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Subexponentials in non-commutative linear logic

Published online by Cambridge University Press:  02 May 2018

MAX KANOVICH
Affiliation:
National Research University Higher School of Economics, Moscow, Russia Email: m.kanovich@qmul.ac.uk
STEPAN KUZNETSOV
Affiliation:
Steklov Mathematical Institute of RAS, Moscow, Russia Email: skuzn@inbox.ru
VIVEK NIGAM
Affiliation:
Federal University of Paraíba, João Pessoa, Brazil Email: vivek.nigam@gmail.com Fortiss GmbH, Munich, Germany
ANDRE SCEDROV
Affiliation:
National Research University Higher School of Economics, Moscow, Russia Email: m.kanovich@qmul.ac.uk University of Pennsylvania, Philadelphia, U.S.A. E-mail: scedrov@math.upenn.edu

Abstract

Linear logical frameworks with subexponentials have been used for the specification of, among other systems, proof systems, concurrent programming languages and linear authorisation logics. In these frameworks, subexponentials can be configured to allow or not for the application of the contraction and weakening rules while the exchange rule can always be applied. This means that formulae in such frameworks can only be organised as sets and multisets of formulae not being possible to organise formulae as lists of formulae. This paper investigates the proof theory of linear logic proof systems in the non-commutative variant. These systems can disallow the application of exchange rule on some subexponentials. We investigate conditions for when cut elimination is admissible in the presence of non-commutative subexponentials, investigating the interaction of the exchange rule with the local and non-local contraction rules. We also obtain some new undecidability and decidability results on non-commutative linear logic with subexponentials.

