Abstract
We present a theorem proving system for abstract relation algebra called RALL (=Relation-Algebraic Language and Logic), based on the generic theorem prover Isabelle. On the one hand, the system is an advanced case study for Isabelle/HOL, and on the other hand, a quite mature proof assistant for research on the relational calculus. RALL is able to deal with the full language of heterogeneous relation algebra including higher-order operators and domain constructions, and checks the type-correctness of all formulas involved. It offers both an interactive proof facility, with special support for substitutions and estimations, and an experimental automatic prover. The automatic proof method exploits an isomorphism between relation-algebraic and predicate-logical formulas, relying on the classical universal-algebraic concepts of atom structures and complex algebras.
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
Backhouse, R.C., Hoogendijk, P., Voermans, E., van der Woude, J.C.S.P: A relational theory of datatypes. Eindhoven University of Technology, Dept. of Mathematics and Computer Science (1992)
Berghammer, R., Hattensperger, C., Schmidt, G.: RALF — A relation-algebraic formula manipulation system and proof checker. In: Nivat, M., Rattray, C., Rus, T., Scollo, G. (Eds.): Proc. 3rd Conference on Algebraic Methodology and Software Technology — AMAST '93. Series: Workshops in Computing, Springer-Verlag (1994) 407–408
Berghammer, R., Schmidt, G.: Relational specifications. In: Rauszer C. (ed.): Algebraic Methods in Logic and in Computer Science. Series: Banach Center Publications 28, Polish Academy of Sciences (1993) 167–190
Birkhoff, G.: Lattice Theory. AMS Colloquium Publications 25 (31967)
Brink, C., Schmidt, G.: Relational methods in computer science. Schloß Dagstuhl, Seminar Nr. 9403, Technischer Bericht Nr. 80 (1994)
Church, A.: A Formulation of the Simple Theory of Types. In: Journal of Symbolic Logic (1940) 56
Desharnais, J., Baltagi, S., Chaib-draa, B.: Simple weak sufficient conditions for sharpness. Université Laval, Quebec, Research Report DIUL-RR-9404 (1994)
Gritzner, T.F.: Die Axiomatik abstrakter Relationenalgebren: Darstellung der Grundlagen und Anwendung auf das Unschärfeproblem relationaler Produkte. Technische Universität München, Diploma Thesis, also available as report: TUM-INFO-04-91 (1991)
Gritzner, T.F.: wp-Kalkül und relationale Spezifikation kommunizierender Systeme. Technische Universität München, Doctoral Dissertation (1995) Herbert Utz Verlag Wissenschaft, ISBN 3-931327-64-7 (1996)
Haeberer, A.M. (Ed.): Relational methods in computer science. Proceedings of PARATI '95. In preparation.
Henkin, L., Monk, J.D., Tarski, A., Cylindric Algebras, Parts I & II, Series: Studies in Logic and the Foundations of Mathematics 64 (1971) & 115 (1985), North-Holland Publ.Co.
Hoare, C.A.R., He Jifeng: The weakest prespecification, parts I&II. In: Fundamenta Informaticae IX (1986) 51–84 & 217–252
Jónsson, B., Tarski, A.: Boolean algebras with operators, parts I&II. In: American Journal of Mathematics 73 (1951) 891–939 & 74 (1952) 127–167
Kawahara, Y., Furusawa, H.: An algebraic formalization of fuzzy relations. Draft paper (April 19, 1995)
Lyndon, R.C., The representation of relation(al) algebras, parts I&II, in: Annals of Mathematics (II) 51 (1950) 707–729 & 63 (1956) 294–307, Princeton University Press
Maddux, R.: Finite integral relation algebras. In: Comer, S.D. (ed.), Universal Algebra and Lattice Theory, Lecture Notes in Mathematics 1149 (1985) 175–197, Springer-Verlag
Nipkow, T.: Term Rewriting and Beyond — Theorem Proving in Isabelle. In: Formal Aspects of Computing 1 (1989) 320–338
Nipkow, T.: Order-Sorted Polymorphism in Isabelle. In: Huet, G., Plotkin, G. (eds.): Logical Environments. Cambridge University Press (1993) 164–188
von Oheimb, D.: Zur Konstruktion eines auf Isabelle gestützten Beweissystems für die Relationenalgebra. Technische Universität München, Praktische Semesterarbeit (1995)
Ounalli, H., Jaoua, A.: On fuzzy difunctional relations. Draft paper presented at 2nd RelMiCS — PARATI '95.
Paulson, L.C.: ML for the Working Programmer. Cambridge University Press (1991)
Paulson, L.C.: Isabelle — A Generic Theorem Prover. Lecture Notes in Computer Science 828 (1994)
Rasmussen, O.: Formalizing Ruby in Isabelle ZF. In: Paulson, L.C. (Ed.): Proceedings of the First Isabelle Users Workshop. University of Cambridge (1995) 246–265
de Roever, W.P.: Recursive Program Schemes: Semantics and Proof Theory. Vrije Universiteit te Amsterdam, Doctoral Dissertation (1974)
Schmidt, G., Ströhlem, Th.: Relations and Graphs. Series: EATCS Monographs in Computer Science, Springer-Verlag (1993)
Tarski, A.: On the calculus of relations. In: Journal of Symbolic Logic 6 (1941) 73–89
Tarski, A., Givant, S.: A Formalization of Set Theory Without Variables. AMS Colloquium Publications 41 (1987)
Zierer, H.: Programmierung mit Funktionsobjekten: Konstruktive Erzeugung semantischer Bereiche und Anwendung auf die partielle Auswertung. Technische Universität München, Doctoral Dissertation, also available as: TUM-I8803 (1988)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
von Oheimb, D., Gritzner, T.F. (1997). RALL: Machine-supported proofs for relation algebra. In: McCune, W. (eds) Automated Deduction—CADE-14. CADE 1997. Lecture Notes in Computer Science, vol 1249. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63104-6_36
Download citation
DOI: https://doi.org/10.1007/3-540-63104-6_36
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63104-0
Online ISBN: 978-3-540-69140-2
eBook Packages: Springer Book Archive