Learning Reductions to Sparse Sets [chapter]

Harry Buhrman, Lance Fortnow, John M. Hitchcock, Bruno Loff
2013 Lecture Notes in Computer Science  
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 ≤ p m LT1). 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
more » ... at ≤ p m LT1 then PH = P NP . We furthermore show that if Sat disjunctive truth-table (or majority truth-table) reduces to a sparse set then Sat ≤ p m LT1 and hence a collapse of PH to P NP also follows. Lastly we show several interesting consequences of Sat ≤ p dtt SPARSE.
doi:10.1007/978-3-642-40313-2_23 fatcat:ocm23fh4lzfyppn3w2qp5q4nyq