Type
Paper
Copyright
© Cambridge University Press 2018 

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References

Abrusci, V.M. (1990). A comparison between Lambek syntactic calculus and intuitionistic linear logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 36 1115.CrossRefGoogle Scholar
Ajdukiewicz, K. (1935). Die syntaktische Konnexität. Studia Philosophica 1 127.Google Scholar
Andreoli, J.-M. (1992). Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2(3) 297347.CrossRefGoogle Scholar
Bar-Hillel, Y. (1953). A quasi-arithmetical notation for syntactic description. Language 29 4758.CrossRefGoogle Scholar
Benthem, J. (1991). Language in Action: Categories, Lambdas and Dynamic Logic, North Holland, Amsterdam.Google Scholar
Braüner, T. and de Paiva, V. (1998). A formulation of linear logic based on dependency relations. In: Proceedings of the International Workshop on Computer Science Logic (CSL '97), Lecture Notes in Computer Science, vol. 1414, Springer, 129–148.Google Scholar
Buszkowski, W. (1982). Some decision problems in the theory of syntactic categories. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 28 539548.CrossRefGoogle Scholar
Buszkowski, W. (2010). Lambek calculus and substructural logics. Linguistic Analysis 36 (1–4) 1548.Google Scholar
Chaudhuri, K. (2010). Classical and intuitionistic subexponential logics are equally expressive. In: Proceedings of the International Workshop on Computer Science Logic (CSL '10), Lecture Notes in Computer Science, vol. 6247, Springer, 185–199.Google Scholar
Danos, V., Joinet, J. and Schellinx, H. (1993). The structure of exponentials: Uncovering the dynamics of linear logic proofs. In: Proceedings of the Kurt Gödel Colloquium on Computational Logic and Proof Theory (KGC '93), Lecture Notes in Computer Science, vol. 713, Springer, 159–171.CrossRefGoogle Scholar
Gentzen, G. (1935). Untersuchungen über das logische Schließen I. Mathematische Zeitschrift 39 176210.CrossRefGoogle Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science 50(1) 1102.CrossRefGoogle Scholar
de Groote, P. (2005). On the expressive power of the Lambek calculus extended with a structural modality. In: Language and Grammar, CSLI Lecture Notes, vol. 168, CSLI, 95111.Google Scholar
Hodas, J. and Miller, D. (1991). Logic programming in a fragment of intuitionistic linear logic. Extended abstract. In: Proceedings of the 6th Annual Symposium on Logic in Computer Science (LICS '91) 32–42.Google Scholar
Hodas, J. and Miller, D. (1994). Logic programming in a fragment of intuitionistic linear logic. Information and Computation 110(2) 327365.CrossRefGoogle Scholar
Kanazawa, M. (1992). The Lambek calculus enriched with additional connectives. Journal of Logic, Language and Information 1(2) 141171.CrossRefGoogle Scholar
Kanazawa, M. (1999). Lambek calculus: Recognizing power and complexity. In: Johannes Franciscus Abraham Karel (JFAK). Essays dedicated to Johan van Benthem to the occasion of his 50th birthday. Vossiuspers, Amsterdam Univ. Press.Google Scholar
Kanovich, M. (1994). Horn fragments of non-commutative logics with additives are PSPACE-complete. In: Extended abstract, presented at the 8th Annual Conference of the European Association for Computer Science Logic (CSL '94), Kazimierz, Poland (unpublished manuscript).Google Scholar
Kanovich, M.I. (1995). The complexity of neutrals in linear logic. In: Proceedings of the 10th Annual Symposium on Logic in Computer Science (LICS '95), IEEE, 486–495.CrossRefGoogle Scholar
Kanovich, M., Kuznetsov, S. and Scedrov, A. (2016a). Reconciling Lambek's restriction, cut-elimination, and substitution in the presence of exponential modalities. arXiv 1608.02254.Google Scholar
Kanovich, M., Kuznetsov, S. and Scedrov, A. (2016b). Undecidability of the Lambek calculus with a relevant modality. In: Proceedings of the International Conference on Formal Grammar (FG '15 and '16), Lecture Notes in Computer Science, vol. 9804, Springer, 240–256.Google Scholar
Kanovich, M., Kuznetsov, S. and Scedrov, A. (2016c). On Lambek's restriction in the presence of exponential modalities. In: Artemov, S. and Nerode, A. (eds.) Proceedings of the International Symposium on Logical Foundations of Computer Science (LFCS '16), Lecture Notes in Computer Science, vol. 9537, Springer, 146–158.Google Scholar
Kanovich, M., Kuznetsov, S. and Scedrov, A. (2017). Undecidability of the Lambek calculus with subexponential and bracket modalities. In: Fundamentals of Computation Theory, Lecture Notes in Computer Science, vol. 10472, Springer, 326–340.CrossRefGoogle Scholar
Kuznetsov, S. and Okhotin, A. (2017). Conjunctive categorial grammars. In: Proceedings of the 15th Meeting on the Mathematics of Language (MoL '17), ACL Anthology, vol. W17-3414, 140–151.CrossRefGoogle Scholar
Kuznetsov, S.L. (2011). On the Lambek calculus with a unit and one division. Moscow University Mathematics Bulletin 66(4) 173175.CrossRefGoogle Scholar
Kuznetsov, S.L. (2016). On translating Lambek grammars with one division into context-free grammars. Proceedings of the Steklov Institute of Mathematics 294 129138.CrossRefGoogle Scholar
Lambek, J. (1958). The mathematics of sentence structure. The American Mathematical Monthly 65 154170.CrossRefGoogle Scholar
Lambek, J. (1961). On the calculus of syntactic types. In: Structure of Language and Its Mathematical Aspects, Proceedings of the Symposia Applied Mathematics, vol. 12, AMS, 166–178.CrossRefGoogle Scholar
