We prove a square-root space simulation for deterministic multitape Turing machines, showing $\mathrm{TIME}[t]\subseteq \mathrm{SPACE}[O(\sqrt{t})]$ \emph{measured in tape cells over a fixed finite alphabet}. The key step is a Height Compression Theorem that uniformly (and in logspace) reshapes the canonical left-deep succinct computation tree for a block-respecting run into a binary tree whose evaluation-stack depth along any DFS path is $O(\log T)$ for $T=\lceil t/b\rceil$, while preserving $O(b)$ workspace at leaves and $O(1)$ at internal nodes. Edges have \emph{addressing/topology} checkable in $O(\log t)$ space, and \emph{semantic} correctness across merges is witnessed by an exact $O(b)$ bounded-window replay at the unique interface. Algorithmically, an Algebraic Replay Engine with constant-degree maps over a constant-size field, together with pointerless DFS, index-free streaming, and a \emph{rolling boundary buffer that prevents accumulation of leaf summaries}, ensures constant-size per-level tokens and eliminates wide counters, yielding the additive tradeoff $S(b)=O(b+t/b)$. Choosing $b=Θ(\sqrt{t})$ gives $O(\sqrt{t})$ space with no residual multiplicative polylog factors. The construction is uniform, relativizes, and is robust to standard model choices. Consequences include branching-program upper bounds $2^{O(\sqrt{s})}$ for size-$s$ bounded-fan-in circuits, tightened quadratic-time lower bounds for $\mathrm{SPACE}[n]$-complete problems via the standard hierarchy argument, and $O(\sqrt{t})$-space certifying interpreters; under explicit locality assumptions, the framework extends to geometric $d$-dimensional models. Conceptually, the work isolates path bookkeeping as the chief obstruction to $O(\sqrt{t})$ and removes it via structural height compression with per-path analysis rather than barrier-prone techniques.

翻译：我们证明了确定性多带图灵机的一种平方根空间模拟，即 $\mathrm{TIME}[t]\subseteq \mathrm{SPACE}[O(\sqrt{t})]$ \emph{（在固定有限字母表上以磁带单元度量）}。关键步骤是一个高度压缩定理，该定理（在对数空间内）均匀地将一个块遵从运行的规范左深简洁计算树重塑为一棵二叉树，使得沿任意深度优先搜索路径的求值栈深度为 $O(\log T)$，其中 $T=\lceil t/b\rceil$，同时在叶子节点保留 $O(b)$ 的工作空间，在内部节点保留 $O(1)$ 的工作空间。边的\emph{寻址/拓扑结构}可在 $O(\log t)$ 空间内验证，而跨越合并操作的\emph{语义}正确性由在唯一接口处进行精确的 $O(b)$ 有界窗口重放来见证。算法上，一个在常数大小域上具有常数度映射的代数重放引擎，结合无指针深度优先搜索、无索引流式处理以及一个\emph{防止叶子节点摘要累积的滚动边界缓冲区}，确保了每层令牌大小为常数，并消除了宽计数器，从而产生了加法权衡 $S(b)=O(b+t/b)$。选择 $b=Θ(\sqrt{t})$ 可得到 $O(\sqrt{t})$ 空间，且无残留的乘法多对数因子。该构造是均匀的、可相对化的，并且对标准模型选择具有鲁棒性。其推论包括：对于规模为 $s$ 的有界扇入电路，分支程序的上界为 $2^{O(\sqrt{s})}$；通过标准层次结构论证，为 $\mathrm{SPACE}[n]$-完全问题收紧的二次时间下界；以及 $O(\sqrt{t})$-空间的认证解释器。在明确的局部性假设下，该框架可扩展至几何 $d$ 维模型。从概念上讲，这项工作将路径簿记识别为实现 $O(\sqrt{t})$ 的主要障碍，并通过结合逐路径分析的结构性高度压缩（而非易产生障碍的技术）将其消除。