LL与CLL之间的无条件分离
Unconditional Separation between LL and CLL
AI总结：
本研究证明双对数尺度下催化图灵机可计算比标准确定性双对数空间机更多的语言，通过构造特定催化图灵机与标准图灵机的空间复杂度差异实现了该无条件分离。
摘要：
催化计算研究的是配备额外催化带（内容任意）的图灵机，机器可使用该带但需在计算结束时恢复带内容。已知催化对数空间机至少与非确定性对数空间机一样强大（甚至更强），但目前尚无无条件渐近分离结果。本研究证明了一个确定性空间复杂度类与其催化对应类之间的无条件分离：在双对数尺度下，催化图灵机可证明能计算比标准确定性双对数空间机更多的语言。该分离由一元语言POW2 = {1^n: n是2的幂}见证，我们构造了一个使用O(log log n)位空间和O(log n)位催化带的催化图灵机来判定POW2，而任何标准确定性图灵机判定该语言需要Ω(log n)位空间。
英文摘要：
Catalytic computation involves the study of Turing machines that are provided an additional catalytic tape (with arbitrary content) that the machine may use while ensuring that the tape contents are restored at the end of the computation. Although it is known that catalytic log-space machines are at least as powerful as non-deterministic log-space (and more), we do not have unconditional asymptotic separations. In this work, we prove an unconditional separation between a deterministic space complexity class and its catalytic counterpart. More precisely, at the doubly logarithmic scale, we show that catalytic Turing machines can provably compute more languages than standard deterministic doubly-logarithmic space machines. The separation is witnessed by the unary language $\mathrm{POW2} = \{1^n : n \text{ is a power of two}\}$. We construct a catalytic Turing machine that uses $O(\log \log n)$ bits of space and $O(\log n)$ bits of catalytic tape to decide $\mathrm{POW2}$, whereas any standard deterministic Turing machine requires $Ω(\log n)$ space to decide it.
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