We investigate alternative notions of approximation for problems inside P (deter-ministic polynomial time), and show that even a slightly nontrivial information about a problem may be as hard to obtain as the solution itself. For example, we prove that if one could eliminate even a single possibility for the value of an arithmetic circuit on a given input, then this would imply that the class P has fast (polygarithmic time) parallel solutions. In other words, this would constitute a proof that there are no inherently sequential problems in P, which is quite unlikely. The result is robust with respect to eliminating procedures that are allowed to err (by excluding the correct value) with small probability. We also show that several fundamental linear algebra problems are hard in this sense. It turns out that it is as hard to substantially reduce the number of possible values for the determinant and rank as to compute them exactly. Finally, we show that (in some precise sense) randomness can be nontrivially substituted for nondeterminism in space. Although it is believed that randomness does not give more than a constant factor advantage in space over determinism, it is not even known whether it is no more powerful than nondeterminism. We will show that the latter is true for a restricted version of probabilistic logspace, where the error is potentially larger than what can be achieved by amplification
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