Finding All Solutions if You can Find One

We address the problem of enumerating (producing) all models of a given theory. We show that the enumeration task can be performed in time proportional to the product of the number of models and the e ort needed to generate each model in isolation. In other words, the requirement of generating a new solution in each iteration does not in itself introduce substantial complexity. Consequently, it is possible to decide whether any tractably satis able formula has more than K solutions in time polynomial in the size of the formula and in K. In the special cases of Horn formulas and 2-CNFs, although counting is #P-complete, to decide whether the count exceeds K, is polynomial in K. This work was partially supported by NSF grant IRI-9157636 and by the Air Force O ce of Scienti c Research, AFOSR 900136.