Model Representation and Evaluation

Abstract

The model building algorithms presented so far either produce finite models or finite representations of infinite models (in particular Herbrand models) . In case of finite models the distinction between the model (as an abstract mathematical object) and its syntactic representation, although necessary from a logical point of view, is rather pointless and sophistic; clearly every finite model is uniquely determined by the enumeration of its domain elements and by the “multiplication”-tables for functions and predicates. The situation becomes less trivial for infinite structures. As every satisfiable formula F of first-order logic has a Herbrand model (defined via the skolemized form), Herbrand interpretations are the most natural candidates. In order to specify a specific Herbrand model of a formula we may always order all models of a formula and take the smallest one, let us call it M this is possible by the well-ordering theorem (which is equivalent to the axiom of choice) . Such a specification is clearly noncomputational in a strong sense, e.g. we have no means to evaluate arbitrary ground atoms over M; note that M is model of F and also a Herbrand interpretation of all skolemized formulae over the signature of F.