Advertisement

A Circuit Complexity Approach to Transductions

  • Michaël CadilhacEmail author
  • Andreas Krebs
  • Michael Ludwig
  • Charles Paperman
Conference paper
First Online:
  • 563 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9234)

Abstract

Low circuit complexity classes and regular languages exhibit very tight interactions that shade light on their respective expressiveness. We propose to study these interactions at a functional level, by investigating the deterministic rational transductions computable by constant-depth, polysize circuits. To this end, a circuit framework of independent interest that allows variable output length is introduced. Relying on it, there is a general characterization of the set of transductions realizable by circuits. It is then decidable whether a transduction is definable in \(\mathrm{AC}^0\) and, assuming a well-established conjecture, the same for \(\mathrm{ACC}^0\).

Keywords

Regular Language Surjective Morphism Input Length Output Gate Output Length 
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.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgment

We thank Michael Blondin, Michael Hahn, and the referees.

References

  1. 1.
    Barrington, D.A.M.: Bounded-width polynomial size branching programs recognize exactly those languages in NC\(^1\). J. Comp. Syst. Sc. 38, 150–164 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barrington, D.A.M., Compton, K., Straubing, H., Thérien, D.: Regular languages in NC\(^1\). J. Comput. Syst. Sci. 44(3), 478–499 (1992)CrossRefzbMATHGoogle Scholar
  3. 3.
    Berstel, J.: Transductions and Context-Free Languages, Leitfäden der Angewandten Mathematik und Mechanik LAMM. Teubner, Stuttgart (1979) CrossRefGoogle Scholar
  4. 4.
    Beyersdorff, O., Datta, S., Krebs, A., Mahajan, M., Scharfenberger-Fabian, G., Sreenivasaiah, K., Thomas, M., Vollmer, H.: Verifying proofs in constant depth. TOCT 5(1), 2 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bojańczyk, M.: Transducers with Origin Information. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 26–37. Springer, Heidelberg (2014) Google Scholar
  6. 6.
    Chaubard, L., Pin, J.É., Straubing, H.: First-order formulas with modular predicates. In: LICS, pp. 211–220. IEEE (2006)Google Scholar
  7. 7.
    Choffrut, C., Schützenberger, M.P.: Counting with rational functions. Theor. Comput. Sci. 58(1–3), 81–101 (1988)CrossRefzbMATHGoogle Scholar
  8. 8.
    Choffrut, C.: A generalization of Ginsburg and Rose’s characterization of G-S-M mappings. In: Maurer, H.A. (ed.) ICALP 1979. LNCS, vol. 71, pp. 88–103. Springer, Heidelberg (1979) CrossRefGoogle Scholar
  9. 9.
    Esik, Z., Ito, M.: Temporal logic with cyclic counting and the degree of aperiodicity of finite automata. Acta Cybern. 16(1), 1–28 (2003)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Filiot, E., Krishna, S.N., Trivedi, A.: First-order definable string transformations. In: Raman, V., Suresh, S.P. (eds.) FSTTCS. LIPIcs, vol. 29, pp. 147–159 (2014)Google Scholar
  11. 11.
    Furst, M., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Theor. Comput. Syst. 17, 13–27 (1984)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Koucký, M., Pudlák, P., Thérien, D.: Bounded-depth circuits: separating wires from gates. In: STOC, pp. 257–265. ACM (2005)Google Scholar
  13. 13.
    Lange, K.-J., McKenzie, P.: On the complexity of free monoid morphisms. In: Chwa, K.-Y., Ibarra, O.H. (eds.) ISAAC 1998. LNCS, vol. 1533, pp. 247–255. Springer, Heidelberg (1998) CrossRefGoogle Scholar
  14. 14.
    Lautemann, C., McKenzie, P., Schwentick, T., Vollmer, H.: The descriptive complexity approach to LOGCFL. J. Comput. Syst. Sci. 62(4), 629–652 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pin, J.É., Straubing, H.: Some results on C-varieties. RAIRO-Theor. Inf. Appl. 39(01), 239–262 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Reutenauer, C., Schützenberger, M.P.: Variétés et fonctions rationnelles. Theor. Comput. Sci. 145(1–2), 229–240 (1995)CrossRefzbMATHGoogle Scholar
  17. 17.
    Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser, Boston (1994)CrossRefzbMATHGoogle Scholar
  18. 18.
    Straubing, H.: On logical descriptions of regular languages. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 528–538. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  19. 19.
    Vollmer, H.: Introduction to Circuit Complexity. Springer-Verlag, Berlin (1999) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Michaël Cadilhac
    • 1
    Email author
  • Andreas Krebs
    • 1
  • Michael Ludwig
    • 1
  • Charles Paperman
    • 2
  1. 1.Wilhelm Schickard InstitutUniversität TübingenTübingenGermany
  2. 2.University of Warsaw and Warsaw Center of Mathematics and Computer ScienceWarsawPoland

Personalised recommendations

Cite paper

Buy options