Skip to main content
SpringerLink
Log in

A Matrix Characterization for Multiplicative Exponential Linear Logic

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

We develop a matrix characterization of logical validity in MELL, the multiplicative fragment of propositional linear logic with exponentials and constants. To prove the correctness and completeness of our characterization, we use a purely proof-theoretical justification rather than semantical arguments. Our characterization is based on concepts similar to matrix characterizations proposed by Wallen for other nonclassical logics. It provides a foundation for developing proof search procedures for MELL by adopting techniques that are based on these concepts and also makes it possible to adopt algorithms that translate the machine-found proofs back into the usual sequent calculus for MELL.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Andreoli, J.-M. (1993) Logic programming with focussing proofs in linear logic, J. Logic Comput. 2(3), 297-347.

    Google Scholar 

  • Andreoli, J.-M. (2001) Focussing and proof construction, Ann. Pure Appl. Logic 107(1), 131-163.

    Google Scholar 

  • Andrews, P. (1981) Theorem-proving via general matings. J. Assoc. Comput. Mach. 28(2), 193-214.

    Google Scholar 

  • Bibel, W. (1981) On matrices with connections, J. Assoc. Comput. Mach. 28, 633-645.

    Google Scholar 

  • Bibel, W. (1986) A deductive solution for plan generation, New Generation Computing 4, 115-132.

    Google Scholar 

  • Bibel, W. (1987) Automated Theorem Proving, 2nd edn, Vieweg Verlag.

  • Cervesato, I., Hodas, J. S. and Pfenning, F. (2000) Efficient resource management for linear logic proof search, Theoret. Comput. Sci. 232(1-2), 133-163.

    Google Scholar 

  • Danos, V. and Regnier, L. (1989) The structure of the multiplicatives, Arch.Math. Logic 28, 181-203.

    Google Scholar 

  • Fronhöfer, B. (1996) The Action-As-Implication Paradigm, CS Press.

  • Galmiche, D. (2000) Connection methods in linear logic and proof nets construction, Theoret. Comput. Sci. 232(2), 231-272.

    Google Scholar 

  • Galmiche, D. and Marion, J. Y. (1995) Semantics proof search methods for ALL-a first approach, in P. Baumgartner, R. Hähnle and J. Posegga (eds), 4thWorkshop on Theorem Proving with Analytic Tableaux and Related Methods-Short Papers & Poster Sessions, Institute of Computer Science, Universität Koblenz.

  • Galmiche, D. and Martin, B. (1997) Proof search and proof nets construction in linear logic, in 4th Workshop on Logic, Language, Information and Computation (WoLLIC'97).

  • Galmiche, D. and Notin, J. (2000) Proof-search and proof nets in mixed linear logic, Electronic Notes in Theoret. Comput. Sci. 37.

  • Galmiche, D. and Notin, J. (2001) Calculi with dependency relations for mixed linear logic, in Workshop on Logic and Complexity in Computer Science, LCCS 2001.

  • Galmiche, D. and Notin, J. (2002) Based-on dependency calculi for non-commutative logic, preprint.

  • Galmiche, D. and Perrier, G. (1992) A procedure for automatic proof nets construction, in A. Voronkov (ed.), LPAR'92, Conference on Logic Programming and Automated Reasoning, St. Petersburgh, Russia, July 1992, Springer-Verlag, pp. 42-53.

    Google Scholar 

  • Galmiche, D. and Perrier, G. (1994) On proof normalization in linear logic, Theoret. Comput. Sci. 135(1), 67-110.

    Google Scholar 

  • Gehlot, V. and Gunter, C. (1991) Normal process representatives, in G. Kahn (ed.), LICS-91-Sixth Annual Symposium on Logic in Computer Science, Amsterdam, Netherlands, July 1991, pp. 200-207.

  • Girard, J.-Y. (1987) Linear logic, Theoret. Comput. Sci. 50, 1-102.

    Google Scholar 

  • Girard, J.-Y. (1996) Proof-nets: The parallel syntax for proof-theory, in Logic and Algebra, pp. 97-124.

