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 _r (SPARSE). Then we are interested in finding upper bounds for the complexity (relative to A) of sparse sets S such that A ∃ R _r (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 R ^p _hd (R ^p _c (SPARSE)) then A is in R ^c _p (R ^p _hd (S)) for a sparse set S ∃ NP(A). As a consequence we can locate R ^p _hd (R ^p _c (SPARSE)) in the EL ^Θ₃ level of the extended low hierarchy. Since R ^p _hd (R ^p _c (SPARSE)) $\supseteq$ R ^p _b (R ^p _c (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 ∃ R ^p _d (SPARSE) there exists a sparse set S ∃ NP(A ⊕ SAT)/Fθ ^p₂ (A) such that A ∃ R ^p _d (S). Based on this we show that R ^p_1−tt (R ^p _d (SPARSE)) is in EL ^Θ₃ .

Finally, we construct for every set A ∃ R ^p _c (TALLY)∩R ^p _d (TALLY) (or equivalently, A ∃ IC[log, poly], as shown in [AHH⁺92]) a tally set T ∃ P(A ⊕ SAT) such that A ∃ R ^p _c (T) ∩ R ^p _d (T). This implies that the class IC[log, poly] of sets with low instance complexity is contained in EL ^Σ₁ .

Work done while visiting UniversitÄt Ulm. Supported in part by an Alexander von Humboldt research fellowship.