Linear Time Monitoring for One Variable TPTL

The temporal logic Timed Propositional Temporal Logic () extends with freeze quantifiers in order to express timing constraints, and is strictly more expressive than Metric Temporal Logic () over future modalities. The monitoring problem is to check whether a particular timed trace satisfies a given temporal logic specification, and monitoring procedures form core subroutines of testing and falsification approaches for Cyber-Physical Systems. In this work, we develop an efficient linear time monitoring algorithm, linear in the length of the trace (for traces that have at most a constant number of sample points in any unit interval), for one variable in the pointwise semantics. This one variable fragment is known to be already more expressive than and thus allows specifications of richer timed properties. Our algorithm carefully combines a divide and conquer approach with dynamic programming in order to achieve a linear time algorithm. As a plus, our algorithm uses only a simple two-dimensional table, and a syntax tree of the formula, as the data structures, and hence can be easily implemented on various platforms. We demonstrate the tractability of our approach with our prototype tool implementation on Matlab; our experiments show the tool scales easily to long trace lengths.