Learning 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 [1] 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 ≤pmLT1 ). They claim that P= NP follows as a consequence, but unfortunately their proof was incorrect. We take up this question and use results from computational learning theory to show that if Sat ≤pmLT1 then PH = PNP. We furthermore show that if Sat disjunctive truth-table (or majority truth-table) reduces to a sparse set then Sat ≤pm LT1 and hence a collapse of PH to PNP also follows. Lastly we show several interesting consequences of Sat ≤pdtt SPARSE.