An algebraic framework for the definition of compositional semantics of normal logic programs

The aim of our work is the definition of compositional semantics for modular units over the class of normal logic programs. In this sense, we propose a declarative semantics for normal logic programs in terms of model classes that is monotonic in the sense that $\text{Mod}\text{(P∪P′) ⊆}\text{Mod}\text{(P)}$, for any programs P and P′ and we show that in the model class associated to every program there is a least model that can be seen as the semantics of the program, which may be built upwards as the least fix point of a continuous immediate consequence operator. In addition, it is proved that this least model is “typical” for the class of models of Clark-Kunen's completion of the program. This means that our semantics is equivalent to Clark-Kunen's completion. Moreover, following the approach defined in a previous paper, it is shown that our semantics constitutes a “specification frame ” equipped with the adequate categorical constructions needed to define compositional and fully abstract (categorical) semantics for a number of program units. In particular, we provide a categorical semantics of arbitrary normal logic program fragments which is compositional and fully abstract with respect to the (standard) union.