发表机构
École Polytechnique（巴黎综合理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究为齐次搜索问题建立自相似约化定理，提出有限编码、多边形化与量词消去结合的方法，证明平面海岸线搜索最优值为可计算实数，并提供相关研究工具箱。
AI 中文摘要
海岸线搜索路径始于原点，必须与一条未知仿射直线相遇，且该直线的法向量与距离均未知。我们首先为齐次搜索问题建立了自相似约化定理，这类问题的历史信息是通过逐点最大值更新的记录剖面。归一化状态的两次准返回界定了一个块，该块可自行更新所需的剖面；一段短连接段将该块闭合为一个单元。因此，每一条有限比率路径都可以通过在所有尺度上重复单个单元来近似，且损失可任意小。\n 随后我们将主链方法付诸实现。状态空间的有限编码可计算地界定了近似最优单元的尺度因子和归一化长度。对于平面海岸线搜索，凸包的支撑函数给出了精确的单元泛函。随后，单侧多边形化将问题简化为数量可计算的顶点，之后通过量词消去法判定是否存在低于有理阈值的多边形单元。\n 由此可得，最优确定性平面海岸线值$C_2^*$是一个可计算实数：对于任意有理$ε>0$，存在算法可终止并输出一个宽度至多为$ε$、包含$C_2^*$的有理区间。其他结果——滑动记忆、Bellman转移、截止期限、几何滤波器和相对平衡——作为认证计算和螺旋刚性研究的工具箱单独呈现，未用于可计算性证明。
英文摘要
A shoreline-search path starts at the origin and must meet an unknown affine line, without knowing either its normal or its distance. We first establish a self-similar reduction theorem for homogeneous search problems whose historical information is a record profile updated by pointwise maximum. Two quasi-returns of the normalized state delimit a block that renews the required profile by itself; a short connector closes this block into a cell. Every finite-ratio path can therefore be approximated, with arbitrarily small loss, by repetitions of a single cell at all scales. The main chain is then made effective. A finite coding of the state space computably bounds the scale factor and normalized length of a nearly optimal cell. For planar Shoreline search, the support function of the convex hull gives an exact cell functional. A one-sided polygonalization then reduces the problem to a computable number of vertices, after which quantifier elimination decides whether a polygonal cell exists below a rational threshold. It follows that the optimal deterministic planar Shoreline value $C_2^*$ is a computable real: for every rational $ε>0$, an algorithm terminates with a rational interval of width at most $ε$ containing $C_2^*$. Additional results---sliding memory, Bellman transitions, deadlines, geometric filters, and relative equilibria---are presented separately as a toolbox for certified computation and for the study of spiral rigidity; they are not used in the computability proof.
Comments23 pages, 1 figure. English revision 8.3
