This rather long paper is a theoretical one, on reasoning about disjunctive information. One starts with a knowledge base built from atoms, a, and their negatives, not a, and structured by some rules of the form a1 ∨ a2 ∨ .... ∨ an ← b1, b2, ... , bm, not c1, not c2, .... , not cp. Let S be a set of positive disjunctions (disjunctions of atoms) and negative ones (disjunctions of negated atoms). Such a set is consistent if it is consistent in the sense of propositional calculus. An example is the set ms(P) of positive disjunctions, which are consequences of the rules. From the point of view of knowledge theory, ms(P) is too narrow and the interest is in larger consistent sets that include it, which are called model states here. The authors revisit three approaches that are very different in nature, slightly modify them, and then show that they lead to the same model state &THgr;.
The first approach is argumentation, which focuses on the negative part of &THgr; (the family of its negative disjunctions). Let us call a hypothesis a set of negative disjunctions. If &Dgr; is a hypothesis, a negative atom, not a, is admissible with respect to &Dgr; if a hypothesis &THgr;, which contradicts it, also contradicts &Dgr;. Then, the set A(&Dgr;) is the set of negative disjunctions &bgr;, such that some of their negative atoms are admissible with respect to &Dgr;. The negative part of &OHgr; (the family of its negative disjunctions) is the least fixed point of A (and the next one can recover the positive part).
The second approach, unfounded sets, focuses again on the negative part of &OHgr;, which in fact is characterized by the set X of atoms a, such that the negative disjunction with the unique term not a is in &OHgr;. One looks after a direct description of X, which can be seen as the least fixed point of a suitable functional.
With transformation of the rules, the third approach, one looks at the rules and defines a family of transformations of them (and so of programs). A model state W of P is invariant by a transformation T if it is also a model state of the transformed program. Then, &OHgr; appears as the smallest model state invariant by all of the transformations of the family.
The intuitive content is not very clear in each of the three characterizations. In fact, the major difficulty is always the same, the processing of the negative information. On the basis of the main result of the paper—the convergence of the three different approaches—the authors suggest that &OHgr; is the semantics of the program, a thesis that has to be tested against concrete data.
This theoretical research paper addresses the experts, with the customary condensed writing that sometimes may be obscure to nonexperts. The authors do, however, provide precise and useful references and carefully analyze their contribution.
