Abstract
Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the meta-theory of linear logic can be used to draw conclusions about the specified sequent calculus. For example, derivability of one proof system from another can be decided by a simple procedure that is implemented via bounded logic programming-style search. Also, simple and decidable conditions on the linear logic presentation of inference rules, called homogeneous and coherence, can be used to infer that the initial rules can be restricted to atoms and that cuts can be eliminated. In the present paper we introduce Llinda, a logical framework based on linear logic augmented with inference rules for definition (fixed points) and induction. In this way, the above properties can be proved entirely inside the framework. To further illustrate the power of Llinda, we extend the definition of coherence and provide a new, semi-automated proof of cut-elimination for Girard’s Logic of Unicity (LU).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2(3), 297–347 (1992)
Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56–68 (1940)
Felty, A., Miller, D.: Specifying theorem provers in a higher-order logic programming language. In: Ninth International Conference on Automated Deduction (1988)
Gentzen, G.: Investigations into logical deductions. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North-Holland Publishing Co., Amsterdam (1969)
Girard, J.-Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)
Girard, J.-Y.: On the unity of logic. Ann. of Pure and Applied Logic 59, 201–217 (1993)
Girard, J.-Y.: On the meaning of logical rules I: syntax vs. semantics. In: Computational Logic, Berger and Schwichtenberg, pp. 215–272. SV (1999)
Guglielmi, A.: A system of Interaction and Structure. ACM Transactions in Computational Logic (2005) (to appear)
Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. Journal of the ACM 40(1), 143–184 (1993)
Miller, D.: Forum: A multiple-conclusion specification language. Theoretical Computer Science 165(1), 201–232 (1996)
McDowell, R., Miller, D.: Cut-elimination for a logic with definitions and induction. Theoretical Computer Science 232, 91–119 (2000)
Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Ann. of Pure and Applied Logic 51, 125–157 (1991)
Miller, D., Pimentel, E.: Linear logic as a framework for specifying sequent calculus. In: Lecture Notes in Logic 17, Logic Colloquium 1999 (2004)
Miller, D., Pimentel, E.: Using linear logic to reason about sequent systems. In: Egly, U., Fermüller, C. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, p. 2. Springer, Heidelberg (2002)
Nadathur, G., Miller, D.: An Overview of λProlog. In: Fifth International Logic Programming Conference, pp. 810–827. MIT Press, Cambridge (August 1988)
Pfenning, F.: Elf: A Language for Logic Definition and Verified Metaprogramming. In: Fourth Annual Symposium on Logic in Computer Science (1989)
Pfenning, F.: Structural Cut Elimination. In: Proceedings, Tenth Annual IEEE Symposium on Logic in Computer Science (1995)
Pfenning, F.: Structural Cut Elimination: I. Intuitionistic and Classical Logic. Information and Computation 157(1-2), 84–141 (2000)
Pimentel, E.G.: Lógica linear e a especificação de sistemas computacionais. PhD thesis, Universidade Federal de Minas Gerais, Belo Horizonte, M.G., Brasil (written in English) (December 2001)
Pimentel, E.G.: Cut elimination for Llinda (2005), Draft, available from http://www.mat.ufmg.br/~elaine
Tiu, A.: A Logical Framework for Reasoning about Logical Specifications. PhD thesis, Penn State University (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pimentel, E., Miller, D. (2005). On the Specification of Sequent Systems. In: Sutcliffe, G., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2005. Lecture Notes in Computer Science(), vol 3835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11591191_25
Download citation
DOI: https://doi.org/10.1007/11591191_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30553-8
Online ISBN: 978-3-540-31650-3
eBook Packages: Computer ScienceComputer Science (R0)