We describe an efficient decentralized algorithm to monitor the execution of a distributed program in order to check for violations of safety properties. The monitoring is based on formulas written in PT-DTL, a variant of past time linear temporal logic that we define. PT-DTL is suitable for expressing temporal properties of distributed systems. Specifically, the formulas of PT-DTL are relative to a particular process and are interpreted over a projection of the trace of global states that

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... sents what that process is aware of. A formula relative to one process may refer to the local states of other processes through remote expressions and remote formulas. In order to correctly evaluate remote expressions, we introduce the notion of knowledge vector and provide an algorithm which keeps a process aware of other processes' local states, if those states may affect the validity of a monitored PT-DTL formula. Both the logic and the monitoring algorithm are illustrated through a number of examples. Finally, we describe our implementation of the algorithm in a tool called DIANA.