PAA：概率艾伦代数——艾伦区间关系的生成式且完备的概率扩展
PAA: The Probabilistic Allen Algebra: A Generative and Complete Probabilistic Extension of Allen's Interval Relations
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中文总结 AI 辅助
针对艾伦区间代数无法处理不确定时间边界的问题，提出概率艾伦代数PAA，从边界分布生成关系概率，实现完备扩展，并验证了分类层级与尺度不变性。
中文摘要 AI 辅助
艾伦区间代数是一种用于时间关系的定性演算，但其十三个基本关系是对精确区间边界的清晰谓词。这对于来自语言、感知、数据库或不确定历史的时间信息而言是不充分的，在这些场景中，时间、持续时间和边界是不确定的，诸如“就在之前”或“大致在期间”等表达具有分级含义。我们开发了概率艾伦代数（PAA）：一种生成式且完备的扩展，其中关系概率源自区间边界的分布，而非作为分数赋值。时间点是高斯的；区间具有高斯中点和截断高斯持续时间。每个关系都是在同一概率空间中的边界排序谓词：点-点关系简化为误差函数，点-区间和区间-区间关系简化为由线性不等式诱导的多元高斯象限概率。接触关系（相遇、开始、结束、相等）通过容差带获得正测度，并且在单一容差下，十三个关系形成一个真正的划分，当容差消失时恢复为清晰的艾伦代数。该构造推导出艾伦的分类法而非假定它：诸如优先、重叠和包含等粗粒度谓词是叶子的并集，其概率为叶子之和，并且当区间坍缩为点且十三个关系减少为五个然后三个时，此层级得以保留。每个关系进一步分解为符合CIDOC CRM精神的相关性感知时间原语。该代数具有尺度不变性，并将诸如“不久之前”等分级表达与接触关系区分开来。所有结果均经过蒙特卡洛验证，并作为开放、经过测试的Python包发布。
英文摘要
Allen's interval algebra is a qualitative calculus for temporal relations, but its thirteen base relations are crisp predicates over exact interval boundaries. This is inadequate for temporal information from language, perception, databases, or uncertain histories, where times, durations, and boundaries are uncertain and expressions such as "just before" or "roughly during" have graded meaning. We develop the probabilistic Allen algebra (PAA): a generative and complete extension in which relation probabilities are derived from distributions over interval boundaries rather than assigned as scores. Time points are Gaussian; intervals have Gaussian midpoints and truncated-Gaussian durations. Every relation is a boundary-ordering predicate in one common probability space: point-point relations reduce to error functions, and point-interval and interval-interval relations to multivariate Gaussian orthant probabilities induced by linear inequalities. Contact relations (meets, starts, finishes, equals) receive positive measure through a tolerance band, and under a single tolerance the thirteen relations form a true partition that recovers crisp Allen as the tolerance vanishes. The construction derives Allen's taxonomy rather than positing it: coarse predicates such as precedence, overlap, and containment are unions of leaves whose probabilities are leaf sums, and this hierarchy is preserved as intervals collapse to points and thirteen relations reduce to five and then three. Each relation further decomposes into correlation-aware temporal primitives in the spirit of CIDOC CRM. The algebra is scale-invariant and separates graded expressions such as "shortly before" from contact relations. All results are Monte-Carlo validated and shipped as an open, tested Python package.
