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

Monotonous and Randomized Reductions to Sparse Sets

An oracle machine is called monotonous, if after a negative answer the machine does not ask further queries to the oracle. For example, one truthtable, conjunctive, and Hausdorff reducibilities are monotonous. We study the consequences of the existence of sparse hard sets for different complexity classes under monotonous and randomized reductions. We prove trade-offs between the randomized time complexity of NP sets that reduce to a set B via such reductions and the density of B as well as the number of queries made by the monotonous reduction. As a consequence, bounded Turing hard sets for NP are not co-rp reducible to a sparse set unless RP = NP. We also prove similar results under the apparently weaker assumption that some solution of the promise problem (1SAT; SAT) reduces via the mentioned reductions to a sparse set