Abstract
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 \(\leq_m^p \mathrm{LT}_1\)). 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 \(\leq_m^p \mathrm{LT}_1\) then PH = PNP.
We furthermore show that if Sat disjunctive truth-table (or majority truth-table) reduces to a sparse set then Sat \(\leq_m^p\) LT1 and hence a collapse of PH to PNP also follows. Lastly we show several interesting consequences of Sat \(\leq_{dtt}^p\) SPARSE.
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Buhrman, H., Fortnow, L., Hitchcock, J.M., Loff, B. (2013). Learning Reductions to Sparse Sets. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_23
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