We describe an efficient decentralized monitoring algorithm that monitors a distributed program's execution to check for violations of safety properties. The monitoring is based on formulae 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 formulae of PT-DTL are relative to a particular process and are interpreted over a projection of the trace of global states that represents what that process is aware of. A formula relative to one process may refer to other processes' local states through remote expressions and remote formulae. In order to correctly evaluate remote expressions, we introduce the notion of KNOWLEDGEVECTOR and provide an algorithm which keeps a process aware of other processes' local states that can 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.

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Introduction Reasoning about knowledge is one of the fundamental problems in multi-agent systems. Usually, dynamic global state spaces partitioned w.r.t. the agents' information are considered [3]. In the theory of distributed systems, knowledge formulas are interpreted over infinite linear or branching runs of the systems [5, 6, 3, 8, 13, 14]. It is clear that capturing changes in state due to actions is crucial for successful modelling of knowledge. Consequently, the knowledge of the agents who do not participate in execution of an action should remain unchanged, whereas the agents executing the action should know the effect of the execution. While these changes are usually present in the frames, logical formalisms quite rarely incorporate them. One of the reasons is that when actions are incorporated into global state formalisms, this leads to high undecidability [7, 8]. The solution to this problem was to interpret formulas on local states of agents [13]. Our frames are de