Towards an Algebra of Routing Tables

  • Peter Höfner
  • Annabelle McIver
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6663)


We use well-known algebraic concepts like semirings and matrices to model and argue about Wireless Mesh Networks. These networks are used in a wide range of application areas, including public safety and transportation. Formal reasoning therefore seems to be necessary to guarantee safety and security. In this paper, we model a simplified algebraic version of the AODV protocol and provide some basic properties. For example we show that each node knows a route to the originator of a message (if there is one).


Control Message Wireless Mesh Network Route Request Route Reply Route Table 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    AODV-UU: An implementation of the AODV routing protocol (IETF RFC 3561), (accessed February 26, 2011)
  2. 2.
    Backhouse, R.: Closure Algorithms and the Star-Height Problem of Regular Languages. Ph.D. thesis, Imperial College, London (1975)Google Scholar
  3. 3.
    Backhouse, R., Carré, B.A.: Regular algebra applied to path-finding problems. Journal of the Institute of Mathematics and Applications (1975)Google Scholar
  4. 4.
    Carré, B.A.: Graphs and Networks. Oxford Applied Mathematics & Computing Science Series. Oxford University Press, Oxford (1980)zbMATHGoogle Scholar
  5. 5.
    Conway, J.H.: Regular Algebra and Finite Machines. Chapman & Hall, Boca Raton (1971)zbMATHGoogle Scholar
  6. 6.
    Desharnais, J., Möller, B., Struth, G.: Modal Kleene algebra and applications — A survey. Journal of Relational Methods in Computer Science 1, 93–131 (2004)zbMATHGoogle Scholar
  7. 7.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fernandes, T., Desharnais, J.: Describing data flow analysis techniques with Kleene algebra. SCP 65, 173–194 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Griffin, T.G., Gurney, A.J.T.: Increasing bisemigroups and algebraic routing. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS/AKA 2008. LNCS, vol. 4988, pp. 123–137. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Griffin, T.G., Sobrinho, J.: Metarouting. SIGCOMM Comp. Com. Rev. 35, 1–12 (2005)CrossRefGoogle Scholar
  11. 11.
    Hoare, C.A.R., Möller, B., Struth, G., Wehrman, I.: Concurrent Kleene algebra. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 399–414. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Höfner, P.: Database for automated proofs of Kleene algebra, (accessed February 26, 2011)
  13. 13.
    Höfner, P., Struth, G.: Automated reasoning in kleene algebra. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 279–294. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Kozen, D.: The Design and Analysis of Algorithms. Springer, Heidelberg (1991)zbMATHGoogle Scholar
  15. 15.
    Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    McCune, W.W.: Prover9 and Mace4, (accessed February 26, 2011)
  17. 17.
    McIver, A.K., Gonzalia, C., Cohen, E., Morgan, C.C.: Using probabilistic Kleene algebra pKA for protocol verification. J. Logic and Algebraic Programming 76(1), 90–111 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Möller, B.: Dijkstra, Kleene, Knuth. Talk at WG2.1 Meeting, slides available online, at (accessed February 26, 2006)
  19. 19.
    Möller, B., Struth, G.: WP is WLP. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 200–211. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Perkins, C., Belding-Royer, E., Das, S.: Ad hoc on-demand distance vector (AODV) routing. RFC 3561 (Experimental) (July 2003),
  21. 21.
    Singh, A., Ramakrishnan, C.R., Smolka, S.A.: A process calculus for mobile ad hoc networks. SCP 75, 440–469 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Sobrinho, J.: Algebra and algorithms for QoS path computation and hop-by-hop routing in the internet. IEEE/ACM Trans. Networking 10(4), 541–550 (2002)CrossRefGoogle Scholar
  23. 23.
    Sobrinho, J.: Network routing with path vector protocols: Theory and applications. In: Applications, Technologies, Architectures, and Protocols for Computer Communications. SIGCOMM 2003, pp. 49–60. ACM Press, New York (2003)Google Scholar
  24. 24.
    Takai, T., Furusawa, H.: Monodic tree kleene algebra. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 402–416. Springer, Heidelberg (2006) (accessed February 26, 2011) Errata available at monodic_kleene_algebra.pdf Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Peter Höfner
    • 1
    • 3
  • Annabelle McIver
    • 2
    • 3
  1. 1.Institut für InformatikUniversität AugsburgGermany
  2. 2.Department of ComputingMacquarie UniversityAustralia
  3. 3.National ICT Australia Ltd. (NICTA)Australia

Personalised recommendations