Unary Coded NP-Complete Languages in ASPACE (log log n)

Abstract

(i) There exists an NP-complete language $\mathcal{L}$ such that its unary coded version un-$\mathcal{L}$ is in ASpace(log log n). (ii) If P ≠ NP, there exists a binary language $\mathcal{L}$ such that its unary version un-$\mathcal{L}$ is in ASpace(log log n), while the language $\mathcal{L}$ itself is not in ASpace(log n). As a consequence, under assumption that P ≠ NP, the standard space translation between unary and binary languages does not work for alternating machines with small space, the equivalence $\mathcal{L} \in$ ASpace(s(n)) ≡ un-$\mathcal{L} \in$ ASpace(s(log n)) is valid only if s(n)∈Ω(n). This is quite different from deterministic and nondeterministic machines, for which the corresponding equivalence holds for each s(n)∈Ω(log n), and hence for s(log n)∈Ω(log log n).