Focusing Strategies in the Sequent Calculus of Synthetic Connectives

  • Kaustuv Chaudhuri
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5330)


It is well-known that focusing striates a sequent derivation into phases of like polarity where each phase can be seen as inferring a synthetic connective. We present a sequent calculus of synthetic connectives based on neutral proof patterns, which are a syntactic normal form for such connectives. Different focusing strategies arise from different polarisations and arrangements of synthetic inference rules, which are shown to be complete by synthetic rule permutations. A simple generic cut-elimination procedure for synthetic connectives respects both the ordinary focusing and the maximally multi-focusing strategies, answering the open question of cut-admissibility for maximally multi-focused proofs.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramsky, S.: Sequentiality vs. concurrency in games and logic. Mathematical Structures in Computer Science 13(4), 531–565 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abramsky, S., Melliès, P.-A.: Concurrent games and full completeness. In: 14th Symp. on Logic in Computer Science, pp. 431–442. IEEE Computer Society Press, Los Alamitos (1999)Google Scholar
  3. 3.
    Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. of Logic and Computation 2(3), 297–347 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baelde, D., Miller, D.: Least and greatest fixed points in linear logic. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS, vol. 4790, pp. 92–106. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Chaudhuri, K., Miller, D., Saurin, A.: Canonical sequent proofs via multi-focusing. In: Ausiello, G., Karhumäki, J., Mauri, G., Ong, L. (eds.) Fifth IFIP International Conference on Theoretical Computer Science. IFIP International Federation for Information Processing, vol. 273, pp. 383–396. Springer, Boston (2008)Google Scholar
  6. 6.
    Chaudhuri, K., Pfenning, F.: Focusing the inverse method for linear logic. In: Luke Ong, C.-H. (ed.) CSL 2005. LNCS, vol. 3634, pp. 200–215. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Chaudhuri, K., Pfenning, F., Price, G.: A logical characterization of forward and backward chaining in the inverse method. J. of Automated Reasoning 40(2-3), 133–177 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Danos, V., Joinet, J.-B., Schellinx, H.: LKT and LKQ: sequent calculi for second order logic based upon dual linear decompositions of classical implication. In: Girard, J.-Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic. London Mathematical Society Lecture Note Series, vol. 222, pp. 211–224. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  9. 9.
    Delande, O., Miller, D.: A neutral approach to proof and refutation in MALL. In: Pfenning, F. (ed.) 23th Symp. on Logic in Computer Science, pp. 498–508. IEEE Computer Society Press, Los Alamitos (2008)Google Scholar
  10. 10.
    Girard, J.-Y.: Locus solum. Mathematical Structures in Computer Science 11(3), 301–506 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kleene, S.C.: Permutabilities of inferences in Gentzen’s calculi LK and LJ. Memoirs of the American Mathematical Society 10, 1–26 (1952)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Laurent, O.: Etude de la polarisation en logique. Thèse de doctorat, Université Aix-Marseille II (March 2002)Google Scholar
  13. 13.
    Laurent, O.: Syntax vs. semantics: a polarized approach. Prépublication électronique PPS//03/04//no 17 (pp), Laboratoire Preuves, Programmes et Systèmes (Submitted) (March 2003)Google Scholar
  14. 14.
    Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science (accepted, 2008)Google Scholar
  15. 15.
    Miller, D., Saurin, A.: From proofs to focused proofs: A modular proof of focalization in linear logic. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 405–419. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Wadler, P.: Call-by-value is dual to call-by-name. In: 8th Int. Conf. on Functional Programming, pp. 189–201 (2003)Google Scholar
  17. 17.
    Zeilberger, N.: Focusing and higher-order abstract syntax. In: Necula, G.C., Wadler, P. (eds.) Proceedings of the 35th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2008, pp. 359–369. ACM, New York (2008)Google Scholar
  18. 18.
    Zeilberger, N.: On the unity of duality. Ann. of Pure and Applied Logic (to appear, 2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Kaustuv Chaudhuri
    • 1
  1. 1.INRIA Saclay - Île-deFrance

Personalised recommendations