Forgetting revisited

In this paper, we propose an alternative notion, called weak forgetting, of forgetting a set of predicates in a first-order theory. One important feature of this new notion is that the result of weak forgetting is always first-order expressible. In contrast, this is not the case for the traditional notion of forgetting, called strong forgetting, introduced by Lin and Reiter. As a consequence, these two notions are not exactly the same. Interestingly, we prove that they coincide when the result of strong forgetting is first-order expressible. We also present a representation theorem to characterize weak forgetting from different aspects.