Between 1998 and 2004, the planning community has seen vast progress in terms of the sizes of benchmark examples that domain-independent planners can tackle successfully. The key technique behind this progress is the use of heuristic functions based on relaxing the planning task at hand, where the relaxation is to assume that all delete lists are empty. The unprecedented success of such methods, in many commonly used benchmark examples, calls for an understanding of what classes of domains these methods are well suited for. In the investigation at hand, we derive a formal background to such an understanding. We perform a case study covering a range of 30 commonly used STRIPS and ADL benchmark domains, including all examples used in the first four international planning competitions. We *prove* connections between domain structure and local search topology -- heuristic cost surface properties -- under an idealized version of the heuristic functions used in modern planners. The idealized heuristic function is called h^+, and differs from the practically used functions in that it returns the length of an *optimal* relaxed plan, which is NP-hard to compute. We identify several key characteristics of the topology under h^+, concerning the existence/non-existence of unrecognized dead ends, as well as the existence/non-existence of constant upper bounds on the difficulty of escaping local minima and benches. These distinctions divide the (set of all) planning domains into a taxonomy of classes of varying h^+ topology. As it turns out, many of the 30 investigated domains lie in classes with a relatively easy topology. Most particularly, 12 of the domains lie in classes where FFs search algorithm, provided with h^+, is a polynomial solving mechanism. We also present results relating h^+ to its approximation as implemented in FF. The behavior regarding dead ends is provably the same. We summarize the results of an empirical investigation showing that, in many domains, the topological qualities of h^+ are largely inherited by the approximation. The overall investigation gives a rare example of a successful analysis of the connections between typical-case problem structure, and search performance. The theoretical investigation also gives hints on how the topological phenomena might be automatically recognizable by domain analysis techniques. We outline some preliminary steps we made into that direction.

In particular, modeling context dependent eects, concurrent execution of actions with dierent duration, and continuous resources are all awkward, or impossible, within the STRIPS language. To overcome the rst of these limitations, Pednault (1989) dened the (nowadays widely accepted) ADL language, which amongst other things allows for conditional eects (eects that only occur when their condition holds true in the state of execution). To overcome (one or both of) the latter two limitations, various proposals have been made (e.g., Ghallab & Laruelle, 1994; Koehler, 1998; Smith & Weld, 1999). The most recent eort in this direction is the PDDL2.1 language dened by Fox and Long (2002) as the input language for the 3rd International Planning Competition (IPC-3). The IPC series is a biennial challenge for the planning community, inviting planning systems to participate in a large scale publicly accessible evaluation. IPC-3 was hosted at AIPS-2002, and stressed planning beyond the STRIPS formalism, featuring tracks for temporal and numeric planners. This article describes the approach behind one of the planners that participated in IPC-3, Metric-FF. Metric-FF is an extension of the FF system (that can handle ADL) to numeric constructs.

Planning with numeric state variables has been a challenge for many years, and was a part of the 3rd International Planning Competition (IPC-3). Currently one of the most popular and successful algorithmic techniques in STRIPS planning is to guide search by a heuristic function, where the heuristic is based on relaxing the planning task by ignoring the delete lists of the available actions. We present a natural extension of ``ignoring delete lists'' to numeric state variables, preserving the relevant theoretical properties of the STRIPS relaxation under the condition that the numeric task at hand is ``monotonic''. We then identify a subset of the numeric IPC-3 competition language, ``linear tasks'', where monotonicity can be achieved by pre-processing. Based on that, we extend the algorithms used in the heuristic planning system FF to linear tasks. The resulting system Metric-FF is, according to the IPC-3 results which we discuss, one of the two currently most efficient numeric planners.

STRIPS: n. 1) An automated planner by Fikes and Nilsson in 1971 2)