UNARY CODED NP-COMPLETE LANGUAGES IN ASPACE(log log n)

Abstract

We shall show that (i) there exists a binary NP-complete language [Formula: see text] such that its unary coded version [Formula: see text] is in ASPACE( log log n), (ii) if P ≠ NP, there exists a binary language [Formula: see text] such that its unary version [Formula: see text] is in ASPACE( log log n), while the language [Formula: see text] 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 [Formula: see text] 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). Under assumption that NP ≠ co-NP, we also show that binary versions of unary languages in ASPACE( log log n) form a complexity class not contained in NP.