Timed Partial Orders (TPOs), originally proposed for workflows, provide an interpretable framework for robot task specification with planning algorithms based on mixed-integer linear programming (MILP). However, TPOs are limited in expressivity, capturing only partial-order events with simple timing constraints. In this paper, we introduce Conditional TPOs (cTPOs), which extend TPOs with richer relative-timing constraints and conditional event activations based on environmental conditions. We show that planning for cTPOs also reduces to an MILP problem; however, the added expressivity results in significantly larger MILPs that can become computationally intractable. To address this challenge, we propose a decomposition algorithm that partitions a cTPO into smaller sub-TPOs, yielding a sequence of smaller MILP problems. We prove that this decomposition is complete and preserves plan optimality while improving the interpretability of complex tasks. Experimental results demonstrate the effectiveness of cTPOs as a task specification framework and the efficiency of our decomposition approach, achieving up to four orders of magnitude speedup over the monolithic MILP.

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