AI 中文总结
研究针对能量缺乏形式化处理的问题，提出能量演算这一组合代数，基于能量元素及三个运算符，通过七条公理构建代数，有归约定理，不确定性可传播，运算符还能扩展到时间-能量帕累托前沿，实现节能策略组合及权衡推理。
AI 中文摘要
能量是人工智能扩展的关键限制，但缺乏像计算、通信和学习那样的形式化处理。近期系统虽有节能成果，但各有特定粒度和结构，无法组合不同系统的节能策略。我们提出能量演算，将能量视为一等原语的组合代数。它基于能量元素构建，能量元素携带能量签名。三个运算符沿计算结构组合签名，代数基于七条公理，具有能量独特的性质。还给出归约定理，不确定性贯穿组合，每个预测有误差界。最后表明运算符可扩展到时间-能量帕累托前沿。
英文摘要
Energy is a binding constraint for AI scaling, yet it lacks the formal treatment that computation, communication, and learning have long enjoyed. Recent systems demonstrate large energy savings, but each targets a specific granularity and structure; one cannot combine frequency scaling from one system with critical-path analysis from another and reason about their joint effect on total energy. Energy remains a monolithic scalar that is measured after the fact and optimized with point solutions that do not generalize. We propose energy calculus, a compositional algebra that treats energy as a first-class primitive. It builds on energy elements, units of computation whose energy we can reliably measure, each carrying an energy signature that comprises its time, its static and dynamic energy, the hardware operating point and execution context under which we measured it, and the associated measurement uncertainty. Three operators (sequential, same-device parallel, and cross-device parallel) compose signatures along the same structure as the computation itself, covering arbitrary DAG-structured executions. The algebra rests on seven axioms that capture how hardware consumes energy, and it exhibits two properties distinctive to energy among computing resources: sequential composition commutes only when elements are mutually context-insensitive, and sequential composition does not distribute over parallel composition. We also present a Reduction Theorem that recovers simple context-independent algebra whenever interactions fall below measurement uncertainty, so practitioners pay for context dependence only where the physics demands it. Uncertainty propagates through every composition, so each prediction carries an error bound. Finally, we show that the same operators extend from energy totals to time--energy Pareto frontiers, so reasoning about tradeoffs composes with the same algebra.
