Invariant Definability and P/poly

  • J. A. Makowsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1584)


We look at various uniform and non-uniform complexity classes within P/poly and its variations L/poly, NL/poly, NP/poly and PSpace/poly, and look for analogues of the Ajtai-Immerman theorem which characterizes AC 0 as the non-uniformly First Order Definable classes of finite structures. We have previously observed that the Ajtai-Immerman theorem can be rephrased in terms of invariant definability: A class of finite structures is FOL invariantly definable iff it is in AC 0. Invariant definability is a notion closely related to but different from implicit definability and Δ-definability. Its exact relationship to these other notions of definability has been determined in [Mak97].

Our first results are a slight generalization of similar results due to Molzan and can be stated as follows: let C be one of L, NL, P, NP, PSpace and \({\cal L}\) be a logic which captures C on ordered structures. Then the non-uniform \({\cal L}\)-invariantly definable classes of (not necessarily ordered) finite structures are exactly the classes in C/poly. We also consider uniformity conditions imposed on invariant definability and relate them to uniformity conditions on the advice sequences. This approach is different from imposing uniformity conditions on the circuit families.

The significance of our investigation is conceptual, rather than technical: We identify exactly the logical analogue of uniform and non-uniform complexity classes.


Linear Order Complexity Class Order Logic Uniformity Condition Relation Symbol 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AB97.
    Atserias, A., Balcázar, J.L.: Refining logical characterizations of advice complexity classes. In: First Panhellenic Symposium on Logic, Cyprus (1997) (to appear)Google Scholar
  2. Ajt83.
    Ajtai, M.: ∑\(_1^1\)formulae on finite structures. Annals of Pure and Applied Logic 24, 1–48 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  3. Ajt89.
    Ajtai, M.: First-order definability on finite structures. Annals of Pure and Applied Logic 45, 211–225 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  4. BCST92.
    Barrington, D.A.M., Compton, K., Straubing, H., Thérien, D.: Regular languages in NC1. Journal of Computer and System Sciences 44, 478–499 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  5. BF85.
    Barwise, J., Feferman, S. (eds.): Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer, Heidelberg (1985)Google Scholar
  6. BI94.
    Barrington, D.A.M., Immerman, N.: Time, hardware and uniformity. In: Proccedings of the 9th Structure in Complexity Theory, pp. 176–185. IEEE Computer Society, Los Alamitos (1994)CrossRefGoogle Scholar
  7. BIS90.
    Barrington, D.A.M., Immerman, N., Straubing, H.: On uniformity in nc1. JCSS 41, 274–306 (1990)zbMATHMathSciNetGoogle Scholar
  8. BS90.
    Boppana, R.B., Sipser, M.: The complexity of finite functions. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, ch. 14, vol. 1. Elsevier Science Publishers, Amsterdam (1990)Google Scholar
  9. EF95.
    Ebbinghaus, H.D., Flum, J.: Finite Model Theory. Perspectives in Mathematical Logic. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  10. FMP99.
    Frick, M., Makowsky, J.A., Pnueli, Y.B.: Oracles and Lindström quantifiers on ordered structures. Submitted to Information and Computation (in revision)Google Scholar
  11. Fri97.
    Frick, M.: Oracles and quantifiers. PhD thesis, Department of Mathematics, University of Freiburg, Freiburg, Germany (1997)Google Scholar
  12. FSS84.
    Furst, M., Saxe, J., Sipser, M.: Parity, circuits, and the polynomial hierarchy. J. Math. Systems Theory 17, 13–27 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  13. GL84.
    Gurevich, Y., Lewis, H.: A logic for constant-depth circuits. Information and Control 61, 65–74 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  14. Got97.
    Gottlob, G.: Relativized logspace and generalized quantifiers over finite ordered structures. Journal of Symbolic Logic 62(2), 545–574 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  15. Grä90.
    Grädel, E.: On logical descriptions of some concepts in structural complexity theory. In: Börger, E., Kleine Büning, H., Richter, M.M. (eds.) CSL 1989. LNCS, vol. 440, pp. 163–175. Springer, Heidelberg (1990)Google Scholar
  16. Imm87.
    Immerman, N.: Languages that capture complexity classes. SIAM Journal on Computing 16(4), 760–778 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  17. Imm9x.
    Immerman, N.: Descriptive Complexity. Springer, Heidelberg (1998)Google Scholar
  18. Joh90.
    Johnson, D.S.: A catalog of complexity classes. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, ch. 2, vol. 1, Elsevier Science Publishers, Amsterdam (1990)Google Scholar
  19. Kol90.
    Kolaitis, P.G.: Implicit definability on finite structures and unambiguous computations. In: LiCS 1990, pp. 168–180. IEEE, Los Alamitos (1990)Google Scholar
  20. Lin92.
    Lindell, S.: A purely logical characterization of circuit uniformity. In: Proceedings of the 7th conference on Structures in Complexity (1992)Google Scholar
  21. Mak.
    Makowsky, J.A.: Translations, interpretations and reductions. Lecture Notes of a course given at ESSLLI 1997 in Aix-en-Provence (August 1997)Google Scholar
  22. Mak97.
    Makowsky, J.A.: Invariant definability. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) KGC 1997. LNCS, vol. 1289, pp. 186–202. Springer, Heidelberg (1997)Google Scholar
  23. Mol90.
    Molzan, B.: Expressibility and nonuniform complexity classes. SIAM Journal of Computing 19(3), 411–423 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  24. MP94.
    Makowsky, J.A., Pnueli, Y.B.: Oracles and quantifiers. In: Meinke, K., Börger, E., Gurevich, Y. (eds.) CSL 1993. LNCS, vol. 832, pp. 189–222. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  25. MP95.
    Makowsky, J.A., Pnueli, Y.: Logics capturing oracle complexity classes uniformly. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 463–479. Springer, Heidelberg (1995)Google Scholar
  26. Saz80.
    Sazonov, V.: Polynomial computability and recursivity in finite domains. Elektronische Informationsverarbeitung und Kybernetik 16, 319–323 (1980)zbMATHMathSciNetGoogle Scholar
  27. Ste91.
    Stewart, I.A.: Comparing the expressibility of languages formed using NPcomplete operators. Journal of Logic and Computation 1(3), 305–330 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  28. Ste92.
    Stewart, I.A.: Using the Hamilton path operator to capture NP. Journal of omputer and Systems Sciences 45, 127–151 (1992)zbMATHCrossRefGoogle Scholar
  29. Ste93a.
    Stewart, I.A.: Logical characterization of bounded query classes i: Polynomial ime oracle machines. Fundamenta Informaticae 18, 93–105 (1993)zbMATHMathSciNetGoogle Scholar
  30. Ste93b.
    Stewart, I.A.: Logical characterizations of bounded query classes II: Polynomial–time oracle machines. Fundamenta Informaticae 18, 93–105 (1993)zbMATHMathSciNetGoogle Scholar
  31. Str94.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Progress in Theoretical Computer Science. Birkhäuser, Basel (1994)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • J. A. Makowsky
    • 1
  1. 1.Department of Computer ScienceTechnion–Israel Institute of TechnologyHaifaIsrael

Personalised recommendations