Unary Coded PSPACE-Complete Languages in ASPACE(loglog n)

We study the class of binary coded versions of unary languages that can be accepted by alternating machines with loglog n space. We show that there exists a binary PSpace -complete language ℒ such that the unary coded version of ℒ is in ASpace (loglog n ). Consequently, the standard translation between unary languages accepted with loglog n space and binary languages accepted with log n space works for alternating machines if and only if P = PSpace . In general, if a binary language is accepted deterministically in 2 n ⋅ n O (1) time and, simultaneously, in n O (1) space—which covers many PSpace -complete problems—then its unary coded version is accepted by an alternating Turing machine using an initially delimited worktape of size loglog n . This unexpected power follows from the fact that, with an auxiliary worktape of size O (loglog n ) on a unary input 1 n , an alternating machine can simulate a stack with log n bits, representing the contents of the stack by its input head position. The standard push/pop operations on the stack are implemented by moving the head along the input.