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We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete black-box derandomization of identity testing problem for depth four circuits with(More)
A finite state Markov chain M is often viewed as a probabilistic transition system. An alternative view - which we follow here - is to regard M as a linear transform operating on the space of probability distributions over its set of nodes. The novel idea here is to discretize the probability value space [0,1] into a finite set of intervals. A concrete(More)
We show that all sets that are c omplete for NP under non-uniform AC 0 reductions are isomorphic under non-uniform AC 0-computable isomorphisms. Furthermore, these sets remain NP-complete even under non-uniform NC 0 reductions. More generally, we show two theorems that hold for any complexity class C closed under uniform NC 1-computable many-one reductions.(More)