Lambek, J. (1969). Deductive systems and categories II. Standard constructions and closed categories. In: Category Theory, Homology Theory and Their Applications I, Lecture Notes in Mathematics, vol. 86, Springer, 76–122.Google Scholar
Lambek, J. (1993). From categorial grammar to bilinear logic. In: Substructural Logics, Studies in Logic and Computations, vol. 2, Clarendon Press, Oxford, 207237.Google Scholar
Lincoln, P., Mitchell, J., Scedrov, A. and Shankar, N. (1992). Decision problems for propositional linear logic. Annals of Pure and Applied Logic 56(1) 239311.CrossRefGoogle Scholar
Markov, A. (1947). On the impossibility of certain algorithms in the theory of associative systems. Doklady Academy of Sciences USSR (N. S.) 55 583586.Google Scholar
Miller, D. (1994). A multiple-conclusion meta-logic. In: Proceedings of the 9th Annual Symposium on Logic in Computer Science(LICS '94), IEEE, 272–281.CrossRefGoogle Scholar
Miller, D. (1996). Forum: A multiple-conclusion specification logic. Theoretical Computer Science 165(1) 201232.CrossRefGoogle Scholar
Moortgat, M. (1996). Multimodal linguistic inference. Journal of Logic, Language and Information 5 (3–4) 349385.CrossRefGoogle Scholar
Moortgat, M. (1997). Categorial type logics. In: van Benthem, J. and ter Meulen, A. (eds.), Handbook of Logic and Language, Elsevier.Google Scholar
Moot, R. (2017). The Grail theorem prover: Type theory for syntax and semantics. In: Modern Perspectives in Type-Theoretical Semantics, Studies in Linguistics and Philosophy, vol. 98, Springer, 247277.CrossRefGoogle Scholar
Moot, R. and Retoré, C. (2012). The Logic of Categorial Grammars: A Deductive Account of Natural Language Syntax and Semantics, Lecture Notes in Computer Science, vol. 6850, Springer.CrossRefGoogle Scholar
Morrill, G. (2017a). Parsing logical grammar: CatLog3. In: Proceedings of the Workshop on Logic and Algorithms in Computational Linguistics (LACompLing '17), Stockholm University, 107–131.Google Scholar
Morrill, G. (2017b). Grammar logicised: Relativisation. Linguistics and Philosophy 40(2) 119163.CrossRefGoogle Scholar
Morrill, G. and Valentín, O. (2015). Computation coverage of TLG: Nonlinearity. In: Proceedings of the Natural Language and Computer Science (NLCS '15), EPiC Series, vol. 32, 5163.Google Scholar
Morrill, G.V. (2011). Categorial Grammar: Logical Syntax, Semantics, and Processing, Oxford Univ. Press.Google Scholar
Nigam, V. (2012). On the complexity of linear authorization logics. In: Proceedings of the 27th Annual Symposium on Logic in Computer Science (LICS '12), IEEE, 511–520.CrossRefGoogle Scholar
Nigam, V. (2014). A framework for linear authorization logics. Theoretical Computer Science 536(0) 2141.CrossRefGoogle Scholar
Nigam, V. and Miller, D. (2009). Algorithmic specifications in linear logic with subexponentials. In: Proceedings of the Principles and Practice of Declarative Programming (PPDP '09) 129–140.Google Scholar
Nigam, V., Olarte, C. and Pimentel, E. (2013). A general proof system for modalities in concurrent constraint programming. In: Proceedings of the International Conference on Concurrency Theory (CONCUR '13), Lecture Notes in Computer Science, vol. 8052, Springer, 410–424.Google Scholar
Nigam, V., Pimentel, E. and Reis, G. (2016). An extended framework for specifying and reasoning about proof systems. Journal of Logic and Computation 26(2) 539576.CrossRefGoogle Scholar
Olarte, C., Pimentel, E. and Nigam, V. (2015). Subexponential concurrent constraint programming. Theoretical Computer Science 606 98120.CrossRefGoogle Scholar
Peirce, C. S. (1885). On the algebra of logic: A contribution to the philosophy of notation. American Journal of Mathematics 7 180202.CrossRefGoogle Scholar
Pentus, M. (1993). Lambek grammars are context-free. In: Proceedings of the 8th Annual Symposium on Logic in Computer Science (LICS '93), IEEE, 429–433.CrossRefGoogle Scholar
Pentus, M. (1998). Free monoid completeness of the Lambek calculus allowing empty premises. In: Proceedings of the Logic Colloquium '96, Lecture Notes Logic, vol. 12, Springer, 171–209.CrossRefGoogle Scholar
Pentus, M. (2006). Lambek calculus is NP-complete. Theoretical Computer Science 357(1) 186201.CrossRefGoogle Scholar
Pfenning, F. and Simmons, R.J. (2009). Substructural operational semantics as ordered logic programming. In: Proceedings of the 24th Annual Symposium on Logic In Computer Science (LICS '09), IEEE Computer Society, 101–110.CrossRefGoogle Scholar
Post, E.L. (1947). Recursive unsolvability of a problem of Thue. Journal of Symbolic Logic 12 111.CrossRefGoogle Scholar
Savitch, W.J. (1970). Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences 4(2) 177192.CrossRefGoogle Scholar
Schellinx, H. (1991). Some syntactical observations on linear logic. Journal of Logic and Computation 1(4) 537559.CrossRefGoogle Scholar
Shieber, S. M. (1985). Evidence against the context-freeness of natural languages. Linguistics and Philosophy 8 333343.CrossRefGoogle Scholar
Thue, A. (1914). Probleme über Veränderungen von Zeichenreihen nach gegebener Regeln. Christiana Videnskabs-Selskabets Skrifter 10 134.Google Scholar
Yetter, D.N. (1990). Quantales and (noncommutative) linear logic. Journal of Symbolic Logic 55(1) 4164.CrossRefGoogle Scholar
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