Elements of Timed Pattern Matching

The rise of machine learning and cloud technologies has led to a remarkable influx of data within modern cyber-physical systems. However, extracting meaningful information from this data has become a significant challenge due to its volume and complexity. Timed pattern matching has emerged as a powerful specification-based runtime verification and temporal data analysis technique to address this challenge. In this paper, we provide a comprehensive tutorial on timed pattern matching that ranges from the underlying algebra and pattern specification languages to performance analyses and practical case studies. Analogous to textual pattern matching, timed pattern matching is the task of finding all time periods within temporal behaviors of cyber-physical systems that match a predefined pattern. Originally we introduced and solved several variants of the problem using the name of match sets, which has evolved into the concept of timed relations over the past decade. Here we first formalize and present the algebra of timed relations as a standalone mathematical tool to solve the pattern matching problem of timed pattern specifications. In particular, we show how to use the algebra of timed relations to solve the pattern matching problem for timed regular expressions and metric compass logic in a unified manner. We experimentally demonstrate that our timed pattern matching approach performs and scales well in practice. We further provide in-depth insights into the similarities and fundamental differences between monitoring and matching problems as well as regular expressions and temporal logic formulas. Finally, we illustrate the practical application of timed pattern matching through two case studies, which show how to extract structured information from temporal datasets obtained via simulations or real-world observations. These results and examples show that timed pattern matching is a rigorous and efficient technique in developing and analyzing cyber-physical systems.