Generalized planning is concerned with how to find a single plan to solve multiple similar planning instances. Abstractions are widely used for solving generalized planning, and QNP (qualitative numeric planning) is a popular abstract model. Recently, Cui et al. showed that a plan solves a sound and complete abstraction of a generalized planning problem if and only if the refined plan solves the original problem. However, existing work on automatic abstraction for generalized planning can hardly guarantee soundness let alone completeness. In this paper, we propose an automatic sound and complete abstraction method for generalized planning with baggable types. We use a variant of QNP, called bounded QNP (BQNP), where integer variables are increased or decreased by only one. Since BQNP is undecidable, we propose and implement a sound but incomplete solver for BQNP. We present an automatic method to abstract a BQNP problem from a classical planning instance with baggable types. The basic idea for abstraction is to introduce a counter for each bag of indistinguishable tuples of objects. We define a class of domains called proper baggable domains, and show that for such domains, the BQNP problem got by our automatic method is a sound and complete abstraction for a generalized planning problem whose instances share the same bags with the given instance but the sizes of the bags might be different. Thus, the refined plan of a solution to the BQNP problem is a solution to the generalized planning problem. Finally, we implement our abstraction method and experiments on a number of domains demonstrate the promise of our approach.

Mutex groups are defined in the context of STRIPS planning as sets of facts out of which, maximally, one can be true in any state reachable from the initial state. The importance of computing and exploiting mutex groups was repeatedly pointed out in many studies. However, the theoretical analysis of mutex groups is sparse in current literature. This work provides a complexity analysis showing that inference of mutex groups is as hard as planning itself (PSPACE-Complete) and it also shows a tight relationship between mutex groups and graph cliques. This result motivates us to propose a new type of mutex group called a fact-alternating mutex group (fam-group) of which inference is NP-Complete. Moreover, we introduce an algorithm for the inference of fam-groups based on integer linear programming that is complete with respect to the maximal fam-groups and we demonstrate how beneficial fam-groups can be in the translation of planning tasks into finite domain representation. Finally, we show that fam-groups can be used for the detection of dead-end states and we propose a simple algorithm for the pruning of operators and facts as a preprocessing step that takes advantage of the properties of fam-groups. The experimental evaluation of the pruning algorithm shows a substantial increase in a number of solved tasks in domains from the optimal deterministic track of the last two planning competitions (IPC 2011 and 2014).