Extending Horn clause theories by reflection principles

Abstract

In this paper, we introduce logical reflection as a principled way to empower the representation and reasoning capabilities of logic programming systems. In particular, reflection 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 infinite) set of axioms in the first place. Moreover, the well-established semantic properties of Horn clauses, carry over to Horn clauses with reflection. In fact, the semantics of Reflective SLD Resolution and the semantics of the Reflective Least Herbrand Model are obtained by making slight variations to, respectively, the procedural and the declarative semantics classically defined for Horn clauses. We present a complete formalization of this concept of reflection, that should constitute a simple way of understanding reflective programs; and a description of how reflection allows one to treat uniformly different 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 reflection principle is shown, which tries to capture the specificity of the problem domain.