Space Hierarchy Theorem Revised

We show that, for an arbitrary function h(n) and each recursive function ‘(n), that are separated by a nondeterministically fully space constructible g(n), such that h(n)∈0(g(n)) but ‘(n) ∈ 0(g(n)), there exists a unary language L in NSPACE(h(n)) that is not contained in NSPACE(‘(n)). The same holds for the deterministic case. The main contribution to the well-known Space Hierarchy Theorem is that (i) the language L separating the two space classes is unary (tally), (ii) the hierarchy is independent of whether h(n) or ‘(n) are in 0(log n) or in o(log n), (iii) the functions h(n) or ‘(n) themselves need not be space constructible nor monotone increasing, (iv) the hierarchy is established both for strong and weak space complexity classes. This allows us to present unary languages in such complexity classes as, for example, NSPACE(log log n · log∗ n) \ NSPACE(log log n), using a plain diagonalization. c © 2002 Elsevier Science B.V. All rights reserved.