On the Discovery and Utility of Precedence Constraints in Temporal Planning

Temporal planning considers temporal dependencies and numeric resources. As a further step towards real-world applications, it has been attracting many attentions and triggered some great work. Recently, most of those works are based on heuristic search, e.g., in the work of Sapa (Do and Kambhampati 2003), LPG (Gerevini et al. 2008), SGP (Chih-wei et al. 2008) and TFD (Eyerich et al. 2009). In this line, the design of reasonable and informative heuristics shows a great progress, especially in the planner TFD. TFD proposed an extension of the SAS+ formulism, which is called temporal numeric SAS+. The formulism basically has two advantages: leading to a smaller search space in comparison with the STRIPS representation, and building a convenient base for designing causal graph (Helmert 2006) based heuristics. Specifically, TFD proposed an extension of the heuristic h (Helmert and Geffner 2008), which is potentially more informative than the causal graph heuristic h (Helmert 2006) and can lead to plans of high quality. Noted that Cai et al. (2009) proposed an extension of h, which is called h. h enhanced h with the so called precedence constraints, and is potentially more informative than h. So, a natural question is, in a temporal and numeric setting, how h can be generalized and what the result is. In this preliminary work, we make an attempt to answer the question and report some initial results. Let’s first consider an example where h could be improved. We follow the example used by Patrick et al (2009). Assume that there are two locations l0 and l1, a robot r1 located at l1 with a water tank whose capacity is