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A Connection-based Characterization of Bi-intuitionistic Validity

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Abstract

We give a connection-based characterization of validity in propositional bi-intuitionistic logic in terms of specific directed graphs called R-graphs. Such a characterization is well-suited for deriving labelled proof-systems with counter-model construction facilities. We first define the notion of bi-intuitionistic R-graphs from which we then obtain a connection-based characterization of propositional bi-intuitionistic validity and derive a sound and complete cut-free free-variable labelled sequent calculus that enjoys variable splitting.

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References

  1. Antonsen, R., Waaler, A.: A labelled system for IPL with variable splitting. In: 21st Int. Conference on Automated Deduction, CADE-21. LNAI, vol. 4603, Bremen, Germany, pp. 132–146. Springer (2007)

  2. Avron, A., Lahav, O.: Kripke semantics for basic sequent systems. In: Int. Conference on Analytic Tableaux and Related Methods, TABLEAUX 2011. LNAI, vol. 6793, Bern, Switzerland, pp. 43–57. Springer (2011)

  3. Crolard, T.: Substractive logic. Theor. Comp. Sci. 254(1–2), 151–185 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Crolard, T.: A formulae-as-types interpretation of substractive logic. J. Log. Comput. 14(4), 529–570 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Galmiche, D., Méry, D.: Characterizing provability in BI’s pointer logic through resource graphs. In: Int. Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2005. LNAI, vol. 3835, Montego Bay, Jamaica, pp. 459–473. Springer (2005)

  6. Galmiche, D., Méry, D.: Resource graphs and countermodels in resource logics. Electron. Notes Theor. Comp. Sci. 125(3), 117–135 (2005)

    Article  Google Scholar 

  7. Galmiche, D., Méry, D.: A connection-based characterization of bi-intuitionistic validity. In: 23rd Int. Conference on Automated Deduction, CADE-23. LNAI, vol. 6803, Wroclaw, Poland, pp. 268–282. Springer (2011)

  8. Goré, R., Postniece, L.: Combining derivations and refutations for cut-free completeness in bi-intuitionistic logic. J. Log. Comput. 20(1), 233–260 (2010)

    Article  MATH  Google Scholar 

  9. Goré, R., Postniece, L., Tiu, A.: Cut-elimination and proof-search for bi-intuitionistic logic using nested sequents. In: Advances in Modal Logic, vol. 7, pp. 43–66. College Publications, London (2008)

    Google Scholar 

  10. Negri, S.: Proof analysis in modal logic. J. Philos. Logic 34(5–6), 507–554 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Otten, J., Kreitz, C.: T-string-unification: unifying prefixes in non classical proof methods. In: 5th Int. Workshop on Analytic Tableaux and Related Methods, TABLEAUX’96. LNAI, vol. 1071, Terrasini, Palermo, Italy, pp. 244–260. Springer (1996)

  12. Pinto, L., Uustalu, T.: Proof search and counter-model construction for bi-intuitionistic propositional logic with labelled sequents. In: Int. Conference on Analytic Tableaux and Related Methods, TABLEAUX 2009. LNAI, vol. 5607, Oslo, Norway, pp. 295–309. Springer (2009)

  13. Postniece, L.: Deep inference in bi-intuitionistic logic. In: Int Workshop on Logic, Language, Information and Computation, WoLLIC 2009. LNAI, vol. 5514, Tokyo, Japan, pp. 320–334. Springer (2009)

  14. Rauszer, C.: An algebraic and Kripke-style approach to a certain extension of intuitionistic logic. Diss. Math. 167, 1–62 (1980)

    MathSciNet  Google Scholar 

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  1. Université de Lorraine - LORIA, Campus Scientifique BP 239, Vandœuvre-lès-Nancy, France

    Didier Galmiche & Daniel Méry

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  1. Didier Galmiche
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  2. Daniel Méry
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Correspondence to Daniel Méry.

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Galmiche, D., Méry, D. A Connection-based Characterization of Bi-intuitionistic Validity. J Autom Reasoning 51, 3–26 (2013). https://doi.org/10.1007/s10817-013-9279-4

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