating them with representative classes of algebras, thus generalizing the Lindenbaum-Tarski process that leads to linking Boolean algebras with classical propositional logic. The theory of AAL then classifies logical systems and their metaproperties, systematically, along bridge theorems that relate them with properties of the associated algebras. Recently, concepts of behavioral algebraic specification have influenced the development of the behavioral approach to AAL. Namely, the notion of behavioral equivalence, imported from computer science, has been used to weaken the traditional...

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References (20)

  1. Andréka, H., Németi, I., Sain, I.: Algebraic logic. in D. M. Gabbay and F. Guenther, (eds.). Handbook of Philosophical Logic, Dordrecht, Kluwer Academic Publishers. 2 (2001) 133-247.
  2. Béziau, J.-Y.: Classical Negation can be Expressed by One of its Halves. Logic Journal of the IGPL. (1999) 145-151.
  3. Blok, W.J., Pigozzi, D.: Algebraizable logics. Memoirs of the American Mathe- matical Society. 396 (1989).
  4. Blok, W.J. and Pigozzi, D.:Algebraic semantics for universal Horn logic without equality. Universal algebra and quasigroup theory, Lect. Conf., Jadwisin/Pol. 1989, Res. Expo. Math. 19 (1992) 1-56.
  5. Blok, W.J., J.: Algebraic semantics for deductive systems. Studia Logica. 74 (2003) 153-180.
  6. Caleiro, C., Gonçalves, R.: On the algebraization of many-sorted logics. in J. Fi- adeiro and P.Y. Schobbens, (eds.), Recent Trends in Algebraic Development Tech- niques -Selected Papers. in Lecture Notes in Computer Science. 4409 (2007) 21-36.
  7. Caleiro, C., Gonçalves, R.: Behavioral algebraization of da Costa's C-systems. Journal of Applied Non-Classical Logics. 19(2) (2009) 127-148.
  8. Caleiro, C., Gonçalves, R.: Abstract valuation semantics. Submitted for publica- tion.
  9. Caleiro, C., Gonçalves, R., Martins, M.A.: Behavioral algebraization of logics. in Studia Logica. 91 (2009) 63-111.
  10. Czelakowski, J.: Protoalgebraic Logics. Trends in logic, Studia Logica, Kluwer Academic Publishers (2001).
  11. Czelakowski, J., Pigozzi, D.: Amalgamation and interpolation in abstract alge- braic logic. in X. Caicedo and C.H. Montenegro, (eds.), Models, algebras and proofs. in Lecture Notes in Pure Appl. Math. 203 (1999) 187-265.
  12. L. Descalço, A. Madeira, and M. A. Martins. Applying abstract algebraic logic to classic automata theory: an exercise. In F. Ferreira, Guerra H., and E. Mayordomo, editors, Programs, Proofs and Processes; Computability in Europe Cie 2010, (2010) 146-157.
  13. Gonçalves, R.: Behavioral algebraization of logics. PhD Thesis. Technical Uni- versity of Lisbon, Instituto Superior Técnico (2008).
  14. J. Malinowski. Equivalence in intensional logic, IFIS PAN (1990). Available at http://www.home.umk.pl/∼jacekm/arlint.pdf.
  15. Martins, M.A.: Closure properties for the class of behavioral models. Theor. Comput. Sci., 379 (2007) 53-83.
  16. M. A. Martins. On the behavioral equivalence between k-data structures. Comp. J., 51(2) (2008) 181-191.
  17. Martins, M.A., Madeira, A. and Barbosa, L.S.: Refinement by interpretation in a general setting. In J. Derrick E. Boiten and S. Reeves, (eds.). Proc. Refinement Workshop, Elsevier (2009) 105-121.
  18. Martins, M.A. and Pigozzi, D.: Behavioural reasoning for conditional equations. Math. Struct. in Comput. Sci. 17 (2007) 1075-1113.
  19. Poças, J.: Leibniz Hierarchy. MSc Thesis. University of Aveiro (2009).
  20. Tarski, A.: Logic, semantics, and metamathematics (2nd Edition). Hackett Pub Co., Indianapolis, Indian (1983).

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