A Logical Framework with Commutative and Non-commutative Subexponentials

  • Max Kanovich
  • Stepan Kuznetsov
  • Vivek NigamEmail author
  • Andre Scedrov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10900)


Logical frameworks allow the specification of deductive systems using the same logical machinery. Linear logical frameworks have been successfully used for the specification of a number of computational, logics and proof systems. Its success relies on the fact that formulas can be distinguished as linear, which behave intuitively as resources, and unbounded, which behave intuitionistically. Commutative subexponentials enhance the expressiveness of linear logic frameworks by allowing the distinction of multiple contexts. These contexts may behave as multisets of formulas or sets of formulas. Motivated by applications in distributed systems and in type-logical grammar, we propose a linear logical framework containing both commutative and non-commutative subexponentials. Non-commutative subexponentials can be used to specify contexts which behave as lists, not multisets, of formulas. In addition, motivated by our applications in type-logical grammar, where the weakenening rule is disallowed, we investigate the proof theory of formulas that can only contract, but not weaken. In fact, our contraction is non-local. We demonstrate that under some conditions such formulas may be treated as unbounded formulas, which behave intuitionistically.



We are grateful to Glyn Morrill, Frank Pfenning, and the anonymous referees.

Financial Support: The work of Max Kanovich and Andre Scedrov was supported by the Russian Science Foundation under grant 17-11-01294 and performed at National Research University Higher School of Economics, Moscow, Russia. The work of Stepan Kuznetsov was supported by the Young Russian Mathematics award, by the Program of the Presidium of the Russian Academy of Sciences No. 01 ‘Fundamental Mathematics and Its Applications’ under grant PRAS-18-01, and by the Russian Foundation for Basic Research grant 18-01-00822. The work of Vivek Nigam was supported by CNPq grant number 304193/2015-1. Sections 1, 2, 3, 7 and 8 were contributed jointly and equally by all co-authors; Sect. 4 was contributed by Scedrov and Kanovich. Section 5 was contributed by Nigam. Section 6 was contributed by Kuznetsov.


