On the Quantitative Semantics of Regular Expressions over Real-Valued Signals

• Published in: Formal Modeling and Analysis of Timed Systems Vol. 10419; pp. 189 - 206
• Main Authors: Bakhirkin, Alexey, Ferrère, Thomas, Maler, Oded, Ulus, Dogan
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2017; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Duration Constraints; Piecewise Linear Signal; Quantitative Semantics; Regular Expression Signatures; Robust Matching
• ISBN: 331965764X; 9783319657646
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-65765-3_11

Signal regular expressions can specify sequential properties of real-valued signals based on threshold conditions, regular operations, and duration constraints. In this paper we endow them with a quantitative semantics which indicates how robustly a signal matches or does not match a given expression. First, we show that this semantics is a safe approximation of a distance between the signal and the language defined by the expression. Then, we consider the robust matching problem, that is, computing the quantitative semantics of every segment of a given signal relative to an expression. We present an algorithm that solves this problem for piecewise-constant and piecewise-linear signals and show that for such signals the robustness map is a piecewise-linear function. The availability of an indicator describing how robustly a signal segment matches some regular pattern provides a general framework for quantitative monitoring of cyber-physical systems.