We define a semantic framework to reason about compositional properties
of SLD-derivations and their abstractions (observables). The framework
allows us to address problems such as the relation between the (top-down) operational
semantics and the (bottom-up) denotational semantics, the existence
of a denotation for a set of definite clauses and their properties (compositionality
w.r.t. various syntactic operators, correctness and minimality). This leads
us to a flexible classification of the observables, where we can reason about
properties such as OR-compositionality and existence of abstract transition
system. Using abstract interpretation techniques to model abstraction allows
us to state very simple conditions on the observables which guarantee the
validity of several general theorems.