Theory of Computing Systems

, Volume 63, Issue 3, pp 615–633 | Cite as

A Unifying Approach to Algebraic Systems Over Semirings

  • Peter KostolányiEmail author


A fairly general definition of canonical solutions to algebraic systems over semirings is proposed. This is based on the notion of summation semirings, traditionally known as \({\Sigma }\)-semirings, and on assigning unambiguous context-free languages to variables of each system. The presented definition applies to all algebraic systems over continuous or complete semirings and to all proper algebraic systems over power series semirings, for which it coincides with the usual definitions of their canonical solutions. As such, it unifies the approaches to algebraic systems over semirings studied in literature. An equally general approach is adopted to study pushdown automata, for which equivalence with algebraic systems is proved. Finally, the Chomsky-Schützenberger theorem is generalised to the setting of summation semirings.


Algebraic system Semiring Summation semiring Template 



  1. 1.
    Bloom, S.L., Ésik, Z., Kuich, W.: Partial conway and iteration semirings. Fundam. Inform. 86, 19–40 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chomsky, N., Schützenberger, M.-P.: The algebraic theory of context-free languages. In: Braffort, P., Hirschberg, D. (eds.) Computer programming and formal systems, pp. 118-161. North Holland, Amsterdam (1963)Google Scholar
  3. 3.
    Davey, B.A., Priestley, H.A.: Introduction to lattices and order 2nd ed. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Droste, M., Kuich, W., Vogler, H.: Handbook of weighted automata. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Droste, M., Kuich, W.: Semirings and Formal Power Series. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of weighted automata. Monographs in theoretical computer science, pp 3–28. Springer, Berlin (2009)Google Scholar
  6. 6.
    Droste, M., Vogler, H.: The chomsky-schützenberger theorem for quantitative context-free languages. Int. J. Found. Comput. Sci. 25, 955–970 (2014)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ésik, Z., Kuich, W.: A unifying kleene theorem for weighted finite automata. In: Calude, C. S., Rozenberg, G., Salomaa, A. (eds.) Maurer Festschrift. LNCS, vol. 6570, pp 76–89. Springer, Berlin Heidelberg (2011)Google Scholar
  8. 8.
    Ésik, Z., Kuich, W.: Inductive *-Semirings. Theor. Comput. Sci. 324, 3–33 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ésik, Z., Kuich, W.: Modern automata theory.
  10. 10.
    Hebisch, U., Weinert, H.J.: Semirings – algebraic theory and applications in computer science. World Scientific, Singapore (1998)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hopcroft, J.E., Ullman, J.D.: Introduction to automata theory, languages, and computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  12. 12.
    Korenjak, A.J., Hopcroft, J.E.: Simple deterministic languages. In: SWAT 1966, pp 36–46. IEEE Computer Society, Washington (1966)Google Scholar
  13. 13.
    Kostolányi, P., Rovan, B.: Automata with auxiliary weights. Int. J. Found. Comput. Sci. 27, 787–807 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kuich, W.: Semirings and formal power series: their relevance to formal languages and automata. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of formal languages, vol. 1, pp 609–677. Springer, Berlin (1997)Google Scholar
  15. 15.
    Kuich, W., Salomaa, A.: Semirings, automata, languages. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  16. 16.
    Lausch, H., Nöbauer, W.: Algebra of polynomials. North Holland, Amsterdam (1973)Google Scholar
  17. 17.
    Petre, I., Salomaa, A.: Algebraic systems and pushdown automata. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of weighted automata. Monographs in theoretical computer science, pp 257–289. Springer, Berlin (2009)Google Scholar
  18. 18.
    Sakarovitch, J.: Elements of automata theory. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Salomaa, A., Soittola, M.: Automata-theoretic aspects of formal power series. Springer, New York (1978)CrossRefzbMATHGoogle Scholar
  20. 20.
    Stanat, D.F.: A homomorphism theorem for weighted context-free grammars. J. Comput. Syst. Sci. 6, 217–232 (1972)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer Science, Faculty of Mathematics, Physics and InformaticsComenius University in BratislavaBratislavaSlovakia

Personalised recommendations