We propose axioms for 1-free omega algebra, typed 1-free omega algebra and typed omega algebra. They are based on Kozen’s axioms for 1-free and typed Kleene algebra and Cohen’s axioms for the omega operation. In contrast to Kleene algebra, several laws of omega algebra turn into independent axioms in the typed or 1-free variants.

We set up a matrix algebra over typed 1-free omega algebras by lifting the underlying structure. The algebra includes non-square matrices and care has to be taken to preserve type-correctness. The matrices can represent programs in total and general correctness. We apply the typed construction to derive the omega operation on two such representations, for which the untyped construction does not work.

We embed typed 1-free omega algebra into 1-free omega algebra, and this into omega algebra. Some of our embeddings, however, do not preserve the greatest element. We obtain that the validity of a universal formula using only +, ·,  + , ω and 0 carries over from omega algebra to its typed variant. This corresponds to Kozen’s result for typed Kleene algebra.


Matrix Algebra Great Element General Correctness Total Correctness Universal Formula 
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  1. 1.
    Bloom, S.L., Ésik, Z.: Iteration Theories: The Equational Logic of Iterative Processes. Springer, Heidelberg (1993)CrossRefzbMATHGoogle Scholar
  2. 2.
    Cohen, E.: Separation and reduction. In: Backhouse, R., Oliveira, J.N. (eds.) MPC 2000. LNCS, vol. 1837, pp. 45–59. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, London (1971)zbMATHGoogle Scholar
  4. 4.
    Dunne, S.: Recasting Hoare and He’s Unifying Theory of Programs in the context of general correctness. In: Butterfield, A., Strong, G., Pahl, C. (eds.) 5th Irish Workshop on Formal Methods. Electronic Workshops in Computing, The British Computer Society (2001)Google Scholar
  5. 5.
    Guttmann, W.: General correctness algebra. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds.) RelMiCS/AKA 2009. LNCS, vol. 5827, pp. 150–165. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Guttmann, W.: Extended designs algebraically (submitted, 2011)Google Scholar
  7. 7.
    Guttmann, W., Möller, B.: Normal design algebra. Journal of Logic and Algebraic Programming 79(2), 144–173 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hayes, I.J., Dunne, S.E., Meinicke, L.: Unifying theories of programming that distinguish nontermination and abort. In: Bolduc, C., Desharnais, J., Ktari, B. (eds.) MPC 2010. LNCS, vol. 6120, pp. 178–194. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Kahl, W.: Refactoring heterogeneous relation algebras around ordered categories and converse. Journal on Relational Methods in Computer Science 1, 277–313 (2004)Google Scholar
  10. 10.
    Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kozen, D.: Typed Kleene algebra. Tech. Rep. TR98-1669, Cornell University (1998)Google Scholar
  12. 12.
    Kozen, D.: On Hoare logic, Kleene algebra, and types. In: Gärdenfors, P., Woleński, J., Kijania-Placek, K. (eds.) In the Scope of Logic, Methodology, and Philosophy of Science, Synthese Library, vol. 315, pp. 119–133. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar
  13. 13.
    Mathieu, V., Desharnais, J.: Verification of pushdown systems using omega algebra with domain. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 188–199. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Möller, B.: The linear algebra of UTP. In: Uustalu, T. (ed.) MPC 2006. LNCS, vol. 4014, pp. 338–358. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Pous, D.: Untyping typed algebraic structures and colouring proof nets of cyclic linear logic. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 484–498. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Schmidt, G., Hattensperger, C., Winter, M.: Heterogeneous relation algebra. In: Brink, C., Kahl, W., Schmidt, G. (eds.) Relational Methods in Computer Science, ch. 3, pp. 39–53. Springer, Wien (1997)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Walter Guttmann
    • 1
  1. 1.Department of Computer ScienceUniversity of SheffieldUK

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