We introduce derivation depth-a computable metric of the reasoning effort needed to answer a query based on a given set of premises. We model information as a two-layered structure linking abstract knowledge with physical carriers, and separate essential core facts from operational shortcuts. For any finite premise base, we define and prove the computability of derivation depth. By encoding reasoning traces and applying information-theoretic incompressibility arguments, we establish fundamental bounds linking depth to the descriptive complexity of queries. For frequently asked, information-rich queries, the minimal description length grows proportionally to depth times the logarithm of the knowledge base size. This leads to a practical storage-computation tradeoff: queries accessed beyond a critical threshold become cheaper to cache than recompute. We formulate optimal cache allocation as a mathematical optimization problem solvable with approximation guarantees and extend the framework to handle noisy or incomplete knowledge bases.

翻译：本文引入推导深度——一种可计算的度量指标，用于衡量基于给定前提集回答查询所需的推理工作量。我们将信息建模为连接抽象知识与物理载体的双层结构，并将核心本质事实与操作捷径相分离。针对任意有限前提基，我们定义并证明了推导深度的可计算性。通过编码推理轨迹并应用信息论不可压缩性论证，我们建立了将深度与查询描述复杂度相关联的基本界限。对于频繁访问且信息密集的查询，其最小描述长度与深度乘以知识库规模的对数成正比。这引出了一个实用的存储-计算权衡关系：当查询访问频率超过临界阈值时，缓存查询结果比重新计算更为经济。我们将最优缓存分配形式化为具有近似保证可解的数学优化问题，并将该框架扩展至处理含噪声或不完整知识库的场景。