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Craig Interpolation in the Presence of Unreliable Connectives

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Abstract

Arrow and turnstile interpolations are investigated in UCL [introduced by Sernadas et al. (J Logic Comput, 2013)], a logic that is a complete extension of classical propositional logic for reasoning about connectives that only behave as expected with a given probability. Arrow interpolation is shown to hold in general and turnstile interpolation is established under some provisos.

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References

  1. Areces C., Blackburn P., Marx M.: Repairing the interpolation theorem in quantified modal logic. Ann. Pure Appl. Logic 124(1-3), 287–299 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Areces, C., Hoogland, E., de Jongh, D.: Interpolation, definability and fixed points in interpretability logics. In: Advances in Modal Logic, vol. 119 of CSLI Lecture Notes, pp. 35–58. CSLI Publications, Stanford (2001)

  3. Beklemishev, L.D.: On the Craig interpolation and the fixed point properties for GLP. In: Proofs, Categories and Computations, vol. 13 of Tributes, pp. 49–60. College Publications, London (2010)

  4. Beth, E.W.: On Padoa’s method in the theory of definition. Nederl. Akad. Wetensch. Proc. Ser. A. 56(Indagationes Math.), 15, 330–339 (1953)

    Google Scholar 

  5. Bicarregui J., Dimitrakos T., Gabbay D., Maibaum T.: Interpolation in practical formal development. Logic J. IGPL. 9(2), 231–243 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonacina, M.P., Johansson, M.: On interpolation in decision procedures. In: Automated Reasoning with Analytic Tableaux and Related Methods, vol. 6793 of Lecture Notes in Computer Science, pp. 1–16. Springer, Berlin (2011)

  7. Carbone, A.: The Craig interpolation theorem for schematic systems. In: Collegium Logicum, Vol. 2, vol. 2 of Coll. Logicum Ann. Kurt-Gödel-Soc., pp. 87–100. Springer, Berlin (1996)

  8. Carbone A.: Interpolants, cut elimination and flow graphs for the propositional calculus. Ann. Pure Appl. Logic. 83(3), 249–299 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carnielli W., Rasga J., Sernadas C.: Interpolation via translations. Math. Logic Q. 55(5), 515–534 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carnielli W.A., Rasga J., Sernadas C.: Preservation of interpolation features by fibring. J. Logic Comput. 18(1), 123–151 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Craig W.: Linear reasoning. A new form of the Herbrand–Gentzen theorem. J. Symb. Logic. 22, 250–268 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  12. Craig W.: Three uses of the Herbrand–Gentzen theorem in relating model theory and proof theory. J. Symb. Logic. 22, 269–285 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  13. Czelakowski J.: Sentential logics and Maehara interpolation property. Studia Log. 44(3), 265–283 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Czelakowski, J., Pigozzi, D.: Amalgamation and interpolation in abstract algebraic logic. In: Models, Algebras, and Proofs, vol. 203 of Lecture Notes in Pure and Applied Mathematics, pp. 187–265. Dekker (1999)

  15. D’Agostino G.: Interpolation in non-classical logics. Synthese. An International Journal for Epistemology, Methodology and Philosophy of Science, 164(3), 421–435 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Diaconescu R.: An institution-independent proof of Craig interpolation theorem. Studia Log. 77(1), 59–79 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Feferman S. Harmonious logic: Craig’s interpolation theorem and its descendants. Synthese. 164(3), 341–357 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Font J.M., Jansana R., Pigozzi D.: A survey of abstract algebraic logic. Studia Log. 74(1-2), 13–97 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gabbay, D.M., Maksimova, L.: Interpolation and Definability, vol. 46 of Oxford Logic Guides. Modal and Intuitionistic Logics. The Clarendon Press, Oxford University Press (2005)

  20. Jhala, R., McMillan, K.L.: Interpolant-based transition relation approximation. Log. Methods Comput. Sci. 3(4), 4:1, (electronic) (2007)

  21. Keisler H.J., Keisler J.M.: Craig interpolation for networks of sentences. Ann. Pure Appl. Logic. 163(9), 1322–1344 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kihara H., Ono H.: Interpolation properties, Beth definability properties and amalgamation properties for substructural logics. J. Logic Comput. 20(4), 823–875 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kracht M., Wolter F.: Properties of independently axiomatizable bimodal logics. J. Symb. Logic. 56(4), 1469–1485 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lyndon R.C.: An interpolation theorem in the predicate calculus. Pac. J. Math. 9, 129–142 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  25. Maehara, S.: On the interpolation theorem of Craig. Math. Soc. Jpn. Sūgakugaku, 12, 235–237 (1960/1961)

    Google Scholar 

  26. Marchioni E., Metcalfe G.: Craig interpolation for semilinear substructural logics. Math. Logic Q. 58(6), 468–481 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Marker, D.: Model Theory: An Introduction, vol. 217 of Graduate Texts in Mathematics. Springer-Verlag, Berlin (2002)

  28. McMillan K.L.: An interpolating theorem prover. Theor. Comput. Sci. 345(1), 101–121 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Miller, C., Kupferschmid, S., Lewis, M., Becker, B.: Encoding techniques, Craig interpolants and bounded model checking for incomplete designs. In: Theory and Applications of Satisfiability Testing—SAT, vol. 6175 of Lecture Notes in Computer Science, pp. 194–208. Springer (2010)

  30. Montagna F.: Δ-core fuzzy logics with propositional quantifiers, quantifier elimination and uniform Craig interpolation. Studia Log. 100(1-2), 289–317 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schumm G.F.: Some failures of interpolation in modal logic. Notre Dame J. Form. Logic. 27(1), 108–110 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sernadas, A., Rasga, J., Sernadas, C., Mateus, P.: Approximate reasoning about logic circuits with single-fan-out unreliable gates. J. Logic Comput., in print (2013)

  33. Sernadas A., Sernadas C., Rasga J.: On meet-combination of logics. J. Logic Comput. 22(6), 1453–1470 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sernadas C., Rasga J., Sernadas A.: Preservation of Craig interpolation by the product of matrix logics. J. Appl. Logic. 11(3), 328–349 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tarski, A.: A Decision Method for Elementary Algebra and Geometry. 2nd edn. University of California Press, (1951)

  36. Teige, T., Fränzle, M.: Generalized Craig interpolation for stochastic Boolean satisfiability problems with applications to probabilistic state reachability and region stability. Log. Methods Comput. Sci. 8(2), 2:16, 32 (2012)

    Google Scholar 

  37. Urquhart A.: Failure of interpolation in relevant logics. J. Philos. Logic. 22(5), 449–479 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department Matemtica, Instituto Superior Tcnico, Universidade de Lisboa, Lisbon, Portugal

    João Rasga, Cristina Sernadas & Amlcar Sernadas

  2. SQIG, Instituto de Telecomunicaes, Lisbon, Portugal

    João Rasga, Cristina Sernadas & Amlcar Sernadas

Authors
  1. João Rasga
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  2. Cristina Sernadas
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  3. Amlcar Sernadas
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Correspondence to Cristina Sernadas.

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Rasga, J., Sernadas, C. & Sernadas, A. Craig Interpolation in the Presence of Unreliable Connectives. Log. Univers. 8, 423–446 (2014). https://doi.org/10.1007/s11787-014-0101-9

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  • DOI: https://doi.org/10.1007/s11787-014-0101-9

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