computational complexity

, Volume 22, Issue 3, pp 517–564 | Cite as

Resource Trade-offs in Syntactically Multilinear Arithmetic Circuits

  • Maurice Jansen
  • Meena Mahajan
  • B. V. Raghavendra Rao


The class of polynomials computable by polynomial size log-depth arithmetic circuits (VNC 1) is known to be computable by constant width polynomial degree circuits (VsSC 0), but whether the converse containment holds is an open problem. As a partial answer to this question, we give a construction which shows that syntactically multilinear circuits of constant width and polynomial degree can be depth-reduced, which in our notation shows that sm-VsSC 0 \({\subseteq}\) sm-VNC 1. We further strengthen this inclusion, by giving a separate construction that provides a width-efficient simulation for constant width syntactically multilinear circuits by constant width syntactically multilinear algebraic branching programs; in our notation, sm-VsSC 0 \({\subseteq}\) sm-VBWBP. We then focus on polynomial size syntactically multilinear circuits and study relationships between classes of functions obtained by imposing various resource (width, depth, degree) restrictions on these circuits. Along the way, we also observe a characterization of the class NC 1 in terms of a restricted class of planar branching programs of polynomial size. Finally, in contrast to the general case, we report closure and stability of coefficient functions for the syntactically multilinear classes studied in this paper.


Arithmetic circuits Valiant’s classes syntactic multilinearity circuit width algebraic branching programs 

