Skip to main content

Δ-Languages for sets and sub-PTIME graph transformers

Contributed Papers

Part of the Lecture Notes in Computer Science book series (LNCS,volume 893)

Abstract

This paper discusses three successively extending versions of a set theoretic Δ-language, as a prototype for “nested” data bases query language. Corresponding finite set operations (data base queries) may be realized in NLOGSPACE under representation of sets by extensional well-founded (acyclic) graphs. (In a previous work for another version of Δ-language an exact correspondence to PTIME-computability was established.) Moreover, each of the mentioned versions of the language is faithfully characterized in terms of corresponding three classes of the graph transformers, the last one being just all transformers definable in the First Order Logic with Transitive Closure operator. For simplicity we are considering here the case of “pure” hereditarily-finite sets, i.e. sets without atoms involved. They are naturally linear ordered, however this order is problematic to formally define in our present case (unlike the case corresponding to PTIME). The related question whether the last class of transformers and corresponding class of queries over HF coincide with all NLOGSPACE-computable ones is left open in this paper.

Supported by Russian Foundation of Fundamental Investigations (project 93-011-16016) and partially by INTAS (project 93-972).

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/3-540-58907-4_11
  • Chapter length: 14 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   99.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-49136-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   129.00
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abiteboul, S., Beery, C.: On the power of languages for the manipulation of complex objects. INRIA research report 846 (1988). Abstract in Proc. International Workshop on Theory and Applications of Nested Relations and Complex Objects. Darmstadt (1987)

    Google Scholar 

  2. Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes. No 14 (1988)

    Google Scholar 

  3. Blakely, J.A., Larson, P.A., Tompa, F.W.: Efficiently updating materialized view. Proc. ACM-SIGMOD. 1986. Intern. Conf. on Management of Data. Washington D.C. May 1986. 61–71

    Google Scholar 

  4. Buss, S.R.: Bounded Arithmetic. Bibliopolis. Napoli. 1986

    Google Scholar 

  5. Dahlhaus, E., Makowsky, J.: The Choice of programming Primitives in SETL-like Languages. ESOP'86. LNCS 213 (1986) 160–172

    Google Scholar 

  6. Dahlhaus, E.: Is SETL a suitable language for parallel programming — a theoretical approach. Börger, E., Kleine Buning, H., Richter, M.M. ed. CSL'87. LNCS 329 (1987) 56–63

    Google Scholar 

  7. Ershov, Yu.L.: Numberings theory. “Nauka”. Glavnaja redakcija phisicomatematicheskoi literatury. Moskva. 1977 (in Russian)

    Google Scholar 

  8. Feferman, S.: Predicatively reducible systems of set theory. Proc. Symp. in Pure Math. Vol. 13, Part II (1974) 11–32

    Google Scholar 

  9. Gandy, R.O.: Set theoretic functions for elementary syntax. Proc. Symp. in Pure Math. Vol. 13, Part II (1974) 103–126

    Google Scholar 

  10. Grumbach, S., Vianu, V.: Tractable query languages for complex object databases. Rapports de Recherche N1573. INRIA. 1991

    Google Scholar 

  11. Gurevich, Y.: Algebras of feasible functions. FOCS 24 (1983) 210–214

    Google Scholar 

  12. Gurevich, Y.: Logic and the challenge of computer science. Trends in Theoretical Computer Science. (E. Borger ed.) Computer Science Press (1988) 1–57

    Google Scholar 

  13. Immerman, N.: Languages which captures complexity classes. SIAM J. Comput., 16 4 (1987) 760–778

    Google Scholar 

  14. Immerman, N.: Descriptive and computational complexity. Proc. Symposia in Applied Math. 38 (1989)

    Google Scholar 

  15. Immerman, N., Patnik, S., Stemple, D.: The expressiveness of a family of finite set languages. 1991

    Google Scholar 

  16. Jensen, R.B.: The fine structure of the constructible hierarchy. Ann. Math. Logic 4 (1972) 229–308

    Google Scholar 

  17. Jensen, R.B., Karp, C.: Primitive recursive set functions. Proc. Sympos. Pure Math., vol. 13 part I (1971) 143–176

    Google Scholar 

  18. Kuper, G.M., Vardi, M.Y.: A new approach to database logic. Proc. 3rd ACM Symp. on Principles of Database Systems 1984

    Google Scholar 

  19. Levy, A.: A hierarchy of formulas in set theory. Mem. Amer. Math. Soc. No. 57 (1965) 76pp. MR 32 N 7399

    Google Scholar 

  20. Parikh, R.: Existence and feasibility in arithmetic. JSL, 36 No 3 (1971) 494–508

    Google Scholar 

  21. Red'ko, V.N., Basarab, I.A.: Data bases and information systems. News in the life, science and technique; series: “Mathematics, Kybernetics” N6 Moscow. Znanije. (1987) (in Russian)

    Google Scholar 

  22. Sazonov, V.Yu.: A logical approach to the problem “P=NP?”. Proc., Math. Found. of Comput. Sci. Lect. Not. Comput. Sci. 88 Springer. New York (1980) 562–575 (An important correction to this paper is given in [23], P.490)

    Google Scholar 

  23. Sazonov, V.Yu.: On existence of complete predicate calculus in metamathematics without exponentiation. Lect. Not. Comput. Sci. 118 Springer. New York (1981) 483–490

    Google Scholar 

  24. Sazonov, V.Yu.: Bounded set theory and polynomial computability. All Union Conf. Appl. Logic. Proc. Novosibirsk (1985) 188–191 (In Russian)

    Google Scholar 

  25. Sazonov, V.Yu.: Bounded set theory, polynomial computability and Δ-programming. Application aspects of mathematical logic. Computing systems 122 (1987) 110–132 (In Russian) Cf. also a short English version of this paper in: Lect. Not. Comput. Sci. 278 Springer (1987) 391–397

    Google Scholar 

  26. Sazonov, V.Yu.: Bounded set theory and inductive definability. Abstracts of Logic Colloquium'90. JSL 56 Nu.3 (1991) 1141–1142

    Google Scholar 

  27. Sazonov, V.Yu.: hereditarily-finite sets, data bases and polynomial-time computability. TCS 119 Elsevier (1993) 187–214

    Google Scholar 

  28. Sazonov, V.Yu.: A bounded set theory with anti-foundation axiom and inductive definability. (Presented to the conference CSL'94, Kazimierz, Poland, September 1994) 1994.

    Google Scholar 

  29. Sazonov, V.Yu., Leontjev, A.V.: On coding of hereditarily-finite sets, polynomialtime computability and Δ-expressibility. Proc. Conf. Appl. Logic Novosibirsk, May 1993. Computing systems 146 (1992) 195–198

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sazonov, V., Lisitsa, A. (1995). Δ-Languages for sets and sub-PTIME graph transformers. In: Gottlob, G., Vardi, M.Y. (eds) Database Theory — ICDT '95. ICDT 1995. Lecture Notes in Computer Science, vol 893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58907-4_11

Download citation

  • DOI: https://doi.org/10.1007/3-540-58907-4_11

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58907-5

  • Online ISBN: 978-3-540-49136-1

  • eBook Packages: Springer Book Archive