Description Logic Programs (dl-programs) proposed by constitute an elegant yet powerful formalism for the integration of answer set programming with description logics, for the Semantic Web. In this paper, we generalize the notions of completion and loop formulas of logic programs to description logic programs and show that the answer sets of a dl-program can be precisely captured by the models of its completion and loop formulas. Furthermore, we propose a new, alternative semantics for dl-programs, called the canonical answer set semantics, which is defined by the models of completion that satisfy what are called canonical loop formulas. A desirable property of canonical answer sets is that they are free of circular justifications. Some properties of canonical answer sets are also explored.
However, the relationships among these different approaches are currently not well understood. For example, although we know how to translate a dl-program without the nonmonotonic operator @math to an MKNF theory while preserving the strong answer set semantics @cite_21 , when @math is involved, no such a translation is known. Similarly, although a variant of Quantified Equilibrium Logic (QEL) captures the existing hybrid approaches, as shown by @cite_12 , it is not clear how one would apply the loop formulas for logic programs with arbitrary sentences @cite_0 to dl-programs, since, to the best of our knowledge, there is no syntactic, semantics-preserving translation from dl-programs to logic programs with arbitrary sentences or to QEL.
@cite_0 @cite_21 @cite_12
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Integrating description logics (DL) and logic programming (LP) would produce a very powerful and useful formalism. However, DLs and LP are based on quite different principles, so achieving a seamless integration is not trivial. In this paper, we introduce hybrid MKNF knowledge bases that faithfully integrate DLs with LP using the logic of Minimal Knowledge and Negation as Failure (MKNF) [Lifschitz, 1991]. We also give reasoning algorithms and tight data complexity bounds for several interesting fragments of our logic. We propose a new translation from normal logic programs with constraints under the answer set semantics to propositional logic. Given a normal logic program, we show that by adding, for each loop in the program, a corresponding loop formula to the program's completion, we obtain a one-to-one correspondence between the answer sets of the program and the models of the resulting propositional theory. In the worst case, there may be an exponential number of loops in a logic program. To address this problem, we propose an approach that adds loop formulas a few at a time, selectively. Based on these results, we implement a system called ASSAT(X), depending on the SAT solver X used, for computing one answer set of a normal logic program with constraints. We test the system on a variety of benchmarks including the graph coloring, the blocks world planning, and Hamiltonian Circuit domains. Our experimental results show that in these domains, for the task of generating one answer set of a normal logic program, our system has a clear edge over the state-of-art answer set programming systems Smodels and DLV. Recently Ferraris, Lee and Lifschitz proposed a new definition of stable models that does not refer to grounding, which applies to the syntax of arbitrary first-order sentences. We show its relation to the idea of loop formulas with variables by Chen, Lin, Wang and Zhang, and generalize their loop formulas to disjunctive programs and to arbitrary first-order sentences. We also extend the syntax of logic programs to allow explicit quantifiers, and define its semantics as a subclass of the new language of stable models by Such programs inherit from the general language the ability to handle nonmonotonic reasoning under the stable model semantics even in the absence of the unique name and the domain closure assumptions, while yielding more succinct loop formulas than the general language due to the restricted syntax. We also show certain syntactic conditions under which query answering for an extended program can be reduced to entailment checking in first-order logic, providing a way to apply first-order theorem provers to reasoning about non-Herbrand stable models. Let P be an arbitrary theory and let X be any given logic. Let M be a set of atoms. We say that M is a X-stable model of P if M is a classical model of P and P∪¬M proves in logic X all atoms in M, this is denoted by P∪¬M ⊩xM. We prove that being an X-stable model is an invariant property for disjunctive programmes under a large class of logics. Two kinds of logics are mainly considered: paraconsistent logics and normal modal logics. For modal logics we use a translation proposed by Gelfond that replaces ¬a with ¬□a. As a consequence we prove that several semantics (recently introduced) for non-monotonic reasoning are equivalent for disjunctive programmes. In addition, we show that such semantics can be characterized by a fixed-point operator in terms of classical logic. We also present a simple translation of a disjunctive programme D into a normal programme N, such that the PStable model semantics of N corresponds to the stable semantics of D over the common language. We present the formal proof of this statement.