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From Display Calculi to Deep Nested Sequent Calculi: Formalised for Full Intuitionistic Linear Logic

  • Jeremy E. Dawson
  • Ranald Clouston
  • Rajeev Goré
  • Alwen Tiu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8705)

Abstract

Proof theory for a logic with categorical semantics can be developed by the following methodology: define a sound and complete display calculus for an extension of the logic with additional adjunctions; translate this calculus to a shallow inference nested sequent calculus; translate this calculus to a deep inference nested sequent calculus; then prove this final calculus is sound with respect to the original logic. This complex chain of translations between the different calculi require proofs that are technically intricate and involve a large number of cases, and hence are ideal candidates for formalisation. We present a formalisation of this methodology in the case of Full Intuitionistic Linear Logic (FILL), which is multiplicative intuitionistic linear logic extended with par.

Keywords

Proof System Logical Rule Proof Theory Structural Rule Sequent Calculus 
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.

References

  1. 1.
    Belnap, N.D.: Display logic. J. Philos. Logic 11, 375–417 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bierman, G.M.: A note on full intuitionistic linear logic. Ann. Pure Appl. Logic 79(3), 281–287 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bräuner, T., de Paiva, V.: A formulation of linear logic based on dependency-relations. In: Nielsen, M. (ed.) CSL 1997. LNCS, vol. 1414, pp. 129–148. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  4. 4.
    Brünnler, K.: Deep sequent systems for modal logic. Arch. Math. Logic 48(6), 551–577 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Clouston, R., Dawson, J.E., Goré, R., Tiu, A.: Annotation-free sequent calculi for full intuitionistic linear logic. In: CSL, pp. 197–214 (2013)Google Scholar
  6. 6.
    Clouston, R., Dawson, J.E., Goré, R., Tiu, A.: Annotation-free sequent calculi for full intuitionistic linear logic - extended version. CoRR, abs/1307.0289 (2013)Google Scholar
  7. 7.
    Dawson, J.E., Goré, R.P.: Formalised cut admissibility for display logic. In: Carreño, V.A., Muñoz, C.A., Tahar, S. (eds.) TPHOLs 2002. LNCS, vol. 2410, pp. 131–147. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Dawson, J.E., Goré, R.: Generic methods for formalising sequent calculi applied to provability logic. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 263–277. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Goré, R.: Substructural logics on display. Log. J. IGPL 6(3), 451–504 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Goré, R., Postniece, L., Tiu, A.: Cut-elimination and proof search for bi-intuitionistic tense logic. In: AiML, pp. 156–177 (2010)Google Scholar
  11. 11.
    Goré, R., Postniece, L., Tiu, A.: On the correspondence between display postulates and deep inference in nested sequent calculi for tense logics. LMCS 7(2) (2011)Google Scholar
  12. 12.
    Hyland, M., de Paiva, V.: Full intuitionistic linear logic (extended abstract). Ann. Pure Appl. Logic 64(3), 273–291 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kashima, R.: Cut-free sequent calculi for some tense logics. Studia Log. 53, 119–135 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Park, J., Seo, J., Park, S.: A theorem prover for boolean BI. In: POPL 2013, pp. 219–232 (2013)Google Scholar
  15. 15.
    Poggiolesi, F.: The method of tree-hypersequents for modal propositional logic. In: Trends in Logic IV, pp. 31–51 (2009)Google Scholar
  16. 16.
    Schellinx, H.: Some syntactical observations on linear logic. J. Logic Comput. 1(4), 537–559 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Straßburger, L.: Cut elimination in nested sequents for intuitionistic modal logics. In: Pfenning, F. (ed.) FOSSACS 2013 (ETAPS 2013). LNCS, vol. 7794, pp. 209–224. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  18. 18.
    Traytel, D., Popescu, A., Blanchette, J.C.: Foundational, compositional (co)datatypes for higher-order logic: Category theory applied to theorem proving. In: LICS, pp. 596–605 (2012)Google Scholar
  19. 19.
    Urban, C.: Nominal techniques in isabelle/HOL. J. Automat. Reason. 40(4), 327–356 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2014

Authors and Affiliations

  • Jeremy E. Dawson
    • 1
  • Ranald Clouston
    • 2
  • Rajeev Goré
    • 1
  • Alwen Tiu
    • 3
  1. 1.Research School of Computer ScienceAustralian National UniversityAustralia
  2. 2.Department of Computer ScienceAarhus UniversityDenmark
  3. 3.School of Computer EngineeringNanyang Technological UniversitySingapore

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