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Kleene Theorem in Partial Conway Theories with Applications

  • Zoltán Ésik
  • Tamás Hajgató
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7020)

Abstract

Partial Conway theories are algebraic theories equipped with a partially defined dagger operation satisfying some natural identities. We prove a Kleene type theorem for partial Conway theories and discuss several applications of this result.

Keywords

Point Identity Theory Operation Commutative Monoid Distinguished Ideal Iteration Theory 
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.

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References

  1. 1.
    de Bakker, J.W., Scott, D.: A theory of programs. IBM Seminar, Vienna (1969)Google Scholar
  2. 2.
    Bekić, H.: Definable operations in genaral algebras and the theory of automata and flowcharts. In: Bekic, H. (ed.) Programming Languages and their Definition. LNCS, vol. 177, pp. 30–55. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  3. 3.
    Bernátsky, L., Ésik, Z.: Semantics of flowchart programs and the free Conway theories. Theoretical Informatics and Applications, RAIRO 32, 35–78 (1998)MathSciNetGoogle Scholar
  4. 4.
    Berstel, J., Reutenauer, C.: Recognizable formal power series on trees. Theoretical Computer Science 18, 115–148 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bloom, S.L., Elgot, C.C., Wright, J.B.: Solutions of the iteration equation and extension of the scalar iteration operation. SIAM J. Computing 9, 26–45 (1980)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bloom, S.L., Elgot, C., Wright, J.B.: Vector iteration in pointed iterative theories. SIAM J. Computing 9, 525–540 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bloom, S.L., Ésik, Z.: Iteration Theories. Springer, Heidelberg (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bloom, S.L., Ésik, Z.: An extension theorem with an application to formal tree series. J. of Automata, Languages and Combinatorics, 145–185 (2003)Google Scholar
  9. 9.
    Bloom, S.L., Ésik, Z.: Axiomatizing rational power series over natural numbers. Information and Computation 207, 793–811 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bloom, S.L., Ésik, Z., Kuich, W.: Partial Conway and iteration semirings. Fundamenta Informaticae 86, 19–40 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bloom, S.L., Ginali, S., Rutledge, J.D.: Scalar and vector iteration. J. Comput. System Sci. 14, 251–256 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bozapalidis, S., Louscou-Bozapalidou, O.: The rank of a formal tree power series. Theor. Comput. Sci. 27, 211–215 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Elgot, C.C.: Monadic computation and iterative algebraic theories. In: Logic Colloquium 1973, Bristol. Studies in Logic and the Foundations of Mathematics, vol. 80, pp. 175–230. North-Holland, Amsterdam (1975)Google Scholar
  14. 14.
    Elgot, C.C.: Matricial theories. J. Algebra 42, 391–421 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ésik, Z.: Identities in iterative and rational algebraic theories. Computational Linguistics and Computer Languages XIV, 183–207 (1980)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ésik, Z.: On generalized iterative algebraic theories. Computational Linguistics and Computer Languages XV, 95–110 (1982)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ésik, Z.: Completeness of Park induction. Theoret. Comput. Sci. 177, 217–283 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ésik, Z.: Group axioms for iteration. Information and Computation 148, 131–180 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ésik, Z.: Partial Conway and iteration semiring-semimodule pairs. In: Kuich, W., Rahonis, G. (eds.) Bozapalidis Festschrift. LNCS, vol. 7020, pp. 72–93. Springer, Heidelberg (2011)Google Scholar
  20. 20.
    Ésik, Z., Hajgató, T.: Iteration grove theories with applications. In: Bozapalidis, S., Rahonis, G. (eds.) CAI 2009. LNCS, vol. 5725, pp. 227–249. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Ésik, Z., Kuich, W.: Formal tree series. J. Autom. Lang. Comb. 8, 219–285 (2003)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ésik, Z., Kuich, W.: A semiring-semimodule generalization of ω-regular languages, Parts 1 and 2. J. Automata, Languages, and Combinatorics 10, 203–242, 243–264 (2005)zbMATHGoogle Scholar
  23. 23.
    Golan, J.: Semirings and Their Applications. Kluwer Academic Publishers, Dordrecht (1999)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Automata Studies, pp. 3–42. Princeton University Press, Princeton (1956)Google Scholar
  25. 25.
    Kuich, W., Salomaa, A.: Semirings, Automata, Languages. Springer, Heidelberg (1986)CrossRefzbMATHGoogle Scholar
  26. 26.
    Lawvere, F.W.: Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. U.S.A. 50, 869–872 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  28. 28.
    Niwinski, D.: Equational μ-calculus. In: Skowron, A. (ed.) SCT 1984. LNCS, vol. 208, pp. 169–176. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  29. 29.
    Niwinski, D.: On fixed-point clones (extended abstract). In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226, pp. 464–473. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  30. 30.
    Park, D.M.R.: Fixed point induction and proofs of program properties. In: Michie, D., Meltzer, B. (eds.) Machine Intelligence, vol. 5, pp. 59–78. Edinburgh Univ. Press, Edinburgh (1970)Google Scholar
  31. 31.
    Schützenberger, M.P.: On the definition of a family of automata. Information and Computation 4, 245–270 (1961)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Schützenberger, M.P.: On a theorem of R. Jungen. Proc. American Mathematical Society 13, 885–890 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Simpson, A., Plotkin, G.: Complete axioms for categorical fixed-point operators. In: 15th Annual IEEE Symposium on Logic in Computer Science, Santa Barbara, CA, pp. 30–41. IEEE Comput. Soc. Press, Los Alamitos (2000)Google Scholar
  34. 34.
    Plotkin, G.: Domains. University of Edinburgh, Edinburgh (1983)Google Scholar
  35. 35.
    Wright, J.B., Thatcher, J.W., Wagner, E.G., Goguen, J.A.: Rational algebraic theories and fixed-point solutions. In: 17th Annual Symposium on Foundations of Computer Science, Houston, Tex, pp. 147–158. IEEE Comput. Soc., Long Beach (1976)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Zoltán Ésik
    • 1
  • Tamás Hajgató
    • 1
  1. 1.Department of Computer ScienceUniversity of SzegedHungary

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