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Unary coded PSPACE-complete languages in $$\mathrm{ASPACE}(\log\log n)$$. (English) Zbl 1425.68199
Summary: We study the class of binary coded versions of unary languages that can be accepted by alternating machines with $$\log\log n$$ space. We show that there exists a binary PSpace-complete language $$\mathcal {L}$$ such that the unary coded version of $$\mathcal {L}$$ is in $$\text{\textsc{ASpace}}(\log\log n)$$. Consequently, the standard translation between unary languages accepted with $$\log\log n$$ space and binary languages accepted with $$\log n$$ space works for alternating machines if and only if $$\mathrm{P} = \text{\textsc{PSpace}}$$. In general, if a binary language is accepted deterministically in $$2^n \cdot n^{O (1)}$$ 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 $$\log\log n$$. This unexpected power follows from the fact that, with an auxiliary worktape of size $$O(\log\log n)$$ 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.
##### MSC:
 68Q45 Formal languages and automata 68Q05 Models of computation (Turing machines, etc.) (MSC2010) 68Q25 Analysis of algorithms and problem complexity
##### Keywords:
computational complexity; alternation; sublogarithmic space
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