Scaled Dimension and the Berman-Hartmanis Conjecture

In 1977, L. Berman and J. Hartmanis [BH77] conjectured that all polynomialtime many-one complete sets for NP are are pairwise polynomially isomorphic. It was stated as an open problem in [LM99] to resolve this conjecture under the measure hypothesis from quantitative complexity theory. In this paper we study the polynomial-time isomorphism degrees within degm(SAT ) in the context of polynomial scaled dimension. Our results are the following: 1. We consider scaled dimensions of order in between −2 and −3. Especially we define scaled dimensions dim (−2,k) p , k ∈ N. 2. Let ISO m(SAT ) denote the polynomial-time isomorphism degree of SAT . While for each k, dim (−2,k) p (deg p m(SAT )) = dim (−2,k) p (NP ), if r is a growth rate function of order smaller than every order (−2, k), then dim (r) p (ISO p m(SAT )) = 0. 3. We consider the class of disjoint unions L1 ⊕ L2 of NP -complete languages L1, L2 such that L1 and L2 are polynomially isomorphic. We show that for |i| ≤ 2 the i-th order scaled dimension of this class equals that of NP . The same holds for the scaled dimensions dim (−2,k) p .