Advertisement

Non-commutative computation, depth reduction, and skew circuits (extended abstract)

  • Meena Mahajan
  • V Vinay
Complexity Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 880)

Abstract

It is known that polynomial size polynomial degree Boolean circuits have equivalent semi-unbounded logarithmic depth circuits. For arithmetic circuits, the additional property of commutativity is sufficient for a similar depth reduction result.

We explore conditions under which depth reduction is possible even in non-commutative settings. Known results so far are of two types: specific operations for which depth reduction is possible, and specific instances where linear lower bounds on circuit depth exist. We show that any polynomial size polynomial degree circuit which has short-right-paths or short-left-paths — a purely syntactic notion which generalizes the notion of pushdown height — has an equivalent depth-reduced circuit. Our proof is a direct circuit construction which we also relate to generalized auxiliary pushdown machines.

Further, we extend the lower bounds on left-skew circuit size in a specific non-commutative setting, {UNION, CONCAT}, to more generalized skew circuits. We also show that the generalization is proper; there are problems hard for left-skew circuits which are easy in this model.

Keywords

Polynomial Degree Arithmetic Circuit Circuit Input Proof Tree Algebraic Degree 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AJ92]
    C. Álvarez and B. Jenner. A note on log space optimization. Technical report, Universitat Politècnica Catalunya, Barcelona, 1992.Google Scholar
  2. [AJ93a]
    E. Allender and J. Jiao. Depth reduction for noncommutative arithmetic circuits. In Proceedings of the Symposium on Theory of Computing (STOC), 1993.Google Scholar
  3. [AJ93b]
    C. Álvarez and B. Jenner. A very hard log-space counting class. Theoretical Computer Science, 107: 3–30, 1993.Google Scholar
  4. [Kos90]
    S. Kosaraju. On the parallel evaluation of classes of circuits. In Proceedings of the 10th FST&TCS Conference, pages 232–237, December 1990. LNCS 472.Google Scholar
  5. [MRK88]
    G. Miller, V. Ramachandran, and E. Kaltofen. Efficient parallel evaluation of straight-line code and arithmetic circuits. SIAM Journal on Computing, 17: 687–695, 1988.Google Scholar
  6. [MV94]
    M. Mahajan and V Vinay. Non-commutative computation, depth reduction, and skew circuits. Technical Report IMSc-94/09, The Institute of Mathematical Sciences, Madras, India, March 1994. Available in ftp/pub/tcs/trimsc at imsc.ernet.in.Google Scholar
  7. [Nis91]
    N. Nisan. Lower bounds for noncommutative computation. In Proceedings of the Symposium on Theory of Computing (STOC), 1991.Google Scholar
  8. [Ven91]
    H. Venkateswaran. Properties that characterise LOGCFL. Journal of Computer and System Sciences, 42: 380–404, 1991. also in Proceedings of 19th STOC 1987, pp 141–150.Google Scholar
  9. [Ven92]
    H. Venkateswaran. Circuit definitions of nondeterministic complexity classes. SIAM Journal on Computing, 21: 655–670, 1992.Google Scholar
  10. [Vin91a]
    V. Vinay. Counting auxiliary pushdown automata and semi-bounded arithmetic circuits. In Proceedings of the 6th Annual Conference on Structure in Complexity Theory, pages 270–284, 1991.Google Scholar
  11. [Vin91b]
    V. Vinay. Semi-unboundedness and complexity classes. PhD thesis, Indian Institute of Science, Bangalore, India, 1991.Google Scholar
  12. [VSBR83]
    L. Valiant, S. Skyum, S. Berkowitz, and C. Rackoff. Fast parallel computation of polynomials using few processors. SIAM Journal on Computing, 12: 641–644, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Meena Mahajan
    • 1
  • V Vinay
    • 2
  1. 1.The Institute of Mathematical SciencesMadrasIndia
  2. 2.Centre for Artificial Intelligence and RoboticsBangaloreIndia

Personalised recommendations