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Relation Algebras, Matrices, and Multi-valued Decision Diagrams

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Relational and Algebraic Methods in Computer Science (RAMICS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7560))

Abstract

In this paper we want to further investigate the usage of matrices as a representation of relations within arbitrary heterogeneous relation algebras. First, we want to show that splittings do exist in matrix algebras assuming that the underlying algebra of the coefficients provides this operation. Second, we want to outline an implementation of matrix algebras using reduced ordered multi-valued decision diagrams. This implementation combines the efficiency of operations based on those data structures with the general matrix approach to arbitrary relation algebras.

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References

  1. Atampore, F., Winter, M.: Relation Algebras, Matrices, and Multi-Valued Decision Diagrams. Brock University, Dep. of Computer Science Report CS-12-03 (2012), http://www.cosc.brocku.ca/research/reports

  2. Bahar, I.R., Frohm, E.A., Gaona, C.M., Hachtel, G.D., Macii, E., Pardo, A., Somenzi, F.: Algebraic Decision Diagrams amd their Applications. In: Proceedings of the International Conference on Computer-Aided Design, pp. 188–191. IEEE (1993)

    Google Scholar 

  3. Berghammer, R., Neumann, F.: RelView – An OBDD-Based Computer Algebra System for Relations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 40–51. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  4. Berghammer, R.: Applying Relation Algebra and RelView to solve Problems on Orders and Lattices. Acta Informatica 45(3), 211–236 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berghammer, R., Winter, M.: Embedding Mappings and Aplittings with Applications. Acta Informatica 47(2), 77–110 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berghammer, R., Rusinowska, A., de Swart, H.C.M.: Applying Relation Algebra and RelView to Measures in a Social Network. European Journal of Operational Research 202, 182–195 (2010)

    Article  MATH  Google Scholar 

  7. Bird, R., de Moor, O.: Algebra of Programming. Prentice Hall (1997)

    Google Scholar 

  8. Brace, K.S., Rudell, L.R., Bryant, E.R.: Efficient Implementation of a BDD Package. In: ACM/IEEE Design Automation Conference, pp. 40–42. IEEE (1990)

    Google Scholar 

  9. Bryant, R.E.: Graph-based Algorithms for Boolean Function Manipulation. IEEE Trans. on Computers 35(8), 677–691 (1986)

    Article  MATH  Google Scholar 

  10. Chin, L.H., Tarski, A.: Distributive and Modular Laws in the Arithmetic of Relation Algebras. University of California Press, Berkley and Los Angeles (1951)

    Google Scholar 

  11. Freyd, P., Scedrov, A.: Categories, Allegories. North-Holland (1990)

    Google Scholar 

  12. Miller, D.M., Drechsler, R.: Implementing a Multiple-Valued Decision Diagram Package. In: Proceedings of the 28th International Symposium on Multiple-Valued Logic (ISMVL 1998), pp. 52–57. IEEE (1998)

    Google Scholar 

  13. Nagayama, S., Sasao, T.: Code Generation for Embedded Systems using Heterogeneous MDDs. In: 12th Workshop on Synthesis and System Integration of Mixed Information Technologies (SASIMI 2003), pp. 258–264 (2003)

    Google Scholar 

  14. Nagayama, S., Sasao, T.: Compact Representations of Logic Functions using Heterogeneous MDDs. In: Proceedings of the 33rd International Symposium on Multiple-Valued Logic (ISML 2003), pp. 3168–3175. IEEE (2003)

    Google Scholar 

  15. Balbiani, P., Orlowska, E.: A Hierarchy of Modal Logics with Relative Accessibility Relations. Journal of Applied Non-Classical Logics 9(2-3) (1999)

    Google Scholar 

  16. Schmidt, G., Ströhlein, T.: Relationen und Graphen. Springer (1989); English version: Relations and Graphs. Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoret. Comput. Sci. Springer (1993)

    Google Scholar 

  17. Schmidt, G.: Relational Mathematics. Encyclopedia of Mathematics and its Applications, vol. 132. Cambridge University Press (2010)

    Google Scholar 

  18. Schmidt, G., Hattensperger, C., Winter, M.: Heterogeneous Relation Algebras. In: Brink, C., Kahl, W., Schmidt, G. (eds.) Relational Methods in Computer Science. Advances in Computer Science. Springer, Vienna (1997)

    Google Scholar 

  19. Somenzi, F.: CUDD: CU Decision Diagram Package; Release 2.5.0. Department of Electrical, Computer, and Energy Engineering, University of Colorado (2012), http://vlsi.colorado.edu/~fabio/CUDD/cuddIntro.html

  20. Tarski, A., Givant, S.: A Formalization of Set Theory without Variables. Amer. Math. Soc. Colloq. Publ.4̃1 (1987)

    Google Scholar 

  21. Winter, M.: Strukturtheorie Heterogener Relationenalgebren mit Anwendung auf Nichtdetermismus in Programmiersprachen. Dissertationsverlag NG Kopierladen GmbH, München (1998)

    Google Scholar 

  22. Winter, M.: Relation Algebras are Matrix Algebras over a suitable Basis. University of the Federal Armed Forces Munich, Report Nr. 1998-05 (1998)

    Google Scholar 

  23. Winter, M.: A Pseudo Representation Theorem for various Categories of Relations. TAC Theory and Applications of Categories 7(2), 23–37 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Winter, M.: Weak Relational Products. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 417–431. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  25. Winter, M.: Goguen Categories - A Categorical Approach to L-fuzzy Relations. Trends in Logic, vol. 25 (2007)

    Google Scholar 

  26. Zierer, H.: Relation Algebraic Domain Constructions. Theoret. Comput. Sci. 87, 163–188 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Brock University, St. Catharines, ON, Canada

    Francis Atampore & Michael Winter

Authors
  1. Francis Atampore
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  2. Michael Winter
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Editor information

Editors and Affiliations

  1. Department of Computing and Software, McMaster University, 1280 Main Street West, L8S 4K1, Hamilton, ON, Canada

    Wolfram Kahl

  2. Computer Laboratory, University of Cambridge, 15 JJ Thomson Avenue, CB3 0FD, Cambridge, UK

    Timothy G. Griffin

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Atampore, F., Winter, M. (2012). Relation Algebras, Matrices, and Multi-valued Decision Diagrams. In: Kahl, W., Griffin, T.G. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2012. Lecture Notes in Computer Science, vol 7560. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33314-9_17

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  • DOI: https://doi.org/10.1007/978-3-642-33314-9_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33313-2

  • Online ISBN: 978-3-642-33314-9

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