Abstract
DFOL is a logic that extends first-order logic with dependent types. We give a translation from DFOL to FOL formalized as an institution comorphism and show that it admits the model expansion property. This property together with the borrowing theorem implies the soundness of borrowing — a result that enables us to reason about entailment in DFOL by using automated tools for FOL. In addition, the translation permits us to deduce properties of DFOL such as completeness, compactness, and existence of free models from the corresponding properties of FOL, and to regard DFOL as a fragment of FOL. We give an example that shows how problems about DFOL can be solved by using the automated FOL prover Vampire. Future work will focus on the integration of the translation into the specification and translation tool HeTS.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Astesiano, E., Bidoit, M., Kirchner, H., Krieg-Brückner, B., Mosses, P., Sannella, D., Tarlecki, A.: CASL: The Common Algebraic Specification Language. Theoretical Computer Science (2002)
Benzmüller, C., Cheikhrouhou, L., Fehrer, D., Fiedler, A., Huang, X., Kerber, M., Kohlhase, M., Konrad, K., Melis, E., Meier, A., Schaarschmidt, W., Siekmann, J., Sorge, V.: ΩMEGA: Towards a mathematical assistant. In: McCune, W. (ed.) Proceedings of the 14th Conference on Automated Deduction, pp. 252–255. Springer, Heidelberg (1997)
Belo, J.: Dependently Sorted Logic. In: Miculan, M., Scagnetto, I., Honsell, F. (eds.) TYPES 2008, pp. 33–50. Springer, Heidelberg (2008)
Benzmller, C., Paulson, L., Theiss, F., Fietzke, A.: The LEO-II Project. In: Automated Reasoning Workshop (2007)
Brown, C.: Combining Type Theory and Untyped Set Theory. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS, vol. 4130, pp. 205–219. Springer, Heidelberg (2006)
Cerioli, M., Meseguer, J.: May I Borrow Your Logic? In: Borzyszkowski, A., Sokolowski, S. (eds.) Mathematical Foundations of Computer Science, pp. 342–351. Springer, Heidelberg (1993)
Claessen, K., Sorensson, N.: New techniques that improve MACE-style finite model finding. In: 19th International Conference on Automated Deduction (CADE-19) Workshop on Model Computation - Principles, Algorithms, Applications (2003)
Goguen, J., Burstall, R.: Institutions: Abstract model theory for specification and programming. Journal of the Association for Computing Machinery 39(1), 95–146 (1992)
Goguen, J., Winkler, T., Meseguer, J., Futatsugi, K., Jouannaud, J.: Introducing OBJ. In: Goguen, J. (ed.) Applications of Algebraic Specification using OBJ, Cambridge (1993)
Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. Journal of the Association for Computing Machinery 40(1), 143–184 (1993)
Jacobs, B., Melham, T.: Translating dependent type theory into higher order logic. In: Bezem, M., Groote, J. (eds.) Typed Lambda Calculi and Applications, pp. 209–229 (1993)
Mac Lane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1998)
Makkai, M.: First order logic with dependent sorts, FOLDS (1997) (Unpublished)
Martin-Löf, P.: An Intuitionistic Theory of Types: Predicative Part. In: Proceedings of the Logic Colloquium 1973, pp. 73–118 (1975)
Mossakowski, T., Maeder, C., Lüttich, K.: The Heterogeneous Tool Set. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 519–522. Springer, Heidelberg (2007)
Nordström, B., Petersson, K., Smith, J.: Programming in Martin-Löf’s Type Theory: An Introduction. Oxford University Press, Oxford (1990)
Oberschelp, A.: Untersuchungen zur mehrsortigen Quantorenlogik. Mathematische Annalen 145, 297–333 (1962)
Paulson, L.: Isabelle: A Generic Theorem Prover. In: Paulson, L.C. (ed.) Isabelle. LNCS, vol. 828. Springer, Heidelberg (1994)
Pitts, A.: Categorical Logic. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science ch. 2. Algebraic and Logical Structures, vol. 5, pp. 39–128. Oxford University Press, Oxford (2000)
Rabe, F.: First-Order Logic with Dependent Types. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS, vol. 4130, pp. 377–391. Springer, Heidelberg (2006)
Rabe, F.: Representing Logics and Logic Translations. PhD thesis, Jacobs University Bremen (2008)
Riazanov, A., Voronkov, A.: The design and implementation of Vampire. AI Communications 15, 91–110 (2002)
Trybulec, A., Blair, H.: Computer assisted reasoning with Mizar. In: Proceedings of the 9th International Joint Conference on Artificial Intelligence, Los Angeles, CA, pp. 26–28 (1985)
Urban, J.: Translating Mizar for first-order theorem provers. In: Asperti, A., Buchberger, B., Davenport, J.H. (eds.) MKM 2003. LNCS, vol. 2594, pp. 203–215. Springer, Heidelberg (2003)
Weidenbach, C., Afshordel, B., Brahm, U., Cohrs, C., Engel, T., Keen, E., Theobalt, C., Topić, D.: System description: SPASS version 1.0.0. In: Ganzinger, H. (ed.) Proceedings of the 16th International Conference on Automated Deduction (CADE-16). LNCS (LNAI), vol. 1632, pp. 314–318. Springer, Heidelberg (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sojakova, K., Rabe, F. (2009). Translating a Dependently-Typed Logic to First-Order Logic. In: Corradini, A., Montanari, U. (eds) Recent Trends in Algebraic Development Techniques. WADT 2008. Lecture Notes in Computer Science, vol 5486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03429-9_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-03429-9_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03428-2
Online ISBN: 978-3-642-03429-9
eBook Packages: Computer ScienceComputer Science (R0)