Extending inverse resolution to build up abstractions
Abstraction is a conceptual framework potentially unifying and integrating different methodologies developed in machine learning and problem solving. According to the common understanding of the term, abstraction is a mapping between languages, and, then, in many learning tasks it can be so formulated. On the contrary, in this paper abstraction is defined as a mapping between models, implicitly extending Tenenberg's restricted predicate mapping in order to include more complex abstraction schemes and to cope with the problem of inconsistency in the abstract model.
In our proposal, the abstraction mapping is axiomatized by means of a theory TA, defining the semantics of the relations in the abstract model starting from the ones in the ground model. Therefore, this form of abstraction is called semantic and must be evaluated using a deductive mechanism instead of a purely syntactic rewriting. Afterwards, a restricted class of semantic abstraction (CP-Abstraction) is characterized: it has the property of preserving concept instances, with respect to a given model, and the more-general-than relation between formulas. CP-Abstraction fits in the paradigm of inverse resolution, already proposed as a framework for constructive learning and a restricted form of absorption rule is introduced to compute it.
Inverse resolution, in its original formulation, does not allow important abstraction types, such as the definition of a compound object starting from its parts, to be defined. A new operation, called term abstraction, is then introduced.
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