Arithmetizing Classes Around $\textsf{NC}$ 1 and $\textsf{L}$ | Theory of Computing Systems Skip to main content
Springer Nature Link
Log in

Arithmetizing Classes Around \(\textsf{NC}\) 1 and \(\textsf{L}\)

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

Abstract

The parallel complexity class \(\textsf{NC}\) 1 has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. (J. Comput. Syst. Sci. 57:200–212, 1992) considered arithmetizations of two of these classes, \(\textsf{\#NC}\) 1 and \(\textsf{\#BWBP}\) . We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in \(\textsf{FLogDCFL}\) , while counting proof-trees in logarithmic width formulae has the same power as \(\textsf{\#NC}\) 1. We also consider polynomial-degree restrictions of \(\textsf{SC}\) i, denoted \(\textsf{sSC}\) i, and show that the Boolean class \(\textsf{sSC}\) 1 is sandwiched between \(\textsf{NC}\) 1 and \(\textsf{L}\) , whereas \(\textsf{sSC}\) 0 equals \(\textsf{NC}\) 1. On the other hand, the arithmetic class \(\textsf{\#sSC}\) 0 contains \(\textsf{\#BWBP}\) and is contained in \(\textsf{FL}\) , and \(\textsf{\#sSC}\) 1 contains \(\textsf{\#NC}\) 1 and is in \(\textsf{SC}\) 2. We also investigate some closure properties of the newly defined arithmetic classes.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agrawal, M., Allender, E., Datta, S.: On TC0, AC0, and arithmetic circuits. J. Comput. Syst. Sci. 60(2), 395–421 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Allender, E.: The division breakthroughs. Bull. Eur. Assoc. Theor. Comput. Sci. 74 (2001)

  3. Allender, E.: Arithmetic circuits and counting complexity classes. In: Krajicek, J. (ed.) Complexity of Computations and Proofs. Quaderni di Matematica, vol. 13, 33–72. Seconda Universita di Napoli, Naples (2004). An earlier version appeared in the Complexity Theory Column, SIGACT News 28(4) 2–15 (1997)

    Google Scholar 

  4. Allender, E., Jiao, J., Mahajan, M., Vinay, V.: Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theor. Comput. Sci. 209, 47–86 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Proceedings of the ACM Symposium on Theory of Computing STOC, pp. 202–211 (2004)

  6. Alur, R., Kumar, V., Madhusudan, P., Viswanathan, M.: Congruences for visibly pushdown languages. In: Proceedings of the 32nd International Colloquium on Automata, Languages, and Programming (ICALP), pp. 1102–1114 (2005)

  7. Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. J. Comput. Syst. Sci. 38(1), 150–164 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Borodin, A., Cook, S., Dymond, P., Ruzzo, W., Tompa, M.: Two applications of inductive counting for complementation problems. SIAM J. Comput. 18(3), 559–578 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Buss, S.: The Boolean formula value problem is in ALOGTIME. In: Proceedings of the ACM Symposium on Theory of Computing STOC, pp. 123–131 (1987)

  10. Buss, S., Cook, S., Gupta, A., Ramachandran, V.: An optimal parallel algorithm for formula evaluation. SIAM J. Comput. 21(4), 755–780 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Caussinus, H., McKenzie, P., Thérien, D., Vollmer, H.: Nondeterministic NC1 computation. J. Comput. Syst. Sci. 57, 200–212 (1998)

    Article  MATH  Google Scholar 

  12. Chiu, A., Davida, G., Litow, B.: Division in logspace-uniform NC1. RAIRO Theor. Inf. Appl. 35, 259–276 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cook, S.: Characterizations of pushdown machines in terms of time-bounded computers. J. Assoc. Comput. Mach. 18, 4–18 (1971)

    MATH  MathSciNet  Google Scholar 

  14. Cook, S.: Deterministic CFL’s are accepted simultaneously in polynomial time and log squared space. In: Proceedings of the ACM Symposium on Theory of Computing STOC, pp. 338–345 (1979)

  15. Dymond, P.W.: Input-driven languages are in log n depth. Inf. Process. Lett. 26, 247–250 (1988)

    Article  MathSciNet  Google Scholar 

  16. Dymond, P.W., Cook, S.: Complexity theory of parallel time and hardware. Inf. Comput. 80, 205–226 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. Fernau, H., Lange, K.-J., Reinhardt, K.: Advocating ownership. In: Proceedings of the 16th Foundations of Software Technology and Theoretical Computer Science Conference FST&TCS. LNCS, vol. 1180, pp. 286–297. Springer, Berlin (1996)

