Circuit Bottom Fan-in and Computational Power the Corresponding Author

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 signiicant 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-oo 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 super-polynomial 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. 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.