Learning Specifications for Labelled Patterns

In this work, we introduce a supervised learning framework for inferring temporal logic specifications from labelled patterns in signals, so that the formulae can then be used to correctly detect the same patterns in unlabelled samples. The input patterns that are fed to the training process are labelled by a Boolean signal that captures their occurrences. To express the patterns with quantitative features, we use parametric specifications that are increasing, which we call Increasing Parametric Pattern Predictor (IPPP). This means that augmenting the value of the parameters makes the predicted pattern true on a larger set. A particular class of parametric specification formalisms that we use is Parametric Signal Temporal Logic (PSTL). One of the main contributions of this paper is the definition of a new measure, called c-count, to assess the quality of the learned formula. This measure enables us to compare two Boolean signals and, hence, quantifies how much the labelling signal induced by the formula differs from the true labelling signal (e.g. given by an expert). Therefore, the c-count can measure the number of mismatches (either false positives or false negatives) up to some error tolerance E. Our supervised learning framework can be expressed by a multicriteria optimization problem with two objective functions: the minimization of false positives and false negatives given by the parametric formula on a signal. We provide an algorithm to solve this multi-criteria optimization problem. Our approach is demonstrated on two case studies involving characterization and classification of labeled ECG (electrocardiogram) data.