Subject classification

68Q05 03D15 68Q15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Eric Allender, Andris Ambainis, David A.Mix Barrington, Samir Datta & Huong LêThanh (1999). Bounded Depth Arithmetic circuits: Counting and Closure. In International Colloquium on Automata, Languages, and Programming ICALP, ICALP’99, 149–158.Google Scholar
  2. David A. Mix Barrington, Neil Immerman & Howard Straubing (1990). On uniformity within NC1. Journal of Computer and System Sciences 41(3), 274–306. ISSN 0022-0000.Google Scholar
  3. Michael Ben-Or & Richard Cleve (1992). Computing Algebraic Formulas Using a Constant Number of Registers. SIAM J. Comput. 21(1), 54–58.Google Scholar
  4. A. Borodin, A. Razborov & R. Smolensky (1993). On lower bounds for read-k-times branching programs. Comput. Complex. 3(1), 1–18. ISSN 1016-3328.Google Scholar
  5. Richard P. Brent (1973). The parallel evaluation of arithmetic expressions in logarithmic time. In Complexity of sequential and parallel numerical algorithms (Proc. Sympos., Carnegie-Mellon Univ., Pittsburgh, Pa., 1973), 83–102. Academic Press, New York.Google Scholar
  6. Peter Bürgisser (2000). Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics. Springer-Verlag.Google Scholar
  7. Caussinus Hervé, McKenzie Pierre, Thérien Denis, Vollmer Heribert (1998) Nondeterministic NC1 Computation. Journal of Computer and System Sciences 57: 200–212MathSciNetzbMATHCrossRefGoogle Scholar
  8. Stephen A. Cook (1979). Deterministic CFL’s Are Accepted Simultaneously in Polynomial Time and Log Squared Space. In Proceedings of the ACM Symposium on Theory of Computing STOC, 338–345.Google Scholar
  9. Istrail Sorin, Zivkovic Dejan (1994) Bounded width polynomial size Boolean formulas compute exactly those functions in AC0. Information Processing Letters 50: 211–216MathSciNetzbMATHCrossRefGoogle Scholar
  10. Maurice Jansen & B.V.Raghavendra Rao (2009). Simulation of Arithmetical Circuits by Branching Programs Preserving Constant Width and Syntactic Multilinearity. In CSR, LNCS Vol. 5675, 179–190.Google Scholar
  11. Maurice J. Jansen (2008). Lower Bounds for Syntactically Multilinear Algebraic Branching Programs. In MFCS LNCS vol. 5162, 407–418.Google Scholar
  12. David S. Johnson (1990). A Catalog of Complexity Classes. In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (A), Jan van Leeuwen, editor, 67–161. Elsevier and MIT Press.Google Scholar
  13. E. Kaltofen & P. Koiran (2008). Expressing a fraction of two determinants as a determinant. In Proceedings, The 19th International Symposium on Symbolic and Algebraic and Computation (ISSAC), 141–146.Google Scholar
  14. Nutan Limaye, Meena Mahajan & B. V. Raghavendra Rao (2010). Arithmetizing Classes Around NC1 and L. Theory of Computing Systems 46(3), 499–522. Preliminary version in STACS 2007, LNCS vol. 4393 pp. 477–488.Google Scholar
  15. Meena Mahajan & B. V. Raghavendra Rao (2008). Arithmetic Circuits, Syntactic Multilinearity and Skew formulae. In MFCS, LNCS vol. 5162, 455–466. Full version in ECCC TR08-048.Google Scholar
  16. Guillaume Malod (2007). The Complexity of Polynomials and Their Coefficient Functions. In IEEE Conference on Computational Complexity, 193–204.Google Scholar
  17. Malod Guillaume, Portier Natacha (2008) Characterizing Valiant’s algebraic complexity classes. Journal of Complexity 24(1): 16–38MathSciNetzbMATHCrossRefGoogle Scholar
  18. Noam Nisan (1991). Lower bounds for non-commutative computation. In STOC ’91: Proceedings of the twenty-third annual ACM symposium on Theory of computing, 410–418. ACM, New York, NY, USA. ISBN 0-89791-397-3.Google Scholar
  19. Nisan Noam, Wigderson Avi (1997) Lower Bounds on Arithmetic Circuits Via Partial Derivatives. Computational Complexity 6(3): 217–234MathSciNetzbMATHCrossRefGoogle Scholar
  20. Ran Raz (2006) Separation of Multilinear Circuit and Formula Size. Theory of Computing 2(1), 121–135. Preliminary version in FOCS 2004.Google Scholar
  21. Ran Raz (2009). Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM 56, 8:1–8:17. ISSN 0004-5411. Preliminary version in STOC 2004.Google Scholar
  22. Ran Raz, Amir Shpilka & Amir Yehudayoff (2008). A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits. SIAM J. Comput. 38(4), 1624–1647.Google Scholar
  23. Ran Raz & Amir Yehudayoff (2008). Balancing Syntactically Multilinear Arithmetic Circuits. Computational Complexity 17(4), 515–535. ISSN 1016-3328.Google Scholar
  24. Seinosuke Toda (1992). Classes of arithmetic circuits capturing the complexity of computing the determinant. IEICE Transactions on Informations and Systems E75-D, 116–124.Google Scholar
  25. Leslie G. Valiant (1982). Reducibility by algebraic projections. Logic and Algorithmic: an International Symposium held in honour of Ernst Specker 30, 365–380.Google Scholar
  26. Leslie G. Valiant, Sven Skyum, S. Berkowitz & Charles Rackoff (1983). Fast Parallel Computation of Polynomials Using Few Processors. SIAM J. Comput. 12(4), 641–644.Google Scholar
  27. V Vinay (1996). Hierarchies of Circuit Classes that are Closed Under Complement. In Proceedings of the 11th Annual IEEE Conference on Computational Complexity CCC, 108–117. IEEE Computer Society, Washington, DC, USA.Google Scholar
  28. H. Vollmer (1999). Introduction to Circuit Complexity: A Uniform Approach. Springer-Verlag New York Inc.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Maurice Jansen
    • 1
  • Meena Mahajan
    • 2
  • B. V. Raghavendra Rao
    • 3
  1. 1.Laboratory for Foundations of Computer Science, School of InformaticsThe University of EdinburghEdinburghUK
  2. 2.The Institute of Mathematical SciencesChennaiIndia
  3. 3.Universität des Saarlandes, InformatikSaarbrückenGermany

Personalised recommendations