Optimal bounds for decision problems on the CRCW PRAM

We prove optimal &OHgr;(log n/log log n) lower bounds on the time for CRCW PRAM's with polynomially bounded numbers of processors or memory cells to compute parity and a number of related problems. We also exhibit a strict time hierarchy of explicit Boolean functions of n bits on such machines which holds up to &Ogr;(log n/log log n) time. Furthermore, we show that almost all Boolean functions of n bits require log n - log log n + &OHgr;(1) time when the number of processors is at most polynomial in n. Our bounds do not place restrictions on the uniformity of the algorithms nor on the instruction sets of the machines.