Parallel RAM from Cyclic Circuits

Abstract: Known simulations of random access machines (RAMs) or parallel RAMs (PRAMs) by Boolean circuits incur significant polynomial blowup, due to the need to repeatedly simulate accesses to a large main memory. Consider a single modification to Boolean circuits that removes the restriction that circuit graphs are acyclic. We call this the cyclic circuit model. Note, cyclic circuits remain combinational, as they do not allow wire values to change over time. We simulate PRAM with a cyclic circuit, and the blowup from our simulation is only polylogarithmic. Consider a PRAM program $P$ that on a length-$n$ input uses an arbitrary number of processors to manipulate words of size $\Theta(\log n)$ bits and then halts within $W(n)$ work. We construct a size-$O(W(n)\cdot \log^4 n)$ cyclic circuit that simulates $P$. Suppose that on a particular input, $P$ halts in time $T$; our circuit computes the same output within $T \cdot O(\log^3 n)$ gate delay. This implies theoretical feasibility of powerful parallel machines. Cyclic circuits can be implemented in hardware, and our circuit achieves performance within polylog factors of PRAM. Our simulated PRAM synchronizes processors via logical dependencies between wires.