Size-Depth Tradeoffs for Boolean Formulae

Size-Depth Tradeoffs for Boolean Fomulae

We present a simplified proof that Brent/Spira restructuring of Boolean formulas can be improved to allow a Boolean formula of size n to be transformed into an equivalent log depth formula of size O(nα) for arbitrary α > 1.

Size-Depth Tradeoffs for Algebraic Formulae

Faster Circuits and Shorter Formulae for Multiple Addition, Multiplication and Symmetric Boolean Functions

A general theory is developed for constructing the shallowest possible circuits and the shortest possible formulae for the carry save addition of n numbers using any given basic addition unit. (A carry save addition produces two numbers whose sum is equal to the sum of the n input numbers). More precisely, it is shown that if BA is a basic addition unit with occurrence matrix N then the shortes...

Faster Circuits and Shorter Formulas for Multiple Addition, Multiplication and Symmetric Boolean Functions

A general theory is developed for constructing the shallowest possible circuits and the shortest possible formulae for the carry save addition of n numbers using any given basic addition unit. (A carry save addition produces two numbers whose sum is equal to the sum of the n input numbers). More precisely, it is shown that if BA is a basic addition unit with occurrence matrix N then the shortes...

Model-Theoretic Characterizations of Boolean and Arithmetic Circuit Classes of Small Depth

In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes NC1, SAC1 and AC1 as well as their arithmetic counterparts #NC1, #SAC1 and #AC1. We build on Immerman’s characterization of constant-depth polynomial-size circuits by formulae of first-order logic, i.e., AC0 = FO, an...