Abstract
We study structural properties of restricted width arithmetical circuits. It is shown that syntactically multilinear arithmetical circuits of constant width can be efficiently simulated by syntactically multilinear algebraic branching programs of constant width, i.e. that sm-VSC 0 ⊆ sm-VBWBP. Also, we obtain a direct characterization of poly-size arithmetical formulas in terms of multiplicatively disjoint constant width circuits, i.e. MD-VSC0 = VNC1.
For log-width weakly-skew circuits a width efficient multilinearity preserving simulation by algebraic branching programs is given, i.e. weaklyskew-sm-VSC1 ⊆ sm-VBP[width=log2 n].
Finally, coefficient functions are considered, and closure properties are observed for sm-VSCi, and in general for a variety of other syntactic multilinear restrictions of algebraic complexity classes.
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References
Ben-Or, M., Cleve, R.: Computing algebraic formulas using a constant number of registers. SIAM J. Comput. 21(1), 54–58 (1992)
Brent, R.P.: The parallel evaluation of arithmetic expressions in logarithmic time. In: Proc. Symp., Carnegie-Mellon Univ., Pittsburgh, pp. 83–102 (1973)
Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics. Springer, Heidelberg (2000)
Caussinus, H., McKenzie, P., Thérien, D., Vollmer, H.: Nondeterministic NC1 computation. Journal of Computer and System Sciences 57, 200–212 (1998)
Jansen, M.J.: Lower bounds for syntactically multilinear algebraic branching programs. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 407–418. Springer, Heidelberg (2008)
Mahajan, M., Raghavendra Rao, B.V.: Arithmetic circuits, syntactic multilinearity, and the limitations of skew formulae. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 455–466. Springer, Heidelberg (2008)
Malod, G.: The complexity of polynomials and their coefficient functions. In: IEEE Conference on Computational Complexity, pp. 193–204 (2007)
Malod, G., Portier, N.: Characterizing valiant’s algebraic complexity classes. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 704–716. Springer, Heidelberg (2006)
Raz, R.: Multi-linear formulas for permanent and determinant are of super-polynomial size. In: STOC, pp. 633–641 (2004)
Raz, R., Yehudayoff, A.: Lower bounds and separations for constant depth multilinear circuits. In: CCC 2008, pp. 128–139 (2008)
Ryser, H.J.: Combinatorial mathematics. In: Carus Mathematical Monograph, vol. 14, p. 23. Mathematical Association of America, Washington (1963)
Valiant, L.G.: Reducibility by algebraic projections. L’Enseignement mathématique 28, 253–268 (1982)
Valiant, L.G., Skyum, S., Berkowitz, S., Rackoff, C.: Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12(4), 641–644 (1983)
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Jansen, M., Rao B.V., R. (2009). Simulation of Arithmetical Circuits by Branching Programs with Preservation of Constant Width and Syntactic Multilinearity. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds) Computer Science - Theory and Applications. CSR 2009. Lecture Notes in Computer Science, vol 5675. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03351-3_18
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DOI: https://doi.org/10.1007/978-3-642-03351-3_18
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