  1. 1.
    Ajdukiewicz, K.: Die syntaktische Konnexität. Studia Philosophica 1, 1–27 (1935)zbMATHGoogle Scholar
  2. 2.
    Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. J. Logic Comput. 2(3), 297–347 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bar-Hillel, Y.: A quasi-arithmetical notation for syntactic description. Language 29, 47–58 (1953)CrossRefGoogle Scholar
  4. 4.
    van Benthem, J.: Language in Action: Categories, Lambdas and Dynamic Logic. Elsevier, North Holland (1991)zbMATHGoogle Scholar
  5. 5.
    Danos, V., Joinet, J.-B., Schellinx, H.: The structure of exponentials: uncovering the dynamics of linear logic proofs. In: Gödel, K. (ed.) Colloquium, pp. 159–171 (1993)Google Scholar
  6. 6.
    Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Harper, R., Honsell, F., Plotkin, G.D.: A framework for defining logics. In: LICS (1987)Google Scholar
  8. 8.
    Hodas, J.S., Miller, D.: Logic programming in a fragment of intuitionistic linear logic: extended abstract. In: LICS (1991)Google Scholar
  9. 9.
    Kanovich, M., Kuznetsov, S., Scedrov, A.: Undecidability of the Lambek calculus with subexponential and bracket modalities. In: Klasing, R., Zeitoun, M. (eds.) FCT 2017. LNCS, vol. 10472, pp. 326–340. Springer, Heidelberg (2017). Scholar
  10. 10.
    Kanovich, M., Kuznetsov, S., Nigam, V., Scedrov, A.: Subexponentials in non-commutative linear logic. Math. Struct. Comput. Sci., FirstView, 1–33 (2018).
  11. 11.
    Kuznetsov, S., Morrill, G., Valentín, O.: Count-invariance including exponentials. In: Proceedings of MoL 2017, volume W17–3413 of ACL Anthology, pp. 128–139 (2017)Google Scholar
  12. 12.
    Lambek, J.: The mathematics of sentence structure. Amer. Math. Mon. 65, 154–170 (1958)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Miller, D.: Forum: a multiple-conclusion specification logic. Theor. Comput. Sci. 165(1), 201–232 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Miller, D., Saurin, A.: From proofs to focused proofs: a modular proof of focalization in linear logic. In: CSL, pp. 405–419 (2007)Google Scholar
  15. 15.
    Moortgat, M.: Multimodal linguistic inference. J. Logic Lang. Inf. 5(3–4), 349–385 (1996)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Moot, R.: The grail theorem prover: type theory for syntax and semantics. In: Chatzikyriakidis, S., Luo, Z. (eds.) Modern Perspectives in Type-Theoretical Semantics. SLP, vol. 98, pp. 247–277. Springer, Cham (2017). Scholar
  17. 17.
    Morrill, G.: CatLog: a categorial parser/theorem-prover. In: LACL System demostration (2012)Google Scholar
  18. 18.
    Morrill, G.: Parsing logical grammar: CatLog3. In: Proceedings of LACompLing (2017)Google Scholar
  19. 19.
    Morrill, G., Valentín, O.: Computation coverage of TLG: nonlinearity. In: NLCS (2015)Google Scholar
  20. 20.
    Morrill, G., Valentín, O.: Multiplicative-additive focusing for parsing as deduction. In: First International Workshop on Focusing (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nigam, V.: Exploiting non-canonicity in the sequent calculus. Ph.D. thesis (2009)Google Scholar
  22. 22.
    Nigam, V.: A framework for linear authorization logics. TCS 536, 21–41 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nigam, V., Miller, D.: Algorithmic specifications in linear logic with subexponentials. In: PPDP, pp. 129–140 (2009)Google Scholar
  24. 24.
    Nigam, V., Miller, D.: A framework for proof systems. J. Autom. Reasoning 45(2), 157–188 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nigam, V., Olarte, C., Pimentel, E.: A general proof system for modalities in concurrent constraint programming. In: CONCUR (2013)CrossRefGoogle Scholar
  26. 26.
    Nigam, V., Pimentel, E., Reis, G.: An extended framework for specifying and reasoning about proof systems. J. Logic Comput. 26(2), 539–576 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Olarte, C., Pimentel, E., Nigam, V.: Subexponential concurrent constraint programming. Theor. Comput. Sci. 606, 98–120 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pentus, M.: Lambek grammars are context-free. In: LICS, pp. 429–433 (1993)Google Scholar
  29. 29.
    Pfenning, F., Simmons, R.J.: Substructural operational semantics as ordered logic programming. In: LICS, pp. 101–110 (2009)Google Scholar
  30. 30.
    Polakow, J.: Linear logic programming with an ordered context. In: PPDP (2000)Google Scholar
  31. 31.
    Shieber, S.M.: Evidence against the context-freeness of natural languages. Linguist. Philos. 8, 333–343 (1985)CrossRefGoogle Scholar
  32. 32.
    Simmons, R.J., Pfenning, F.: Weak focusing for ordered linear logic. Technical report CMU-CS-10-147 (2011)Google Scholar
  33. 33.
    Watkins, K., Cervesato, I., Pfenning, F., Walker, D.: A concurrent logical framework: the propositional fragment. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 355–377. Springer, Heidelberg (2004). Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Max Kanovich
    • 1
  • Stepan Kuznetsov
    • 1
    • 2
  • Vivek Nigam
    • 3
    • 4
    Email author
  • Andre Scedrov
    • 1
    • 5
  1. 1.National Research University Higher School of EconomicsMoscowRussia
  2. 2.Steklov Mathematical Institute of RASMoscowRussia
  3. 3.Federal University of ParaíbaJoão PessoaBrazil
  4. 4.fortissMunichGermany
  5. 5.University of PennsylvaniaPhiladelphiaUSA

Personalised recommendations