Abstract
Integer division has been known since 1986 [4 13 12] to be in slightly non-uniform TC0, i.e., computable by polynomial-size, constant depth threshold circuits. This has been perhaps the outstanding natural problem known to be in a standard circuit complexity class, but not known to be in its uniform version. We show that indeed division is in uniform TC0. A key step of our proof is the discovery of a first-order formula expressing exponentiation modulo any number of polynomial size.
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References
M. Ajtai and M. Ben-Or. A theorem on probabilistic constant depth computations. In ACM Symposium on Theory of Computing (STOC’ 84), pages 471–474, 1984. ACM Press.
E. Allender, D. A. Mix Barrington, and W. Hesse. Uniform circuits for division: Consequences and problems. To appear in Proceedings of the 16th Annual IEEE Conference on Computational Complexity (CCC-2001), 2001. IEEE Computer Society.
D. A. M. Barrington, N. Immerman, and H. Straubing. On uniformity within NC1. Journal of Computer and System Sciences, 41:274–306, 1990.
P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994–1003, 1986.
A. Chiu, G. Davida, and B. Litow. NC1 division. online at http://www.cs.jcu.edu.au/~bruce/papers/crr00_3.ps.gz.
G. I. Davida and B. Litow. Fast Parallel Arithmetic via Modular Representation. SIAM Journal of Computing, 20(4):756–765, 1991.
R. Fagin, M. M. Klawe, N. J. Pippenger, and L. Stockmeyer. Bounded-depth, polynomial-size circuits for symmetric functions. Theoretical Computer Science, 36(2-3):239–250, 1985.
M. Furst, J. B. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. In 22nd Annual Symposium on Foundations of Computer Science, 260–270, 1981. IEEE.
J. Hastad. Almost optimal lower bounds for small depth circuits. In Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, 6–20, 1986.
N. Immerman. Descriptive Complexity. Springer-Verlag, New York, 1999.
N. Immerman and S. Landau. The complexity of iterated multiplication. Information and Computation, 116(1):103–116, 1995.
J. H. Reif. On threshold circuits and polynomial computation. In Proceedings, Structure in Complexity Theory, Second Annual Conference, pages 118–123, IEEE Computer Society Press.
J. H. Reif and S. R. Tate. On threshold circuits and polynomial computation. SIAM Journal on Computing, 21(5):896–908, 1992.
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Hesse, W. (2001). Division Is In Uniform TC0 . In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds) Automata, Languages and Programming. ICALP 2001. Lecture Notes in Computer Science, vol 2076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48224-5_9
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DOI: https://doi.org/10.1007/3-540-48224-5_9
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