On Safety Properties and Their Monitoring

Abstract

Various definitions of safety properties as sets of execution traces have been introduced in the literature, some over finite traces, others over infinite traces, yet others over both finite and infinite traces. By employing cardinality arguments, this paper first shows that these notions of safety are ultimately equivalent, by showing each of them to have the cardinal of the continuum. It is then shown that all safety properties can be characterized as "always past" properties, and then that the problem of monitoring a safety property can be arbitrarily hard. Finally, two decidable specification formalisms for safety properties are discussed, namely extended regular expressions and past time LTL. It is shown that monitoring the former requires non-elementary space. An optimal monitor synthesis algorithm is given for the latter; the generated monitors run in space linear with the number of temporal operators and in time linear with the size of the formula.