International Symposium on Mathematical Foundations of Computer Science

MFCS 2015: Mathematical Foundations of Computer Science 2015 pp 445-458 | Cite as

Parallel Identity Testing for Skew Circuits with Big Powers and Applications

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9235)


Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labelled with powers \(x^n\) for binary encoded numbers n. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to \(\mathsf {coRNC}^2\), which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented: (i) Equivalence of higher-dimensional straight-line programs can be tested in \(\mathsf {coRNC}^2\); this result is even new in the one-dimensional case, where the straight-line programs produce strings. (ii) The compressed word problem (or circuit evaluation problem) for certain wreath products belongs to \(\mathsf {coRNC}^2\). Full proofs can be found in the long version [13].


Word Problem Polynomial Ring Wreath Product Edge Label Multiplication Gate 
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  1. 1.
    Agrawal, M., Biswas, S.: Primality and identity testing via Chinese remaindering. J. Assoc. Comput. Mach. 50(4), 429–443 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beaudry, M., McKenzie, P., Péladeau, P., Thérien, D.: Finite monoids: from word to circuit evaluation. SIAM J. Comput. 26(1), 138–152 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berman, P., Karpinski, M., Larmore, L.L., Plandowski, W., Rytter, W.: On the complexity of pattern matching for highly compressed two-dimensional texts. J. Comput. Syst. Sci. 65(2), 332–350 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Eberly, W.: Very fast parallel polynomial arithmetic. SIAM J. Comput. 18(5), 955–976 (1989)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fich, F.E., Tompa, M.: The parallel complexity of exponentiating polynomials over finite fields. J. Assoc. Comput. Mach. 35(3), 651–667 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: \({P}\)-Completeness Theory. Oxford University Press, Oxford (1995)MATHGoogle Scholar
  7. 7.
    Hesse, W., Allender, E., Barrington, D.A.M.: Uniform constant-depth threshold circuits for division and iterated multiplication. J. Comput. Syst. Sci. 65, 695–716 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hirshfeld, Y., Jerrum, M., Moller, F.: A polynomial algorithm for deciding bisimilarity of normed context-free processes. Theor. Comput. Sci. 158(1&2), 143–159 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ibarra, O.H., Moran, S.: Probabilistic algorithms for deciding equivalence of straight-line programs. J. Assoc. Comput. Mach. 30(1), 217–228 (1983)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the STOC 1997, pp. 220–229. ACM Press (1997)Google Scholar
  11. 11.
    Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complex. 13(1–2), 1–46 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    König, D., Lohrey, M.: Evaluating matrix circuits. In: Xu, D., Du, D., Du, D. (eds.) COCOON 2015. LNCS, vol. 9198, pp. 235–248. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  13. 13.
    König, D., Lohrey, M.: Parallel identity testing for algebraic branching programs with big powers and applications. (2015).
  14. 14.
    Lohrey, M.: Word problems and membership problems on compressed words. SIAM J. Comput. 35(5), 1210–1240 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lohrey, M.: Algorithmics on SLP-compressed strings: asurvey. Groups Complex. Cryptol. 4(2), 241–299 (2012)MathSciNetMATHGoogle Scholar
  16. 16.
    Lohrey, M.: The Compressed Word Problem for Groups. SpringerBriefs in Mathematics. Springer, New York (2014)CrossRefMATHGoogle Scholar
  17. 17.
    Lohrey, M., Steinberg, B., Zetzsche, G.: Rational subsets and submonoids of wreath products. Inf. Comput. 243, 191–204 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mehlhorn, K., Sundar, R., Uhrig, C.: Maintaining dynamic sequences under equality tests in polylogarithmic time. Algorithmica 17(2), 183–198 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Amer. Math. Soc. Transl. Ser. 2(9), 1–122 (1958)MATHGoogle Scholar
  20. 20.
    Plandowski, W.: Testing equivalence of morphisms on context-free languages. In: van Leeuwen, J. (ed.) ESA 1994. LNCS, vol. 855, pp. 460–470. Springer, Heidelberg (1994) CrossRefGoogle Scholar
  21. 21.
    Vollmer, H.: Introduction to Circuit Complexity. Springer, New York (1999)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Universität SiegenSiegenGermany

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