Abstract
The logic of definitional reflection is extended with a theory of free equality. Based on this equality theory a sequent-style notion of the completion of a definition is motivated. Definitional reflection with free equality turns out to be equivalent to the completion in this sense.
I would like to thank Roy Dyckhoff, Torkel Franzén, an anonymous reviewer and especially Robert Stärk for many helpful comments and suggestions. — This work was supported by DFG grants Schr 275/8-1 and Schr 275/11-1, and by Esprit Basic Research Working Group 7232 (GENTZEN).
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References
Aronsson, M., GCLA: The Design, Use, and Implementation of a Program Development System. Ph.D. thesis, University of Stockholm 1993.
Aronsson, M., Eriksson, L.-H., Gäredal, A., Hallnäs, L. & Olin, P. The programming language GCLA: A definitional approach to logic programming. New Generation Computing, 4 (1990), 381–404.
Clark, K. L. Negation as failure. In: Gallaire, H. & Minker, J. (Eds.), Logic and Data Bases, New York 1978, 293–322.
Coquand, T. Pattern matching with dependent types. In: Workshop on Logical Frameworks, Båstad 1992, Proceedings, available by ftp from ftp.cs.chalmers.se as /pub/cs/reports/baastad.92/procSS.
Eriksson, L.-H. A finitary version of the calculus of partial inductive definitions. In: Eriksson, L.-H., Hallnäs, L. & Schroeder-Heister, P. (Eds.), Extensions of Logic Programming. Second International Workshop, ELP-91, Stockholm, January 1991, Proceedings. Springer LNCS, Vol. 596, Berlin 1992, 89–134.
Eriksson, L.-H. Finitary Partial Inductive Definitions and General Logic. Ph.D. thesis, Royal Institute of Technology, Stockholm 1993.
Girard, J.-Y. Linear logic. Theoretical Computer Science, 50 (1987), 1–102.
Girard, J.-Y. A fixpoint theorem for linear logic. In: P. Lincoln (Ed.), Linear Logic Mailing List, linear@cs.stanford.edu, 5 February 1992 (Reply: ibid., 19 February 1992).
Hallnäs, L. & Schroeder-Heister, P. A proof-theoretic approach to logic programming. I. Clauses as rules. Journal of Logic and Computation, 1 (1990), 261–283; II. Programs as definitions, ibid. 1 (1991), 635–660. Originally published as SICS Research Report 88005, 1988.
Jäger, G. & Stärk, R. F. A proof-theoretic framework for logic programming. In: S. Buss (Ed.), Handbook of Proof Theory (forthcoming).
Kanger, S. A simplified proof method for elementary logic. In: Braffort, P. & Hirschberg, D. (Eds.), Computer Programming and Formal Systems, Amsterdam 1963, 87–94.
Kreuger, P. Axioms in definitional calculi. This volume.
Lifshits, V.A. Normal form for deductions in predicate calculus with equality and functional symbols. In: Slisenko, A.O. (Ed.), Studies in Constructive Mathematics and Mathematical Logic I, New York 1969, 21–23.
Martelli, A. & Montanari, U. An efficient unification algorithm. ACM Transactions on Programming Languages and Systems, 4 (1982), 259–282.
Richter, M.M. Logikkalküle. Teubner, Stuttgart 1978.
Sahlin, D., Franzén, T. & Haridi, S. An intuitionistic predicate logic theorem prover. Journal of Logic and Computation 2 (1992), 619–656.
Schroeder-Heister, P. Cut-elimination in logics with definitional reflection. In: D. Pearce & H. Wansing (Eds.), Nonclassical Logics and Information Processing. International Workshop, Berlin 1990, Proceedings. Springer LNCS, Vol. 619, Berlin 1992, 146–171.
Schroeder-Heister, P. Rules of definitional reflection. In: 8th Annual IEEE Symposium on Logic in Computer Science (Montreal 1993). IEEE Computer Society Press, Los Alamitos 1993, 222–232.
Schroeder-Heister, P. Cut elimination for logics with definitional reflection and restricted initial sequents. Manuscript, available by ftp as /pub/LS/resini from gopher.informatik.uni-tuebingen.de.
Snyder, W. A Proof Theory for General Unification. Birkhäuser, Basel 1991.
Stärk, R. F. Cut-property and negation as failure. Technical report, Institut für Informatik und angewandte Mathematik, Universität Bern, 1992 (available by ftp as /pub/staerk/cut from ftp.cis.uni-muenchen.de).
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Schroeder-Heister, P. (1994). Definitional reflection and the completion. In: Dyckhoff, R. (eds) Extensions of Logic Programming. ELP 1993. Lecture Notes in Computer Science, vol 798. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58025-5_65
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DOI: https://doi.org/10.1007/3-540-58025-5_65
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