Abstract
When one views (multi-sorted) existential fixed-point logic (EFPL) as a database query language, it is natural to extend it by allowing universal quantification over certain sorts. These would be the sorts for which one has the “closed-world” information that all entities of that sort in the real world are represented in the database. We investigate the circumstances under which this extension of EFPL retains various pleasant properties. We pay particular attention to the pleasant property of preservation by the inverse-image parts of geometric morphisms of topoi, because, as we show, this preservation property implies many of the other pleasant properties of EFPL.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Apt, K.: Ten Years of Hoare’s Logic: A Survey – Part I. ACM Trans. Prog. Lang. and Systems 3, 431–483 (1981)
Artin, M., Grothendieck, A., Verdier, J.-L.: Théorie des Topos et Cohomologie Étale des Schémas. In: Séminaire de Géométrie Algébrique du Bois Marie 1963–64 (SGA) 4, vol. 1, Lecture Notes in Mathematics, vol. 269. Springer, Heidelberg (1972)
Bell, J.: Toposes and Local Set Theories. Oxford Logic Guides, vol. 14. Oxford University Press, Oxford (1988)
Blass, A.: Topoi and Computation. Bull. European Assoc. Theoret. Comp. Sci. 36, 57–65 (1988)
Blass, A.: Geometric Invariance of Existential Fixed-Point Logic. In: Gray, J., Scedrov, A. (eds.) Categories in Computer Science and Logic. Contemp. Math., vol. 92, pp. 9–22. Amer. Math. Soc., Providence (1989)
Blass, A., Gurevich, Y.: Existential Fixed-Point Logic. In: Börger, E. (ed.) Computation Theory and Logic. LNCS, vol. 270, pp. 20–36. Springer, Heidelberg (1987)
Blass, A., Gurevich, Y.: The Underlying Logic of Hoare Logic. Bull. European Assoc. Theoret. Comp. Sci. 70, 82–110 (2000); Reprinted in Paun, G., Rozenberg, G., Salomaa, A.: Current Trends in Theoretical Computer Science: Entering the 21st Century, pp. 409–436. World Scientific, Singapore (2001)
Blass, A., Ščedrov, A. (later simplified to Scedrov): Classifying Topoi and Finite Forcing. J. Pure Appl. Algebra 28, 111–140 (1983)
Chandra, A., Harel, D.: Horn Clause Queries and Generalizations. J. Logic Programming 2, 1–15 (1985)
Cook, S.: Soundness and Completeness of an Axiom System for Program Verification. SIAM J. Computing 7, 70–90 (1978)
Immerman, N.: Relational Queries Computable in Polynomial Time. Information and Control 68, 86–104 (1986); Preliminary version in 14th ACM Symp. on Theory of Computation (STOC), pp. 147–152 (1982)
Johnstone, P.: Topos Theory. London Mathematical Society Monographs, vol. 10. Academic Press, London (1977)
Vardi, M.: Complexity of Relational Query Languages. In: 14th ACM Symp. on Theory of Computation (STOC), pp. 137–146 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Blass, A. (2010). Existential Fixed-Point Logic, Universal Quantifiers, and Topoi. In: Blass, A., Dershowitz, N., Reisig, W. (eds) Fields of Logic and Computation. Lecture Notes in Computer Science, vol 6300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15025-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-15025-8_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15024-1
Online ISBN: 978-3-642-15025-8
eBook Packages: Computer ScienceComputer Science (R0)