Completeness and Weak Completeness Under Polynomial-Size Circuits

This paper investigates the distribution and nonuniform complexity of problems that are com plete or weakly complete for ESPACE under nonuniform reductions that are computed by polynomial size circuits P Poly Turing reductions and P Poly many one reductions A tight exponential lower bound on the space bounded Kolmogorov complexities of weakly P Poly Turing complete problems is established A Small Span Theorem for P Poly Turing reductions in ESPACE is proven and used to show that every P Poly Turing degree including the com plete degree has measure in ESPACE In contrast it is known that almost every element of ESPACE is weakly P many one complete Every weakly P Poly many one complete prob lem is shown to have a dense exponential nonuniform complexity core 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