    Google Scholar 

  18. Holzer, M., Lange, K.-J.: On the complexities of linear LL(1) and LR(1) grammars. In: Proceedings of the 9th International Symposium on Fundamentals of Computation Theory FCT. LNCS, pp. 299–308. Springer, Berlin (1993)

    Google Scholar 

  19. Istrail, S., Zivkovic, D.: Bounded width polynomial size Boolean formulas compute exactly those functions in AC0. Inf. Process. Lett. 50, 211–216 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Johnson, D.S.: A catalog of complexity classes. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science. Algorithms and Complexity (A), vol. A, pp. 67–161. Elsevier/MIT Press, Cambridge (1990)

    Google Scholar 

  21. Lange, K.-J.: Complexity and structure in formal language theory. In: Proceedings of the IEEE Structure in Complexity Theory Conference, pp. 224–230 (1993)

  22. Limaye, N., Mahajan, M., Rao, B.V.R.: Arithmetizing classes around NC1 and AC1. In: Proceedings of the 24th Annual Symposium on Theoretical Aspects of Computer Science STACS. LNCS, vol. 4393, pp. 477–488. Springer, Berlin (2007)

    Google Scholar 

  23. Limaye, N., Mahajan, M., Meyer, A.: On the complexity of membership and counting in height-deterministic pushdown automata. In: Proceedings of the 3nd International Computer Science Symposium in Russia. LNCS, vol. 5010. Springer, Berlin (2008)

    Google Scholar 

  24. Mahajan, M.: Polynomial size log depth circuits: between NC1 and AC1. Bull. Eur. Assoc. Theor. Comput. Sci. 91 (2007)

  25. Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL-recognition. In: Proceedings of the 7th International Colloquium on Automata, Languages, and Programming ICALP. LNCS, pp. 422–432. Springer, Berlin (1980)

    Google Scholar 

  26. Niedermeier, R., Rossmanith, P.: Unambiguous auxiliary pushdown automata and semi-unbounded fan-in circuits. Inf. Comput. 118(2), 227–245 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  27. Nisan, N.: RL ⊆ SC. Comput. Complex. 4(11), 1–11 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ruzzo, W.L.: Tree-size bounded alternation. J. Comput. Syst. Sci. 21, 218–235 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  29. Sudborough, I.: On the tape complexity of deterministic context-free language. J. Assoc. Comput. Mach. 25(3), 405–414 (1978)

    MATH  MathSciNet  Google Scholar 

  30. Venkateswaran, H.: Properties that characterize \(\textsf{LogCFL}\) . J. Comput. Syst. Sci. 42, 380–404 (1991)

    Article  MathSciNet  Google Scholar 

  31. Vinay, V.: Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In: Proceedings of 6th IEEE Structure in Complexity Theory Conference, pp. 270–284 (1991)

  32. Vinay, V.: Hierarchies of circuit classes that are closed under complement. In: Proceedings of the 11th Annual IEEE Conference on Computational Complexity CCC, Washington, DC, USA, pp. 108–117 (1996)

  33. Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, New York (1999)

    Google Scholar 

  34. Von Braunmühl, B., Verbeek, R.: Input-driven languages are recognized in log n space. In: Proceedings of the Fundamentals of Computation Theory Conference FCT. LNCS, pp. 40–51. Springer, Berlin (1983)

    Google Scholar 

  35. Von zur Gathen, J.: Parallel linear algebra. In: Reif, J.H. (ed.) Synthesis of Parallel Algorithms, pp. 573–617. Morgan Kaufmann, San Mateo (1993)

    Google Scholar 

  36. Von zur Gathen, J., Seroussi, G.: Boolean circuits versus arithmetic circuits. Inf. Comput. 91(1), 142–154 (1991)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Institute of Mathematical Sciences, Chennai, 600113, India

    Nutan Limaye, Meena Mahajan & B. V. Raghavendra Rao

Authors
  1. Nutan Limaye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Meena Mahajan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. V. Raghavendra Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Meena Mahajan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Limaye, N., Mahajan, M. & Rao, B.V.R. Arithmetizing Classes Around \(\textsf{NC}\) 1 and \(\textsf{L}\) . Theory Comput Syst 46, 499–522 (2010). https://doi.org/10.1007/s00224-009-9233-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-009-9233-3

Keywords

Search

Navigation