An extension of Khrapchenko's theorem

Uri Zwick*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)215-217
Number of pages3
JournalInformation Processing Letters
Volume37
Issue number4
DOIs
StatePublished - 28 Feb 1991
Externally publishedYes

Funding

FundersFunder number
Tel Aviv University

    Keywords

    • Formula complexity
    • Huffman coding
    • computational complexity
    • lower bound
    • parity function

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