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Arithmetizing Classes Around NC 1 and L

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STACS 2007 (STACS 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4393))

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Abstract

The parallel complexity class NC 1 has many equivalent models such as bounded width branching programs. Caussinus et.al [10] considered arithmetizations of two of these classes, #NC 1 and #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 has the same power as #BWBP, while counting proof-trees in logarithmic width formulae has the same power as #NC 1. We also consider polynomial-degree restrictions of \({\sf SC}^{i}\), denoted \({\sf sSC}^{i}\), and show that the Boolean class \({\sf sSC}{^1}\) lies between NC 1 and L, whereas \({\sf sSC}^0\) equals \({\sf NC}^1\). On the other hand, \({\sf \#}{\sf sSC}^0\) contains #BWBP and is contained in FL, and #sSC 1 contains #NC 1 and is in \({\sf SC}^{2}\). We also investigate some closure properties of the newly defined arithmetic classes.

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References

  1. Agrawal, M., Allender, E., Datta, S.: On TC0, AC0, and arithmetic circuits. Journal of Computer and System Sciences 60(2), 395–421 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Allender, E.: The division breakthroughs. BEATCS: Bulletin of the European Association for Theoretical Computer Science 74 (2001)

    Google Scholar 

  3. Allender, E.: Arithmetic circuits and counting complexity classes. In: Complexity of Computations and Proofs. Quaderni di Matematica, vol. 13, pp. 33–72 (2004)

    Google Scholar 

  4. Allender, E., et al.: Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theoretical Computer Science 209, 47–86 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alur, R., Madhusudan, P.: Visibly pushdown languages. In: STOC, pp. 202–211 (2004)

    Google Scholar 

  6. Barrington, D.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC1. JCSS 38(1), 150–164 (1989)

    MATH  MathSciNet  Google Scholar 

  7. Borodin, A., et al.: Two applications of inductive counting for complementation problems. SIAM Journal of Computation 18(3), 559–578 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Braunmuhl, B.V., Verbeek, R.: Input-driven languages are recognized in log n space. In: Karpinski, M. (ed.) FCT 1983. LNCS, vol. 158, pp. 40–51. Springer, Heidelberg (1983)

    Google Scholar 

  9. Buss, S., et al.: An optimal parallel algorithm for formula evaluation. SIAM J. Comput. 21(4), 755–780 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Caussinus, H., et al.: Nondeterministic NC1 computation. JCSS 57, 200–212 (1998)

    MATH  Google Scholar 

  11. Chiu, A., Davida, G., Litow, B.: Division in logspace-uniform NC1. RAIRO Theoretical Informatics and Applications 35, 259–276 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cook, S.A.: Deterministic CFL’s are accepted simultaneously in polynomial time and log squared space. In: STOC, pp. 338–345 (1979)

    Google Scholar 

  13. Dymond, P., Cook, S.: Complexity theory of parallel time and hardware. Information and Computation 80, 205–226 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dymond, P.W.: Input-driven languages are in logn depth. Information Processing Letters 26, 247–250 (1988)

    Article  MathSciNet  Google Scholar 

  15. Fernau, H., Lange, K.-J., Reinhardt, K.: Advocating ownership. In: Chandru, V., Vinay, V. (eds.) FSTTCS 1996. LNCS, vol. 1180, pp. 286–297. Springer, Heidelberg (1996)

    Google Scholar 

  16. Istrail, S., Zivkovic, D.: Bounded width polynomial size Boolean formulas compute exactly those functions in AC0 . Infor. Proc. Letters 50, 211–216 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Johnson, D.S.: A catalog of complexity classes. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. A, pp. 67–161 (1990)

    Google Scholar 

  18. Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL-recognition. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 422–432. Springer, Heidelberg (1980)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Nisan, N.: RL ⊆ SC. Computational Complexity 4(11), 1–11 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ruzzo, W.: Tree-size bounded alternation. Journal of Computer and System Sciences 21, 218–235 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  23. Sudborough, I.: On the tape complexity of deterministic context-free language. Journal of Association of Computing Machinery 25(3), 405–414 (1978)

    MATH  MathSciNet  Google Scholar 

  24. Venkateswaran, H.: Properties that characterize LogCFL. Journal of Computer and System Sciences 42, 380–404 (1991)

    Article  MathSciNet  Google Scholar 

  25. Vinay, V.: Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In: Proc. Structure in Complexity Theory Conference, pp. 270–284 (1991)

    Google Scholar 

  26. Vinay, V.: Hierarchies of circuit classes that are closed under complement. In: Proc. 11th Annual IEEE Conference on Computational Complexity, pp. 108–117. IEEE Computer Society Press, Los Alamitos (1996)

    Google Scholar 

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

    Google Scholar 

  28. von zur Gathen, J., Seroussi, G.: Boolean circuits versus arithmetic circuits. Information and Computation 91(1), 142–154 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The Institute of Mathematical Sciences, Chennai 600 113, India

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

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  1. Nutan Limaye
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  2. Meena Mahajan
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  3. B. V. Raghavendra Rao
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Wolfgang Thomas Pascal Weil

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Limaye, N., Mahajan, M., Rao, B.V.R. (2007). Arithmetizing Classes Around NC 1 and L . In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_41

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  • DOI: https://doi.org/10.1007/978-3-540-70918-3_41

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70917-6

  • Online ISBN: 978-3-540-70918-3

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