PRAMs over integers do not compute maxflow efficiently

This paper presents a new semantic method for proving lower bounds in computational complexity. We use it to prove that $\mathbf{maxflow}$, a $\mathbf{Ptime}$-complete problem, is not computable in polylogarithmic time on parallel random access machines (PRAMs) working with integers, showing that $\mathbf{NC}_{\mathbb{Z}}\neq\mathbf{Ptime}$, where $\mathbf{NC}_{\mathbb{Z}}$ is the complexity class defined by such machines, and $\mathbf{Ptime}$ is the standard class of polynomial time computable problems (on, say, a Turing machine). On top of showing this new separation result, we show our method captures previous lower bounds results from the literature: Steele and Yao's lower bounds for algebraic decision trees, Ben-Or's lower bounds for algebraic computation trees, Cucker's proof that $\mathbf{NC}_{\mathbb{R}}$ is not equal to $\mathbf{Ptime}_{\mathbb{R}}$, and Mulmuley's lower bounds for "PRAMs without bit operations".