Describing the properties of complex systems that evolve over time is a crucial requirement for monitoring and understanding them. Signal Temporal Logic (STL) is a framework that proved to be effective for this aim because it is expressive and allows state properties as human-readable formulae. Crafting STL formulae that fit a particular system is, however, a difficult task. For this reason, a few approaches have been proposed recently for the automatic learning of STL formulae starting from observations of the system. In this paper, we propose BUSTLE (Bi-level Universal STL Evolver), an approach based on evolutionary computation for learning STL formulae from data. BUSTLE advances the state-of-the-art because it (i) applies to a broader class of problems, in terms of what is known about the state of the system during its observation, and (ii) generates both the structure and the values of the parameters of the formulae employing a bi-level search mechanism (global for the structure, local for the parameters). We consider two cases where (a) observations of the system in both anomalous and regular state are available, or (b) only observations of regular state are available. We experimentally evaluate BUSTLE on problem instances corresponding to the two cases and compare it against previous approaches. We show that the evolved STL formulae are effective and human-readable: the versatility of BUSTLE does not come at the cost of lower effectiveness.
