Bart Selman, Hector J. Levesque
Of all the possible ways of computing abductive explanations, the ATMS procedure is one of the most popular. While this procedure is known to run in exponential time in the worst case, the proof actually depends on the existence of queries with an exponential number of answers. But how much of the difficulty stems from having to return these large sets of explanations? Here we explore abduction tasks similar to that of the ATMS, but which return relatively small answers. The main result is that although it is possible to generate some non-trivial explanations quickly, deciding if there is an explanation containing a given hypothesis is NP-hard, as is the task of generating even one explanation expressed in terms of a given set of assumption letters. Thus, the method of simply listing all explanations, as employed by the ATMS, probably cannot be improved upon. An interesting result of our analysis is the discovery of a subtask that is at the core of generating explanations, and is also at the core of generating extensions in Reiter’s default logic. Moreover, it is this subtask that accounts for the computational difficulty of both forms of reasoning. This establishes for the first time a strong connection between computing abductive explanations and computing extensions in default logic.