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This paper studies the computational complexity of temporal planning, as represented by PDDL 2.1, interpreted over dense time. When time is considered discrete, the problem is known to be EXPSPACE-complete. However, the official PDDL 2.1 semantics, and many implementations, interpret time as a dense domain. This work provides several results about the complexity of the problem, studying a few interesting cases: whether a minimum amount ϵ of separation between mutually exclusive events is given, in contrast to the separation being simply required to be non-zero, and whether or not actions are allowed to overlap already running instances of themselves. We prove the problem to be PSPACE-complete when self-overlap is forbidden, whereas, when allowed, it becomes EXPSPACE-complete with ϵ-separation and undecidable with non-zero separation. These results clarify the computational consequences of different choices in the definition of the PDDL 2.1 semantics, which were vague until now.