Clausal Deductive Databases and a General Framework for Semantics in Disjunctive Deductive Databases

Abstract

In this paper we will investigate the novel concept of clausal deductive databases (cd-databases), which are special normal deductive databases — i.e. deductive databases which may contain default negation in rule bodies — over a meta-language $\mathcal{L}^{{\text{cd}}}$ with a fixed set of predicate symbols, namely dis, con, and some built-in predicate symbols. The arguments of the literals in $\mathcal{L}^{{\text{cd}}}$ are given by disjunctive and conjunctive clauses of a basic first-order language $\mathcal{L}$ (which are considered as terms in $\mathcal{L}^{{\text{cd}}}$). On the other hand, disjunctive deductive databases (dd-databases) extend normal deductive databases by allowing for disjunctions (rather than just single atoms or literals) in rule heads — next to default negation in rule bodies.

We will present an embedding of dd-databases into cd-databases: a dd-database $\mathcal{D}$ is transformed into a cd-database $\mathcal{D}^{{\text{cd}}}$, which talks about the clauses of $\mathcal{D}$ — rather than just the literals. Thus, cd-databases provide a flexible framework for declaratively specifying the semantics of dd-databases. We can fix a standard control strategy, e.g. stable model or well-founded semantics, and vary the logical description $\mathcal{D}^{{\text{cd}}}$ for specifying different semantics. The transformed database $\mathcal{D}^{{\text{cd}}}$ usually consists of a part $\mathcal{D}^ \otimes$ which naturally expresses the rules of $\mathcal{D}$, and two generic parts which are independent of $\mathcal{D}{\text{: }}\mathcal{D}^{logic}$ specifies logical inference rules like resolution and subsumption, and $\mathcal{D}^{cwa}$ specifies non-monotonic inference rules like closed-world-assumptions.

In particular we will show that the hyperresolution consequence operator $\mathcal{T}_\mathcal{D}^s$ for dd-databases without default negation can be expressed as a standard consequence operator $\mathcal{T}_{\mathcal{D}^{cd} }$, for a suitable transformed database $\mathcal{D}^{cd}$, where $\mathcal{D}^{logic} = \mathcal{D}^{cwa} = 0/$. For dd-databases with default negation we can show that the semantics of stable models can be characterized by adding suitable sets $\mathcal{D}^{logic}$ and $\mathcal{D}^{cwa}$. Moreover, we will define a new semantics for dd-databases which we will call stable state semantics; it is based on Herbrand states rather than Herbrand interpretations.