Abstract
This paper describes the logical and philosophical background of an extension of logic programming which uses a general schema for introducing assumptions and thus presents a new view of hypothetical reasoning. The detailed proof theory of this system is given in [7], matters of implementation and control of the corresponding programming language GCLA with detailed examples can be found in [1, 2]. In Section 1 we consider the local rule-based approach to a notion of atomic consequence as opposed to the global logical approach. Section 2 describes our system and characterises the inference schema of definitional reflection which is central for our approach. In Section 3 we motivate the computational interpretation of this system. Finally, Section 4 relates our approach to the idea of logical frameworks and the way elimination inferences for logical constants are treated therein, and thus to the notions of logic and structure. It shows that from a certain perspective, logical reasoning is nothing but a special case of reasoning in our system.
I would like to thank Michael Morreau for helpful suggestions.
Preview
Unable to display preview. Download preview PDF.
References
Aronsson, M., Eriksson, L.-H., Gäredal, A., Hallnäs, L. & Olin, P. The programming language GCLA: A definitional approach to logic programming. New Generation Computing, 7 (1990), 381–404.
Aronsson, M., Eriksson, L.-H., Hallnäs, L. & Kreuger, P. A survey of GCLA: A definitional approach to logic programming (this volume).
Beeson, M. Some applications of Gentzen's proof theory in automated deduction (this volume).
Belnap, N. D. Display logic. Journal of Philosophical Logic, 11 (1982), 375–417.
Gabbay, D.M. & Reyle, U. N-PROLOG: An extension of PROLOG with hypothetical implications: I., Journal of Logic Programming, 1 (1984), 319–355.
Gentzen, G. Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39 (1935), 176–210, 405–431, English translation in: M.E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, Amsterdam: North Holland, 1969, 68–131.
Hallnäs, L. & Schroeder-Heister, P. A proof-theoretic approach to logic programming. SICS Research Report, no. 88005, 1988. To appear in revised form in Journal of Logic and Computation.
Hermes, H. Zum Inversionspirnzip der operativen Logik. In: A. Heyting (ed.), Constructivity in Mathematics, Amsterdam: North-Holland, 1961, 62–68.
Lorenzen, P. Einführung in die operative Logik und Mathematik, Berlin: Springer, 1955.
Martin-Löf, P. Hauptsatz for the intuitionistic theory of iterated inductive definitions. In: J. E. Fenstad (ed.), Proceedings of the Second Scandinavian Logic Symposium, Amsterdam: North Holland, 1971, 179–216.
Miller, D. A theory of modules for logic programming. In: Proceedings of the 1986 Symposium on Logic Programming (Salt Lake City Utah), IEEE Computer Society Press, Washington, 1986.
Schroeder-Heister, P. The role of elimination inferences in a structural framework. In: G. Huet (ed.), Proceedings of the Esprit BRA Logical Frameworks Workshop, Sophia Antipolis 1990.
Schroeder-Heister, P. A natural extension of natural deduction. Journal of Symbolic Logic, 49 (1984), 1284–1300.
Schroeder-Heister, P. Logic programming with weak structural rules. In preparation.
Schroeder-Heister, P. Structural Frameworks with Higher-Level Rules: Proof-Theoretic Investigations. Habilitationsschrift. Universität Konstanz, 1987.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schroeder-Heister, P. (1991). Hypothetical reasoning and definitional reflection in logic programming. In: Schroeder-Heister, P. (eds) Extensions of Logic Programming. ELP 1989. Lecture Notes in Computer Science, vol 475. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0038701
Download citation
DOI: https://doi.org/10.1007/BFb0038701
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-53590-4
Online ISBN: 978-3-540-46879-0
eBook Packages: Springer Book Archive