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 ≤_m^pLT₁). 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 ≤_m^pLT₁ then PH = P^NP.

We furthermore show that if SAT disjunctive truth-table (or majority truth-table) reduces to a sparse set then SAT ≤_m^pLT₁ and hence a collapse of PH to P^NP also follows. Lastly we show several interesting consequences of SAT ≤_dtt^p SPARSE.