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

Abstract

We show that there exists a binary PSpace-complete language ${\mathcal {L}}$ such that its unary coded version ${\mathcal {L}}'$ is in ${\textsc {ASpace}}^{\mathrm {dm}}(\log \log n)$, i.e., ${\mathcal {L}}'$ is accepted by an alternating Turing machine using an initially delimited worktape of size $\log \log n$. As a consequence, 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 ${\textsc {P}}= {\textsc {PSpace}}$.

In general, if a binary language ${\mathcal {L}}$ is in ${\textsc {DTimeSpace}}(2^{n}\!\cdot \!n^{\scriptscriptstyle O(1)},n^{\scriptscriptstyle O(1)})$, i.e., if ${\mathcal {L}}$ is accepted by a deterministic Turing machine in $2^{n}\!\cdot \!n^{\scriptscriptstyle O(1)}$ time and, simultaneously, in $n^{\scriptscriptstyle O(1)}$ space, then its unary coded version ${\mathcal {L}}'$ is in ${\textsc {ASpace}}^{\mathrm {dm}}(\log \log n)$. In turn, if a unary ${\mathcal {L}}'$ is in ${\textsc {ASpace}}^{\mathrm {dm}}(\log \log n)$, then its binary coded version ${\mathcal {L}}$ is in ${\textsc {DTime}}(2^{n}\!\cdot \!n^{\scriptscriptstyle O(1)})\cap {\textsc {DSpace}}(n^{\scriptscriptstyle O(1)})$, and also in ${\textsc {NTimeSpace}}(2^{n}\!\cdot \!n^{\scriptscriptstyle O(1)},n^{\scriptscriptstyle O(1)})$.

This unexpected power of sublogarithmic space follows from the fact that, with a worktape of size $\log \log n$ 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 are implemented by moving the head along the input.

V. Geffert—Supported by the Slovak grant contracts VEGA 1/0142/15 and APVV-15-0091.