Skip to main content
SpringerLink
Log in

A Unifying Approach to Algebraic Systems Over Semirings

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. To be more precise, weighted context-free grammars over complete semirings are considered in [6]. These can nevertheless be viewed as algebraic systems.

  2. We shall dispense with a formally defined notion of weighted context-free grammars over semirings. Nevertheless, an idea of such grammars is useful for gaining intuition.

  3. Semiring-polynomials can be introduced as a specialisation of the notion of polynomials over a universal algebra [16]. With this definition, a semiring-polynomial can be seen as a congruence class of a suitable congruence defined on the algebra of representations of semiring-polynomials as defined below. However, let us stress that this distinction is unimportant for our purposes, since similarly as in [9], we shall only be interested in mappings induced by (representations of) semiring-polynomials.

  4. Note that the resulting algebra is not a semiring. On the other hand, the algebra of semiring-polynomials, which can be obtained as a factor algebra of the algebra of their representations, constitutes a semiring [16].

  5. Commutativity of S is a commonplace assumption when dealing with algebraic systems over S〈〈Σ∗〉〉 [17, 19]. However, let us note that it is not strictly necessary [17] and that the approach presented later in this article subsumes the noncommutative case as well.

  6. This generalised partition can be of any cardinality, as \((0 ~|~ i \in I^{\prime })\) is in \(\mathcal {F}\) for any \(I^{\prime }\).

  7. More precisely, we should write \(m^{\prime } = \{c_{m,0}\} y_{1} \{c_{m,1}\} {\ldots } \{c_{m,r-1}\}y_{r} \{c_{m,r}\}\). However, we follow here the common practice of identifying a singleton set \(\{c\}\) with c itself.

  8. The polynomials of a template system will in fact all be \(2_{1}^{\Sigma }\)-polynomials. However, the sets \(\emptyset \) and \(\{\varepsilon \}\) are the zero and the unity in the semiring \(2^{{\Sigma }^{*}}\), so they have to be included in order to satisfy the technical condition imposed in Definition 3.

  9. Droste and Vogler [6] have in fact dealt with grammars over valuation monoids, thus going beyond semirings.

  10. Here, N stands for a nonempty finite alphabet of nonterminals, \({\Sigma }\) stands for a nonempty finite alphabet of terminals, \(N \cap {\Sigma } = \emptyset \), \(P \subseteq N \times (N \cup T)^{*}\) is a set of production rules, and \(\sigma \) in N is the initial nonterminal.

  11. For the sake of correctness, let us note that the following definition is sound only because it is consistent with the usual definition for automata over semirings of formal languages. This is a hidden proposition, which is nevertheless easy to prove.

References

  1. Bloom, S.L., Ésik, Z., Kuich, W.: Partial conway and iteration semirings. Fundam. Inform. 86, 19–40 (2008)

    MathSciNet  MATH  Google Scholar 

  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)

  3. Davey, B.A., Priestley, H.A.: Introduction to lattices and order 2nd ed. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  4. Droste, M., Kuich, W., Vogler, H.: Handbook of weighted automata. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  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)

  6. Droste, M., Vogler, H.: The chomsky-schützenberger theorem for quantitative context-free languages. Int. J. Found. Comput. Sci. 25, 955–970 (2014)

    Article  MATH  Google Scholar 

  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)

  8. Ésik, Z., Kuich, W.: Inductive *-Semirings. Theor. Comput. Sci. 324, 3–33 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ésik, Z., Kuich, W.: Modern automata theory. http://www.dmg.tuwien.ac.at/kuich/mat.pdf

  10. Hebisch, U., Weinert, H.J.: Semirings – algebraic theory and applications in computer science. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  11. Hopcroft, J.E., Ullman, J.D.: Introduction to automata theory, languages, and computation. Addison-Wesley, Reading (1979)

    MATH  Google Scholar 

  12. Korenjak, A.J., Hopcroft, J.E.: Simple deterministic languages. In: SWAT 1966, pp 36–46. IEEE Computer Society, Washington (1966)

  13. Kostolányi, P., Rovan, B.: Automata with auxiliary weights. Int. J. Found. Comput. Sci. 27, 787–807 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  15. Kuich, W., Salomaa, A.: Semirings, automata, languages. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  16. Lausch, H., Nöbauer, W.: Algebra of polynomials. North Holland, Amsterdam (1973)

  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)

  18. Sakarovitch, J.: Elements of automata theory. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  19. Salomaa, A., Soittola, M.: Automata-theoretic aspects of formal power series. Springer, New York (1978)

    Book  MATH  Google Scholar 

  20. Stanat, D.F.: A homomorphism theorem for weighted context-free grammars. J. Comput. Syst. Sci. 6, 217–232 (1972)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, Bratislava, 842 48, Slovakia

    Peter Kostolányi

Authors
  1. Peter Kostolányi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Kostolányi.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the grants VEGA 2/0165/16 and UK/176/2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kostolányi, P. A Unifying Approach to Algebraic Systems Over Semirings. Theory Comput Syst 63, 615–633 (2019). https://doi.org/10.1007/s00224-018-9895-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-018-9895-9

Keywords

Search

Navigation