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Which bases admit non-trivial shrinkage of formulae?

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Abstract.

We show that the shrinkage exponent, under random restrictions, of formulae over a finite complete basis B of Boolean functions, is strictly greater than 1 if and only if all the functions in B are unate, i.e., monotone increasing or decreasing in each of their variables. As a consequence, we get non-linear lower bounds on the formula complexity of the parity function over any basis composed only of unate functions.

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Authors and Affiliations

  1. School of Computer Science and Engineering, The Hebrew University, Givat Ram, Jerusalem 91904, Israel, e-mail: hanac@cs.huji.ac.il, http://www.cs.huji.ac.il/~hanac, , , , , , IL

    H. Chockler

  2. School of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel, e-mail: zwick@cs.tau.ac.il, http://www.cs.tau.ac.il/~zwick, , , , , , IL

    U. Zwick

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  1. H. Chockler
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  2. U. Zwick
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Received: June 15, 2000.

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Chockler, H., Zwick, U. Which bases admit non-trivial shrinkage of formulae?. Comput. complex. 10, 28–40 (2001). https://doi.org/10.1007/PL00001610

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  • DOI: https://doi.org/10.1007/PL00001610

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