A Direct Version of Shamir and Snir's Lower Bounds on Monotone Circuit Depth

Abstract We present direct proofs of the following results of Shamir and Snir [Mathematical System Theory 13 (1980) 301-322] on the depth of monotone arithmetic circuits over rings of characteristic 0: (1) an ω((log p)(log n)) lower bound for computing the product of p n × n matrices; and (2) an ω(n) lower bound for computing the permanent of an n × n matrix.