The problem of testing whether a finite execution trace of events generated by an executing program violates a linear temporal logic (LTL) formula occurs naturally in runtime analysis of software. Two efficient algorithms for this problem are presented in this paper, both for checking safety formulae of the form always P, where P is a past-time LTL formula. The first algorithm is implemented by rewriting, and the second synthesizes efficient code from formulae. Further optimizations of the second algorithm are suggested, reducing space and time consumption. Special operators suitable for writing succinct specifications are discussed and shown to be equivalent to the standard past-time operators. This work is part of NASAs PathExplorer project, the objective of which is to construct a flexible framework for efficient monitoring and analysis of program executions.