One query reductibilities between partial information classes

Abstract

A partial information algorithm for A computes for input words (x 1,...,xm) a set of bitstrings containing ξA(x1,..., xm). For a family D of sets of bitstrings of length m, A ∈ P[D] if there is a polynomial time partial information algorithm that always outputs a set from D. For the case m = 2 we investigate whether for families D 1 and D2 the languages in P[D1] are reducible to languages in PX [D2] for some X in PH or in EXP. Beigel, Fortnow and Pavan, and Tantau already achieved some results in this respect. We achieve results for all remaining non-trivial pairs of classes P[D] for m = 2. We show: 1. 2-cheatable languages are 1-tt Δ2P-reducible to languages in Δ2P[MIN 2]. 2. Languages in P[SEL2U{xor2}] are 1-tt Δ2P-reducible to some Δ2P-selective languages. 3. 2-countable languages are 1-tt Δ3P-reducible to Δ3P strongly 2-membership comparable languages. 4. 2-membership comparable languages are 1-tt EXP-reducible to EXP strongly 2-membership comparable languages. 5. There are easily 2-countable languages not Turing reducible to languages in EXP[BOTTOM2]. 6. There are languages in P[BOTTOM2] not 1-tt EXP-reducible to EXP-selective languages. © Springer-Verlag 2004.