In this paper, we introduce Interval Real Logic (IRL), a two-sorted logic that interprets knowledge such as sequential properties (traces) and event properties using sequences of real-featured data. We interpret connectives using fuzzy logic, event durations using trapezoidal fuzzy intervals, and fuzzy temporal relations using relationships between the intervals' areas. We propose Interval Logic Tensor Networks (ILTN), a neuro-symbolic system that learns by propagating gradients through IRL. In order to support effective learning, ILTN defines smoothened versions of the fuzzy intervals and temporal relations of IRL using softplus activations. We show that ILTN can successfully leverage knowledge expressed in IRL in synthetic tasks that require reasoning about events to predict their fuzzy durations. Our results show that the system is capable of making events compliant with background temporal knowledge.

翻译：本文提出区间实值逻辑（IRL），这是一种双排序逻辑，通过实值特征序列对顺序属性（迹）和事件属性等知识进行解释。我们使用模糊逻辑解释连接词，利用梯形模糊区间表示事件持续时间，并通过区间面积之间的关系定义模糊时态关系。我们提出区间逻辑张量网络（ILTN），这是一种通过梯度传播学习IRL的神经符号系统。为支持高效学习，ILTN采用softplus激活函数定义了IRL中模糊区间与模糊时态关系的平滑版本。实验表明，在需要推理事件以预测模糊持续时间的合成任务中，ILTN能有效利用IRL表达的知识。研究结果显示，该系统能够使事件符合背景时态知识约束。