A small span theorem for P/Poly-Turing reductions

Abstract

This paper investigates the structure of ESPACE under nonuniform Turing reductions that are computed by polynomial-size circuits (P/Poly-Turing reductions). A small span theorem is proven for such reductions. This result says that every language A in ESPACE satisfies at least one of the following two conditions. (i) The lower P/Poly-Turing span of A (consisting of all languages that are P/Poly-Turing reducible to A) has measure 0 in PSPACE. (ii) The upper P/Poly-Turing span of A (consisting of all languages to which A is P/Poly-Turing reducible) has pspace-measure 0, hence measure 0 in ESPACE. The small span theorem implies that every P/Poly-Turing degree has measure 0 in ESPACE, and that there exist languages that are weakly P-many-one complete, but not P/Poly-Turing complete for ESPACE. The method of proof is a significant departure from earlier proofs of small span theorems for weaker types of reductions. P/Poly-Turing span of A (consisting of all languages to which A is P/Poly-Turing reducible) has pspace-measure 0, hence measure 0 in ESPACE. The small span theorem implies that every P/Poly-Turing degree has measure 0 in ESPACE, and that there exist languages that are weakly P-many-one complete, but not P/Poly-Turing complete for ESPACE. The method of proof is a significant departure from earlier proofs of small span theorems for weaker types of reductions.