# Completeness and weak completeness under polynomial-size circuits

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## Abstract

This paper investigates the distribution and nonuniform complexity of problems that are complete or weakly complete for ESPACE under nonuniform many-one reductions that are computed by polynomial-size circuits (P/Poly-many-one reductions). Every weakly P/Poly-many-one-complete problem is shown to have a dense, exponential, nonuniform complexity core. An exponential lower bound on the space-bounded Kolmogorov complexities of weakly P/Poly-Turing-complete problems is established. More importantly, the P/Poly-many-one-complete problems are shown to be *unusually simple* elements of ESPACE, in the sense that they obey nontrivial *upper* bounds on nonuniform complexity (size of nonuniform complexity cores and space-bounded Kolmogorov complexity) that are violated by almost every element of ESPACE. More generally, a Small Span Theorem for P/Poly-many-one reducibility in ESPACE is proven and used to show that every P/Poly-many-one degree -including the complete degree — has measure 0 in ESPACE. (In contrast, almost every element of ESPACE is weakly P-many-one complete.) All upper and lower bounds are shown to be tight.

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