Learning Weak Reductions to Sparse Sets

We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind~\cite{AA:96} who study the consequences of \SAT being many-one reducible to functions computable by non-uniform circuits consisting of a single weighted threshold gate. (\SAT \leq_m^p \LT). They claim that as a consequence \PTIME = \NP follows, but unfortunately their proof was incorrect. We take up this question and use results from computational learning theory to show that if \SAT \leq_m^p \LT then \PH = \PTIME^\NP. We furthermore show that if \SAT disjunctive truth-table (or majority truth-table) reduces to a sparse set then \SAT \leq_m^p \LT and hence a collapse of \PH to \PTIME^\NP also follows. Lastly we show several interesting consequences of \SAT \leq_{dtt}^p \SPARSE

Black holes as mirrors: quantum information in random subsystems

We study information retrieval from evaporating black holes, assuming that the internal dynamics of a black hole is unitary and rapidly mixing, and assuming that the retriever has unlimited control over the emitted Hawking radiation. If the evaporation of the black hole has already proceeded past the "half-way" point, where half of the initial entropy has been radiated away, then additional quantum information deposited in the black hole is revealed in the Hawking radiation very rapidly. Information deposited prior to the half-way point remains concealed until the half-way point, and then emerges quickly. These conclusions hold because typical local quantum circuits are efficient encoders for quantum error-correcting codes that nearly achieve the capacity of the quantum erasure channel. Our estimate of a black hole's information retention time, based on speculative dynamical assumptions, is just barely compatible with the black hole complementarity hypothesis.Comment: 18 pages, 2 figures. (v2): discussion of decoding complexity clarifie

NP-hard sets are exponentially dense unless coNP C NP/poly

textabstractWe show that hard sets $S$ for \NP must have exponential density, i.e. $|S_{=n}| \geq 2^{n^\epsilon}$ for some $\epsilon > 0$ and infinitely many $n$, unless \coNP \subseteq \NP/\poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make $n^{1-\epsilon}$ queries. In addition we study the instance complexity of \NP-hard problems and show that hard sets also have an exponential amount of instances that have instance complexity $n^\delta$ for some $\delta>0$. This result also holds for Turing reductions that make $n^{1-\epsilon}$ queries

Bounds for the variance of an inverse binomial estimator

Summary  Best [1] found the variance of the minimum variance unbiased estimator of the parameter p of the negative binomial distribution. Mikulski and Sm [2] gave an upper bound to it, easier to calculate than Best's expression and a good approximation for small values of p and large values of r (the number of successes). In this paper both lower bounds and closer upper bounds are derived. Copyrigh