Integrating Abstraction, Explanation-based Learning from Multiple Examples, and Hierarchical Clustering with a Performance Component for Planning

World Wa: Concrete World Wc: Generic Abstraction Theory Tg sa sa sa 0 1 n sc sc sc sc sc 0 1 2 3 m Tg Tg Tg Figure 1: Demonstration of the Plan Abstraction Methodology description consists of a set of essential sentences R [Lifschitz, 1987] which describe the dynamic aspects of a world state, a static theory T which is assumed to be true in all states of the world, and a set of STRIPS-operators Op speci ed by precondition, add-list, and delete-list [Fikes et al., 1972]. Furthermore, we assume additional background knowledge which states how the terms of the abstract world , i.e. the essential sentences, can be de ned in terms of the concrete world. Generic abstraction theories for semantic abstraction, as introduced by Giordana, Roverso and Saitta [Giordana et al., 1991] relate atomic formulae of an abstract language to terms of a corresponding concrete language. In an adaptation of this idea, a generic state abstraction theory Tg in our model is de ned as a set of axioms of the form $ D1 _D2 _ : : : _Dn, where is an essential sentence of the abstract world and D1; : : : ; Dn are conjunctions of sentences of the concrete world. A plan is composed of operations which are executed in a speci c order and thereby successively change the state of a world. Within this planning model, abstraction has two independent dimensions: On the rst dimension a change in the level-of-detail for the representation of single states is described. On the second dimension a change in the level-of-detail is declared by reducing the number of states contained in a plan. As a consequence, a change of the representation of the state description and a change of the operations which describe the state transitions is required. Both dimensions of abstraction are essential to achieve a reduction of the complexity for planning. The problem of plan abstraction can now be described as transforming a plan pc from the concrete world Wc into a plan pa in the abstract world Wa as shown in Figure 1. This transformation must ensure, that all states created by the abstract plan can be derived with the generic abstraction theory Tg from some of the states created by the concrete plan. Thereby, a deductively justi ed plan abstraction is achieved. Abstractions which are more useful for reducing planning complexity are those, which are shared by a larger set of concrete plans rather than by a single plan. An abstraction which is shared by a larger class of concrete solution plans and therefore represents a problem decomposition which is applicable for several problem cases, has a higher application probability (ApplicFreq) for new problems and is consequently of higher utility. Intuitively, an abstract plan which holds for a set of plans P must be an abstraction for each of the plans. A formal de nition of this abstraction methodology can be found in [Bergmann, 1992c] and an extension for shared plan abstractions from several plans is formalized in [Bergmann, sc sc sc sc sc sc sc sc 1 2_ _ sc 1 _ _2 sc