Circuit Bottom Fan-in and Computational Power

We investigate the relationship between circuit bottom fan-in and circuit size when circuit depth is xed. We show that in order to compute certain functions, a moderate reduction in circuit bottom fan-in will cause signi cant increase in circuit size. In particular, we prove that there are functions that are computable by circuits of linear size and depth k with bottom fan-in 2 but require exponential size for circuits of depth k with bottom fan-in 1. A general scheme is established to study the trade-o between circuit bottom fan-in and circuit size. Based on this scheme, we are able to prove, for example, that for any integer c, there are functions that are computable by circuits of linear size and depth k with bottom fan-in O(log n) but require exponential size for circuits of depth k with bottom fan-in c, and that for any constant> 0, there are functions that are computable by circuits of linear size and depth k with bottom fan-in log n but require superpolynomial size for circuits of depth k with bottom fan-in O(log 1; n). A consequence of these results is that the three input read-modes of alternating Turing machines proposed in the literature are all distinct

An average-case depth hierarchy theorem for Boolean circuits

We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of $\mathsf{AND}$, $\mathsf{OR}$, and $\mathsf{NOT}$ gates. Our hierarchy theorem says that for every $d \geq 2$, there is an explicit $n$-variable Boolean function $f$, computed by a linear-size depth-$d$ formula, which is such that any depth-$(d-1)$ circuit that agrees with $f$ on $(1/2 + o_n(1))$ fraction of all inputs must have size $\exp({n^{\Omega(1/d)}}).$ This answers an open question posed by H{\aa}stad in his Ph.D. thesis. Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus answering a question posed by O'Donnell, Kalai, and Hatami. A key ingredient in our proof is a notion of \emph{random projections} which generalize random restrictions

Circuit Bottom Fan-in and Computational Power

\Gamma ffl n). A consequence of these results is that the three input read-modes of alternating Turing machines proposed in the literature are all distinct. Warning: Essentially this paper has been published in SIAM Journal on Computing and is hence subject to copyright restrictions. It is for personal use only