We propose a novel passive learning approach, TeLex, to infer signal temporal logic (STL) formulas that characterize the behavior of a dynamical system using only observed signal traces of the system. First, we present a template-driven learning approach that requires two inputs: a set of observed traces and a template STL formula. The unknown parameters in the template can include time-bounds of the temporal operators, as well as the thresholds in the inequality predicates. TeLEx finds the value of the unknown parameters such that the synthesized STL property is satisfied by all the provided traces and it is tight. This requirement of tightness is essential to generating interesting properties when only positive examples are provided and there is no option to actively query the dynamical system to discover the boundaries of legal behavior. We propose a novel quantitative semantics for satisfaction of STL properties which enables TeLEx to learn tight STL properties without multidimensional optimization. The proposed new metric is also smooth. This is critical to enable the use of gradient-based numerical optimization engines and it produces a 30x to 100x speed-up with respect to the state-of-art gradient-free optimization. Second, we present a novel technique for automatically learning the structure of the STL formula by incrementally constructing more complex formula guided by the robustness metric of subformula. We demonstrate the effectiveness of the overall approach for learning STL formulas from only positive examples on a set of synthetic and real-world benchmarks.

中文翻译：

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Telex：使用紧密度度量从正例中学习信号时间逻辑

我们提出了一种新颖的被动学习方法 TeLex，以仅使用观察到的系统信号轨迹来推断表征动态系统行为的信号时间逻辑 (STL) 公式。首先，我们提出了一种模板驱动的学习方法，它需要两个输入：一组观察到的轨迹和一个模板 STL 公式。模板中的未知参数可以包括时间运算符的时间范围，以及不等式谓词中的阈值。Telex 找到未知参数的值，使得所有提供的迹线都满足合成的 STL 属性并且它是紧密的。当仅提供正面示例且无法主动查询动态系统以发现合法行为的边界时，这种紧密性要求对于生成有趣的属性至关重要。我们提出了一种新的量化语义来满足 STL 属性，它使 Telex 能够在没有多维优化的情况下学习紧密的 STL 属性。提议的新指标也是平滑的。这对于启用基于梯度的数值优化引擎至关重要，并且相对于最先进的无梯度优化，它产生了 30 到 100 倍的加速。其次，我们提出了一种新技术，通过在子公式的稳健性度量指导下逐步构建更复杂的公式，自动学习 STL 公式的结构。我们展示了仅从一组合成和现实世界基准的正面示例中学习 STL 公式的整体方法的有效性。