Disciplina Modelado del Negocio

a semantics for P and M an interpretation of P . Then, S_PQ means that Q is universally entailed by P according to semantics S iff

S_PQ⇔ ∀M(SP(M ) ⇒ M |=S Q)

The negative counterpart now is S_P¬Q which means Q is not true in any model M of

P (according to semantics S), hence ¬Q can be unconditionally inferred. Formally,

S_P¬Q⇔ ∀_M(S_P(M ) ⇒ M 6|=_S Q)

With a semantics allowing for several alternative models we can see each of them as a set of hypotheses plus their corresponding necessary conclusions — a semantics allowing for only one model would not allow any hypothesizing freedom.

In [129] the authors argue in favour of viewing default negated literals in bodies of rules as abducibles, or assumable hypotheses. They defend that negation (and in logic programs in particular, default negation) must play a central role in endowing negated elements with eligibility to hypothesization. We take this hypotheses assumption perspective which has also been taken by several semantics for logic programs before, namely, the Stable Models semantics with which we compare our approach in subsequent chapters. There it will become clear how specifically we embed and semantically implement this hypotheses assumption principle.

1.2.2 Induction

Induction usually refers to a type of reasoning that involves producing a general rule from a set of specific facts. It can also be seen as a form of learning or theory-building, in which specific facts are used to create a theory that explains relationships between the facts and allows prediction of future knowledge. In the context of logic programming, Inductive Logic Programming [136, 162, 163] has been a productive research area in the past years. Inductive reasoning lies outside the scope of this thesis as it is an orthogonal issue to the one focused on here: finding a 2-valued semantics, with a specific set of properties, for all NLPs.

1.2.3 Abduction

Abductive reasoning is discussed in Chapter 10 and is commonly understood as hypoth- esizing plausible reasons sufficient for justifying given observations or supporting desired goals. An abductive problem can be stated in the following way: let KB be a Knowl- edge Base, Q a goal (also referred to as the abductive query), and ∆ a set of adoptable (abducible) hypotheses. In this case we say δ ⊆ ∆ is an abductive solution to Q given

KB iff KB ∪ δ |= Q and δ |= ICs where ICs is the set of Integrity Constraints pertaining KB. We may, of course, be also interested in the side-effects of abducing δ in KB. In

Chapter 10 we introduce an efficient method to check if some literal’s truth-value becomes determined as a side-effect of a given abductive solution to a goal.

For centuries, a central guideline of rational scientific reasoning has been the heuristic known as Ockham’s razor. The Encyclopedia Britannica [1] presents, amongst others, the following formulation of this reasoning principle: “(. . . )the simplest explanation of

an entity is to be preferred.” This principle which has maximal skepticism at its heart,

is also known as the law of economy, or law of parsimony and has been formulated in many different ways including “the explanation requiring the fewest assumptions is most

likely to be correct.” This correlates to abduction, or hypotheses assumption, in the sense

that Ockham’s razor states that we should always strive for minimal sets of hypotheses explaining the observations, or entailing the query.

In turn, hypotheses assumption relates to credulous entailment (or existential query answering) in the sense that, as stated at the end of 1.2.1.3, with “a semantics allowing for several alternative models we can see each of them as a set of hypotheses plus their corresponding necessary conclusions”. In the context of semantics for logic programs allowing for more than one model, Abductive and Deductive reasoning end up being almost equivalent as we show and further detail in Chapter 10.

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The classical approach to semantics rests on demanding minimality of models, i.e., minimality of the set of hypotheses plus their conclusions; but as far as theOckham’s razor principle is concerned, thelaw of parsimony is only focused on the set-inclusion minimality of the hypotheses, letting the orthogonal issue of minimizing their consequences as an optional undertaking. Indeed, there may be non-minimal sets of hypotheses conducive to minimal models when the former might be incompatible amongst themselves. This issue occurs in logic programming due to default negation or integrity constraints, e.g. Example 8.3. Minimality of hypotheses, while resolving away their incompatibility, will be a major issue addressed in this thesis in the context of logic programming. In Chapter 8 we introduce a semantics focused only on minimality of the hypotheses.

1.3 Reasoning Scope

In this thesis we focus on using logic programs as the knowledge representation formalism to encode knowledge bases [131]. Under this setting, and depending on the nature of the problem at hand and the semantics considered, finding an intended solution for a query may require taking into account all the formulas in the KB or, on the other hand, it may suffice to consider only the fraction of the KB relevant to the query formula. In the first case, finding a solution to the query requires the identification of a whole model for the entire KB, whilst in the second case identifying a part of a model might suffice. When solutions to the problem at hand involve by necessity whole models, we say we perform complete knowledge reasoning when finding such solutions. Conversely, partial knowledge reasoning refers to finding parts of model being enough, it being implicit that such part models are extendible to a whole model in the program semantics used.

1.3.1 Complete Knowledge Reasoning

Complete knowledge reasoning pertains to identifying models of the whole knowledge base that conform to some user-specified requirements. From a practical standpoint, this corresponds to whole KB model computation. It is the role of the chosen semantics to determine which interpretations of a KB are accepted as models. Depending on the properties of the envisaged underlying semantics, whole model computation may be the only possibility for any kind of problem solving. In particular, such complete knowledge reasoning may be in general inevitable when the semantics lacks the relevance property. This is discussed in further detail in subsection 6.5.5.

1.3.2 Partial Knowledge Reasoning

Reasoning with large KBs represented as NLPs can be computationally very expensive, especially if the KB can be updated, whether by external agents or through self-updates. When KBs are comprised of rules concerning several fields of knowledge, we can say we have some modularization of the knowledge. This can be the case either when the KB itself is used to represent an ontology plus the set of rules to derive new knowledge, or when the KB is just such a set of rules and is associated with an ontology originating somewhere else, as in [115]. In such cases, it can be quite common for the user to want to perform some form of reasoning concerning just a fragment of the overall knowledge. Whole model computation can then be computationally overwhelming and downright unnecessary. It would also be putting too much of a burden on the user to require her/him to specify the (sub-) module(s) of knowledge to be considered when finding a solution to the query. Again, taking advantage of such a modularization naturally occurring in knowledge can only be afforded when the indicated underlying semantics enjoys the relevance property. Partial knowledge reasoning, which can be seen as partial model computation, using only the fragments of knowledge strictly necessary to find an answer, can have significant impact on real applications’s efficiency.

Having introduced some of the general concepts of knowledge representation and reasoning we now take in Chapter 2 a closer look at the syntactic structure of a set of KBs induced by their interdependencies.