Two queries

AbstractWe consider the question whether two queries SAT are as powerful as one query. We show that if PNP[1]=PNP[2] then: Locally either NP=coNP or NP has polynomial-size circuits; PNP=PNP[1]; Σp2⊆Πp2/1; Σp2=UPNP[1]∩RPNP[1]; PH=BPPNP[1]. Moreover, we extend the work of Hemaspaandra, Hemaspaandra, and Hempel to show that if PΣp2[1]=PΣp2[2] then Σp2=Πp2. We also give a relativized world, where PNP[1]=PNP[2], but NP≠coNP

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

Inverting Onto Functions and Polynomial Hierarchy

In this paper we construct an oracle under which the polynomial hierarchy is infinite but there are non-invertible polynomial time computable multivalued onto functions