Other Semantics for NLPs

As we have seen in Chapter 8, and in particular in Theorem 8.1, all stable models are MH models. Since MH models are always guaranteed to exist for every NLP (cf. Theo- rem 8.5) and SMs are not, it follows immediately that the Minimal Hypotheses semantics is a strict model conservative generalization (cf. Definition 6.38) of the Stable Models semantics. For Normal Logic Programs, the Stable Models semantics coincides with the Answer-Set semantics (which is a generalization of SMs to Extended Logic Programs), where the latter is known (cf. [110]) to correspond to Reiter’s default logic. Hence, all Reiter’s default extensions have a corresponding Minimal Hypotheses model. Also, since

Moore’s expansions of an autoepistemic theory [160] are known to have a one-to-one cor- respondence with the stable models of the NLP version of the theory, we conclude that for every such expansion there is a matching Minimal Hypotheses model for the same NLP.

As shown in Theorem 8.2, at least one MH model of a program complies with its well-founded model, although not necessarily all MH models do. E.g., the program from Example 6.2 has the two MH models {a, b, not c} and {a, not b, c}, whereas the W F M (P ) imposes W F M+(P ) = {a, b}, W F Mu(P ) = ∅, and W F M−(P ) = {c}. This, as we already know, is due to the set of Hypotheses Hyps(P ) of P being taken from ˚P (which is based

on the layered support notion) instead of being taken from P (which is based upon theb

classical notion of support).

Not all Minimal Hypotheses models are Minimal Models of a program. As we have already noted in Chapter 8, the rationale behind MH semantics is minimality of hypothe- ses, but not necessarily minimality of consequences, the latter being enforceable, if so desired, as an additional requirement, although at the expense of increased complexity.

In [187, 204] (the latter being our M.Sc. thesis) we defined and studied the Revised Stable Models (RSMs) semantics for NLPs. This semantics had the same motivational drives as the ones for this thesis, but the definition of the RSM semantics turned out to be quite hard to grasp and to explain. The RSM semantics built upon the notion ofReductio

ad Absurdum (RAA): intuitively, if some atom is assumed false in an interpretation and

the same atom comes out true as a consequence of that assumption plus the rest of the interpretation, then, by reductio ad absurdum, that interpretation is rejected as a candidate model.

Although we have not yet undergone a thorough comparative analysis between the MH and the RSM semantics, in every example program we tried we noticed that all RSMs were also MH models, although the converse was not true.

Example 9.1. Revised Stable Models versus Minimal Hypotheses models.

Let P be

c ← not m m ← not b

b ← m, not c

This program has three MH models: M1= {b, c, not m}, M2= {not b, c, m}, and M3 =

{b, not c, m}. Of these, only M1 and M2 are RSMs.

In a very informal way, it seems that the RAA approach of RSMs can be seen as a restricted version of the hypotheses assumption of MH semantics, but the rigorous comparison of these two semantics is still to be done and, as such, is a topic for future work. Moreover, it turned out that the RSM semantics does not enjoy Cumulativity as

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we would desire, and this was one of the additional factors that motivated the definition of the MH semantics. E.g., in the Example 9.1 above, according to the RSM semantics,

c is true in all RSMs, but if we add c as a fact to the program then P ∪ {c} admits only M2= {not b, c, m} as an RSM thus rendering m also true in all its RSMs. Since m was

not true in all RSMs of P , the RSM semantics fails Cumulativity.

9.2 Argumentation

The relation between logic programs and argumentation systems has been considered for a long time now ([14, 42, 91, 173] amongst many others) and we have also taken steps to understand and further that relationship [188, 189, 190].

In 6.4.2.4 and 6.4.2.5 we saw different approaches to the argumentation perspective of semantics for NLPs.

We are not making a comprehensive comparative analysis of the MH semantics with the Revision Complete Scenarios but just noticing the philosophical similarities between them, namely concerning the minimal assumption of hypotheses producing, as a conse- quence, a 2-valued complete model. The MH semantics explicitly demands minimality of positive hypotheses, whereas the Revision Complete Scenario argumentation approach makes the non-redundant and unavoidable requirements on a set of positive hypotheses. These are not the same as explicit minimality, but the rationale behind them is similar. The Approved Models semantics, on the other hand, strives for maximality (in the sense of Definition 5.10) of negative hypotheses M−. One feature of the Approved Models se- mantics is that it generalizes Dung’s Preferred Extensions to 2-valued complete models. It remains for future work to analyse the Minimal Hypotheses semantics from an argu- mentation perspective, thoroughly comparing it to the Revision Complete Scenarios and the Approved Models semantics.

In [167], the author (with whom we shared our goals and vision of semantics for NLPs a few years ago) also aims at semantics for LPs, guaranteeing model existence, enjoying relevance, and seamlessly encompassing the argumentation approach. He also compares his own approach to our own [187] thereby showing that other researchers in our scientific community have recognized the importance and non-triviality of the issues and concerns we have been addressing.