Asserting Lemmas in the Stable Model Semantics

Non–monotonic reasoning, semantics and foundations, stable model semantics From a logic programming point of view, the stable model semantics for normal programs has the problem that logical consequences of programs cannot, in general, be stored as lemmas. This is because the set of stable models of the resulting program may change. In fact, logical consequence under the stable model semantics does not enjoy an important property required of non–monotonic entailment relations, i.e. cumulativity. We argue that it is possible to assert a conclusion A as a lemma in the stable model semantics, if asserting at the same time a set of facts supporting the conclusion (that we call a base set for A). The effect on the meaning of the program is that of selecting some of the stable models containing A. The collection of all base sets for A generates all the stable models containing A. We formalize this intuition by reformulating the definition of cumulativity accordingly. We propose a characterization of base sets that identifies the minimal ones, i.e. the fewest and smallest base sets for A. Any proof procedure for the stable model semantics (including the abductive ones) should be able, with slight modifications, to return the base sets, by applying the criteria that we propose.