Representing (0, 1)-matrices by boolean circuits

A boolean circuit represents an n by n (0,1)-matrix A if it correctly computes the linear transformation y = Ax over GF(2) on all n unit vectors. If we only allow linear boolean functions as gates, then some matrices cannot be represented using fewer than (n2 /ln n) wires. We first show that using non-linear gates one can save a lot of wires: any matrix can be represented by a depth-2 circuit with O(nln n) wires using multilinear polynomials over GF(2) of relatively small degree as gates. We then show that this cannot be substantially improved: If any two columns of an n by n (0,1)-matrix differ in at least d rows, then the matrix requires (d ln n/lnln n) wires in any depth-2 circuit, even if arbitrary boolean functions are allowed as gates. Key words: Boolean circuits; Linear circuits; Hadamard matrices; Lower bounds; Sunflower lemma

• Type: Journal
• Year: 2010
• Where: DM
• Authors: Stasys Jukna
• Keywords: Boolean Circuits, Boolean Functions, DM 2010, Linear Boolean Functions