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 maxflow, a Ptime complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. This improves, albeit slightly, on a result of Mulmuley since the class of machines considered extends the class "prams without bit operations", making more precise the relationship between Mulmuley's result and similar lower bounds on real prams. More importantly, we show our method captures previous lower bounds results from the literature, thus providing a unifying framework for "topological" proofs of lower bounds: Steele and Yao's lower bounds for algebraic decision trees, Ben-Or's lower bounds for algebraic computation trees, Cucker's proof that NC is not equal to Ptime in the real case, and Mulmuley's lower bounds for "prams without bit operations".

翻译：本文提出了一种证明计算复杂性下界的新抽象方法。基于动力系统的拓扑熵与可测熵概念，该方法被证明可推广文献中代数复杂性领域的三个先前下界结果。我们运用该方法证明了maxflow（一个Ptime完全问题）无法在操作实数的并行随机存取机器（prams）上以多对数时间计算。这改进了Mulmuley的结果（尽管幅度有限），因为所考虑的机器类别扩展了“无位操作prams”类，从而更精确地阐明了Mulmuley的结果与实数prams上类似下界之间的关系。更重要的是，我们证明了该方法能涵盖文献中先前的下界结果，从而为下界的“拓扑”证明提供了统一框架：包括Steele和Yao对代数决策树的下界、Ben-Or对代数计算树的下界、Cucker在实数情形下证明NC不等于Ptime的结果，以及Mulmuley针对“无位操作prams”的下界。