On the power of euclidean division: Lower bounds for algebraic machines, semantically

This paper presents a new abstract method for proving lower bounds in computational complexity. Based on the notion of topological and measurable entropy for dynamical systems, it is shown to generalise three previous lower bounds results from the literature in algebraic complexity. We use it to prove that $\mathtt{maxflow}$, a $\mathrm{Ptime}$ complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. This improves on a result of Mulmuley since the class of machines considered extends the class "prams without bit operations". We further improve on this result by showing that euclidean division cannot be computable in polylogarithmic time using division on the reals, pinpointing that euclidean division provides a significant boost in expressive power. 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 $\mathrm{NC}$ is not equal to $\mathrm{Ptime}$ in the real case, and Mulmuley's lower bounds for prams without bit operations.