Abstract
Using Dedekind categories as an algebraic structure for (binary) set-theoretic relations without complements, we present purely algebraic definitions of “to be bipartite” and “to possess no odd cycles” and prove that both notions coincide. This generalises D. König’s well-known theorem from undirected graphs to abstract relations, and, hence, to models such as L-relations that are different from set-theoretic relations. One direction of this generalisation is shown by specifying a bipartition for the relation in question in form of a pair of disjoint relational vectors. For set-theoretic relations this immediately leads to relational programs for computing bipartitions. We also discuss how the algebraic proofs can be mechanised using theorem proving tools.
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.
Similar content being viewed by others
References
Berghammer, R., Neumann, F.: RelView – An OBDD-based Computer Algebra system for relations. In: Gansha, V.G., Mayr, E.W., Vorozhtsov, E. (eds.) Computer Algebra in Scientific Computing. LNCS, vol. 3718, pp. 40–51. Springer, Heidelberg (2005)
Berghammer, R., Winter, M.: Embedding mappings and splittings with applications. Acta Informatica 47, 77–110 (2010)
Berghammer, R., Struth, G.: On automated program construction and verification. In: Bolduc, C., Desharnais, J., Ktari, B. (eds.) MPC 2010. LNCS, vol. 6120, pp. 22–41. Springer, Heidelberg (2010)
Berghammer, R., Höfner, P., Stucke, I.: Automated verification of relational while-programs. In: Widłak, W. (ed.) Molecular Biology - Not Only for Bioinformatics. LNCS, vol. 8248, pp. 309–326. Springer, Heidelberg (2013)
Diestel, R.: Graph theory, 3rd edn. Springer (2005)
Foster, S., Struth, G., Weber, T.: Automated engineering of relational and algebraic methods in Isabelle/HOL. In: de Swart, H. (ed.) Relational and Algebraic Methods in Computer Science. LNCS, vol. 6663, pp. 52–67. Springer, Heidelberg (2011)
Freyd, P., Scedrov, A.: Categories, allegories. North-Holland (1990)
Furusawa, H., Kawahara, Y., Winter, M.: Dedekind categories with cutoff operators. Fuzzy Sets and Systems 173, 1–24 (2011)
Höfner, P., Struth, G.: On automating the calculus of relations. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 50–66. Springer, Heidelberg (2008)
König, D.: Über Graphen und ihre Anwendung in der Determinantentheorie und Mengenlehre. Mathematische Annalen 77, 453–465 (1916)
Ng, K.C., Tarski, A.: Relation algebras with transitive closure. Abstract 742-02-09, Notices Amer. Math. Soc. 24, A29-A30 (1977)
Oliver, J.P., Serrato, D.: Categories de Dedekind: Morphismes dans les categories de Schröder. C.R. Acad. Sci.Paris 290, 939–941 (1980)
Pous, D.: Kleene algebra with tests and Coq tools for while programs. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 180–196. Springer, Heidelberg (2013)
Pous, D.: Relation algebra and KAT in Coq. http://perso.ens-lyon.fr/damien.pous/ra/
Rubin, H., Rubin, J.E.: Equivalents of the Axiom of Choice. North-Holland (1970)
Schmidt, G., Ströhlein, T.: Relations and graphs, Discrete mathematics for computer scientists, EATCS Monographs on Theoretical Computer Science, Springer (1993)
Schmidt, G.: Relational mathematics. Encyclopedia of Mathematics and its Applications, vol. 132. Cambridge University Press (2010)
Tarski, A.: On the calculus of relations. Journal of Symbolic Logic 6, 73–89 (1941)
Tarski, A., Givant, S.: A formalization of set theory without variables. Colloquium Publications 41. American Mathematical Society (1987)
Winter, M.: Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdetermismus in Programmiersprachen. Dissertation, Fakultät für Informatik, Universität der Bundeswehr München, Dissertationsverlag NG Kopierladen GmbH (1998)
Winter, M.: An ordered category of processes. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS/AKA 2008. LNCS, vol. 4988, pp. 367–381. Springer, Heidelberg (2008)
Winter, M.: Complements in distributive allegories. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds.) RelMiCS 2009. LNCS, vol. 5827, pp. 337–350. Springer, Heidelberg (2009)
Coq-homepage: http://coq.infia.fr
Prover9-homepage: http://www.prover9.org
RelView-homepage: http://www.informatik.uni-kiel.de/~progsys/relview/
Input files, proof scripts: http://media.informatik.uni-kiel.de/Ramics2015/
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Berghammer, R., Stucke, I., Winter, M. (2015). Investigating and Computing Bipartitions with Algebraic Means. In: Kahl, W., Winter, M., Oliveira, J. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2015. Lecture Notes in Computer Science(), vol 9348. Springer, Cham. https://doi.org/10.1007/978-3-319-24704-5_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-24704-5_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24703-8
Online ISBN: 978-3-319-24704-5
eBook Packages: Computer ScienceComputer Science (R0)