# The VLSI complexity of Boolean functions

• M. R. Kramer
• J. van Leeuwen
Section VI: Complexity Of Boolean Functions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 171)

## Abstract

It is well-known that all Boolean functions of n variables can be computed by a logic circuit with O(2n/n) gates (Lupanov's theorem) and that there exist Boolean functions of n variables which require logic circuits of this size (Shannon's theorem). We present corresponding results for Boolean functions computed by VLSI circuits, using Thompson's model of a VLSI chip. We prove that all Boolean functions of n variables can be computed by a VLSI circuit of O(2n) area and period 1, and we prove that there exist Boolean functions of n variables for which every (convex) VLSI chip must have Ω(2n) area.

## Keywords and phrases

logic circuit Boolean function Lupanov's theorem Shannon's theorem VLSI chip area period

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