Abstract
The problem of minimizing the depth of formulas by equivalence preserving transformations is formalized in a general algebraic setting. For a particular algebraic system ∑0 specific methods of a dynamic programming nature are developed for proving lower bounds on depth. Such lower bounds for ∑0 automatically imply the same results for the systems of (i) arithmetic computations with addition and multiplication only, and (ii) computations over finite languages using union and concatenation. The specific lower bounds obtained are (i) depth 2n−o(n) for the permanent, (ii) depth (0.25+o(1))log2 n for the symmetric polynomials and (iii) depth 1.16logn for a problem of formula sizen.
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References
R. P. Brent, The parallel evaluation of arithmetic expressions in logarithmic time,Complexity of Sequential and Parallel Numerical Algorithms, J. F. Traub (ed.), Academic Press, New York, 83–102 (1973).
W. F. McColl, Some results on circuit depth. Ph. D. Thesis, Univ. of Warwick, 1977.
W. F. McColl, The circuit depth of symmetric Boolean functions,JCSS 17, 108–115 (1978).
G. B. Goodrich, R. E. Ladner, and M. J. Fischer, Straight-line programs to compute finite languages, Proc. of a conference on Theoretical Computer Science, Univ. of Waterloo, 221–229 (1977).
W. Miller, Computational complexity and numerical stability,SIAM J. Computing 4, 97–107 (1975).
W. Miller, Computer search for numerical stability,J. ACM 22, 512–521 (1975).
D. E. Muller and F. P. Preparata, Restructuring of arithmetic expressions for parallel evaluation,J. ACM 23, 534–543 (1976)
N. Pippenger, Short monotone formulae for threshold functions,IBM Tech. Rep. RC 5405, 1975.
E. Shamir and M. Snir, Lower bounds on the number of multiplications and the number of additions in monotone computations,IBM Tech. Rep. RC 6757, 1977.
E. Shamir and M. Snir, Lower bounds on depth complexity in monotone arithmetic computations,IBM Tech. Rep. RC 7055, 1978.
M. Snir, On the complexity of formula simplification, in preparation.
L. G. Valiant, The complexity of computing the permanent,Theor. Comp. Sci. 8, 189–201 (1979).
L. Hyafil, On the parallel evaluation of multivariate polynomials,SIAM J. Computing 8, 120–123 (1979).
J. I. Munro, The parallel complexity of arithmetic computations, Proceedings of the 1977 International FCT-Conference on Fundamentals of Computations Theory, M. Karpinski (ed.) Springer, Berlin, 466–475 (1977).
M. Snir, On the size of monotone formulas,Univ. of Edinburgh Tech. Rep. CSR-46-79, 1979.
H. J. Ryser, Combinatorial mathematics, Carus Math. Monograph No. 14 (1963).
M. Jerrum and M. Snir, Some exact complexity results for straight-line computations over semirings, Univ of Edinburgh CRS-58-80, 1980.
L. Csansky, Fast parallel matrix inversion algorithms,SIAM J. Comput. 5, 618–623 (1976).
D. E. Muller and F. P. Preparata, Bounds to complexities of networks for sorting and switching,J. ACM 22, 195–201 (1975).
L. Valiant, private communication.
B. Commentz-Walter, Size-depth tradeoff in monotone Boolean formulae,Acta Informat. 12, 227–243 (1979).
B. Commentz-Walter and J. Sattler, Size-depth tradeoff in non-monotone Boolean formulae,Acta Informat. (in press) (1980).
L. S. Khasin, Complexity bounds for the realization of monotonic symmetric functions.Sov. Phys. Dokl, 14, 1149–1151 (1970), orig. inDok. Akad. Nauk SSSR 189, 752–755 (1969).
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Part of the work was done while the authors were at IBM T. J. Watson Research Center, Yorktown Heights.
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Shamir, E., Snir, M. On the depth complexity of formulas. Math. Systems Theory 13, 301–322 (1979). https://doi.org/10.1007/BF01744302
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DOI: https://doi.org/10.1007/BF01744302