  • Harland, J. and Pym, D. (1997) Resource-distribution via Boolean constraints, in W. McCune (ed.), 14th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 1249, Springer-Verlag, pp. 222-236.

  • Hodas, J. and Miller, D. (1994) Logic programming in a fragment of linear logic, J. Inform. Comput. 110(2), 327-365.

    Google Scholar 

  • Kanovich, M. I. (1991) The multiplicative fragment of linear logic is NP-complete, ITLI Prepublication Series X-91-13, University of Amsterdam.

  • Kreitz, C., Mantel, H., Otten, J. and Schmitt, S. (1997) Connection-based proof construction in linear logic, in W. McCune (ed.), 14th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 1249, Springer-Verlag, pp. 207-221.

  • Kreitz, C. and Otten, J. (1999) Connection-based theorem proving in classical and non-classical logics, J. Universal Comput. Sci. 5(3), 88-112.

    Google Scholar 

  • Kreitz, C. and Schmitt, S. (2000) A uniform procedure for converting matrix proofs into sequent-style systems, J. Inform. Comput. 162(1-2), 226-254.

    Google Scholar 

  • Lincoln, P. and Winkler, T. (1994) Constant-only multiplicative linear logic is NP-complete, Theoret. Comput. Sci. 135, 155-169.

    Google Scholar 

  • Mantel, H. and Kreitz, C. (1998) A matrix characterization for MELL, in U. F. J. Dix and F. L. Del Cerro (eds), 6th European Workshop on Logics in AI (JELIA-98), Lecture Notes in Artificial Intelligence 1489, Springer-Verlag, pp. 169-183.

  • Mantel, H. and Otten, J. (1999) linTAP: A tableau prover for linear logic, in N. Murray (ed.), Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX-99), Lecture Notes in Artificial Intelligence 1617, Springer-Verlag, pp. 217-231.

  • Masseron, M., Tollu, C. and Vauzeilles, J. (1991) Generating plans in linear logic, in Foundations of Software Technology and Theoretical Computer Science, Springer-Verlag, pp. 63-75.

  • McCarthy, J. and Hayes, P. (1969) Some philosophical problems from the standpoint of artificial intelligence, Machine Intelligence 4, 463-502.

    Google Scholar 

  • Miller, D. (1996) FORUM: A multiple-conclusion specification logic, Theoret. Comput. Sci. 165(1), 201-232.

    Google Scholar 

  • Otten, J. and Kreitz, C. (1996) T-string-unification: Unifying prefixes in non-classical proof methods, in U. Moscato (ed.), 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, Lecture Notes in Artificial Intelligence 1071, Springer-Verlag, pp. 244-260.

  • Schmitt, S., Lorigo, L., Kreitz, C. and Nogin, A. (2001) JProver: Integrating connection-based theorem proving into interactive proof assistants, in: R. Gore, A. Leitsch and T. Nipkow (eds), International Joint Conference on Automated Reasoning, Lecture Notes in Artificial Intelligence 2083, Springer-Verlag, pp. 421-426.

  • Tammet, T. (1994) Proof strategies in linear logic, J. Automated Reasoning 12, 273-304.

    Google Scholar 

  • Waaler, A. (2001) Connections in nonclassical logics, in J. A. Robinson and A. Voronkov (eds), Handbook of Automated Reasoning, Elsevier and MIT Press, Chapt. 22, pp. 1487-1578.

  • Wallen, L. (1990) Automated Deduction in Nonclassical Logics, MIT Press.

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Cornell University, Ithaca, NY, 14853-7501, U.S.A.

    Christoph Kreitz

  2. ETH Zürich, Information Security, Haldeneggsteig 4, CH-8092, Zurich, Switzerland

    Heiko Mantel

Authors
  1. Christoph Kreitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heiko Mantel
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kreitz, C., Mantel, H. A Matrix Characterization for Multiplicative Exponential Linear Logic. Journal of Automated Reasoning 32, 121–166 (2004). https://doi.org/10.1023/B:JARS.0000029976.22387.ac

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JARS.0000029976.22387.ac

Search

Navigation