Ampliication by Read-once Formulae

Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone read-once formulae of size O(n +2) (where = log p 5?1 2 ' 3:27) that amplify (? 1 n ; + 1 n) (where = p 5?1 2 ' 0:62) to (2 ?n ; 1 ? 2 ?n) and deduced as a consequence the existence of monotone formulae of the same size that compute the majority of n bits. Boppana has shown that any monotone read-once formula that ampliies (p ? 1 n ; p + 1 n) to (1 4 ; 3 4) (where 0 < p < 1 is constant) has size (n) and that any monotone, not necessarily read-once, contact network (and in particular any monotone formula) that ampliies (1 4 ; 3 4) to (2 ?n ; 1 ? 2 ?n) has size (n 2). We extend Boppana's results in two ways. We rst show that his two lower bounds hold for general read-once formulae, not necessarily monotone, that may even include exclusive-or gates. We are then able to join his two lower bounds together and show that any read-once, not necessarily monotone, formula that ampliies (p ? 1 n ; p + 1 n) to (2 ?n ; 1 ? 2 ?n) has size (n +2). This result does not follow from Boppana's arguments and it shows that the amount of ampliication achieved by Valiant is the maximal achievable using read-once formulae. In a companion paper we construct monotone read-once contact networks of size O(n 2:99) that amplify (1 2 ? 1 n ; 1 2 + 1 n) to (1 4 ; 3 4). This shows that Boppana's lower bound for the rst ampliication stage does not apply to contact networks, even if they are required to be both monotone and read-once.