TY - JOUR

T1 - An extension of Khrapchenko's theorem

AU - Zwick, Uri

N1 - Funding Information:
* Partially supported by a Joseph and Jeanne Nissim post-doctoral grant from Tel Aviv University. * * Current address: The Mathematical Institute, University of Warwick, Coventry, United Kingdom CV4 7AL.

PY - 1991/2/28

Y1 - 1991/2/28

N2 - Khrapchenko's theorem is a classical result yielding lower bounds on the formula complexity of Boolean functions over the unate basis. In particular, it can yield a tight n2 lower bound on the formula complexity of the parity function. In this note we consider an extended definition of formula size in which each variable may be assigned a different cost. We then generalize Khrapchenko's theorem to cover this new definition and in particular derive a {norm of matrix}c{norm of matrix} 1 2=(∑ci 1 2)2 lower bou nd on the generalized formula size complexity of the parity function of n variables with cost vector c=(c1,...,cn). This bound is shown to be tight to within a factor of 2 by methods similar to Huffman coding. The extended definition of formula size arises naturally in cases where formulae for compound functions like f{hook}(g1,...;,gn) are sought.

AB - Khrapchenko's theorem is a classical result yielding lower bounds on the formula complexity of Boolean functions over the unate basis. In particular, it can yield a tight n2 lower bound on the formula complexity of the parity function. In this note we consider an extended definition of formula size in which each variable may be assigned a different cost. We then generalize Khrapchenko's theorem to cover this new definition and in particular derive a {norm of matrix}c{norm of matrix} 1 2=(∑ci 1 2)2 lower bou nd on the generalized formula size complexity of the parity function of n variables with cost vector c=(c1,...,cn). This bound is shown to be tight to within a factor of 2 by methods similar to Huffman coding. The extended definition of formula size arises naturally in cases where formulae for compound functions like f{hook}(g1,...;,gn) are sought.

KW - Formula complexity

KW - Huffman coding

KW - computational complexity

KW - lower bound

KW - parity function

UR - http://www.scopus.com/inward/record.url?scp=0026106054&partnerID=8YFLogxK

U2 - 10.1016/0020-0190(91)90191-J

DO - 10.1016/0020-0190(91)90191-J

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AN - SCOPUS:0026106054

SN - 0020-0190

VL - 37

SP - 215

EP - 217

JO - Information Processing Letters

JF - Information Processing Letters

IS - 4

ER -