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Size-depth tradeoff in monotone Boolean formulae

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Summary

Formula size and depth are two important complexity measures of Boolean functions. We study the tradeoff between those two measures: We give an infinite set of Boolean functions and show for nearly each of them: There is no monotone formula computing it optimal with respect to both measures. We give a lower and upper bound on the product of size and depth of monotone formulae computing our functions. That implies, moreover, a logarithmic lower bound on circuit depth.

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Abbreviations

ℕ:

the set of natural numbers {1,2,...}

⌊ ⌉:

for x>0, x =max{yεℕ∪{0}¦y¦<=x}

log:

logarithm to the base 2

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Author notes

  1. Beate Commentz-Walter

    Present address: IBM Scientific Center Heidelberg, Tiergartenstrasse 15, D-6900, Heidelberg

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  1. FB 10-Informatik, Universität des Saarlandes, D-6600, Saarbrücken, Germany (Fed. Rep.)

    Beate Commentz-Walter

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Commentz-Walter, B. Size-depth tradeoff in monotone Boolean formulae. Acta Informatica 12, 227–243 (1979). https://doi.org/10.1007/BF00264580

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