Extending Horn Clause Theories by Reflection Principles

In this paper, we introduce logical reeection as a principled way to empower the representation and reasoning capabilities of logic programming systems. In particular, reeection principles take the role of axiom schemata of a particular form that, once added to a given logic program (the basic theory, or the initial axioms), enlarge the set of consequences sanctioned by those initial axioms. The main advantage of this approach is that it is much easier to write a basic theory and then to augment it with condensed axiom schemata, than it is to write a corresponding large (or even innnite) set of axioms in the rst place. Moreover, the well-established semantic properties of Horn clauses, carry over to Horn clauses with reeection. In fact, the semantics of Reeective SLD Resolution and the semantics of the Reeective Least Herbrand Model are obtained by making slight variations to, respectively , the procedural and the declarative semantics classically deened for Horn clauses. We present a complete formalization of this concept of reeection, that should constitute a simple way of understanding re-ective programs; and a description of how reeection allows one to treat uniformly diierent application areas. To support this claim, the following three case studies will be discussed: metalevel reasoning; reasoning with multiple communicating theories (agents); and analogical reasoning. For each of these areas, the choice of a suitable reeection principle is shown, which tries to capture the speciicity of the problem domain.