Seeing is Believing: Monitoring Future Time Temporal Logic

Runtime monitors for future-time unbounded temporal logics like RVLTL, LTL 3 and FLTL, have double-exponential (2^2^n) worst-case space complexity bounds in size of the input formula. The semantics of these logics require monitors to perform general satisfiability solving for LTL expressions, a well-studied problem whose computational complexity is NP-hard and PSPACE-complete. This paper introduces an unbounded future-time linear temporal logic defined over a lattice. We call our logic an incremental temporal logic as it can be viewed as incrementally constructing proofs about the trace. On this account, we view online runtime monitoring as a decision procedure for proofs systems about incrementally growing traces. We demonstrate that our incremental temporal logic allows monitor construction to void satisfiability solving while still soundly detecting when the property is violated in an online fashion. This enables asymptotic improvements in space complexity. As proof, we provide a procedure to construct monitors that utilize linear space and time in the size of the input formula, while remaining constant in the size of the input stream and suitable for online monitoring. We further demonstrate, through several examples, that our incremental temporal logic is straightforward to adopt and practical for runtime verification.