Lowness and the complexity of sparse and tally descriptions

Abstract

We investigate the complexity of obtaining sparse descriptions for sets in various reduction classes to sparse sets. Let A be a set in a certain reduction class R,.(SPARSE). Then we are interested in finding upper bounds for the complexity (relative to A) of sparse sets S such that A ε Rr(S). By establishing such upper bounds we are able to derive the lowness of A. In particular, we show that if a set A is in the class Rphd(Rpc(SPARSE)) then A is in Rphd(Rpc(S)) for a sparse set S ε NP(A). As a consequence we can locate Rphd(Rpc(SPARSE)) in the EL⊖3 level of the extended low hierarchy. Since Rphd(Rpc(SPARSE)) ⊇ Rphd(Rpc(SPARSE)) this solves the open problem of locating the closure of sparse sets under bounded truth-table reductions optimally in the extended low hierarchy. Furthermore, we show that for every A ε Rpd(SPARSE) there exists a sparse set S ε NP(A ⊕ SAT)/F⊖p2(A) such Lhat A ε Rpd(S). Based on this we show that Rp1-tt(Rpd(SPARSE)) is in EL⊖3 Finally, we construct for every set A ε Rpc(TALLY) ∩Rpd(TALLY) (or equivalently, A ε IC[log, poly], as shown in [AHH+92]) a tally set T ε P(A ⊕ SAT) such that A 9 Rpc(T) ∩ Rpd(T). This implies that the class IC[log, poly] of sets with low instance complexity is contained in EL∑1.