This paper presents a new semantic method for proving lower bounds in computational 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 integers, showing that NCZ \neq PTIME, where NCZ is the complexity class defined by such machines, and 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 NC is not equal to PTIME on the reals, and Mulmuley's lower bounds for "PRAMs without bit operations".

翻译：本文展示了一种新的语义方法来证明计算复杂度的下限。 我们用它来证明最大流( PTIME 完整的问题), 在平行随机访问机器( PRAMS) 与整数同时运行的多元度时间上无法计算, 显示 NCZ\ neq PTIME, 其中NCZ 是这类机器定义的复杂等级, PTIME 是多数值时可比较问题的标准等级( 例如, 图灵机器 ) 。 除了显示这个新的分离结果外, 我们展示了我们的方法捕捉了以前从文献中得出的下限结果: Steele 和 Yao 的代数决定树下限, Ben- Or 的代数计算树下限, Cucker 证明 NC并不等同于真数的 PTIME, Mululey 的下限为“ 没有位操作的 PRAMs ” 。