##
###
Probabilistic description logic programs

Thomas Lukasiewicz

2007
*
International Journal of Approximate Reasoning
*

Towards sophisticated representation and reasoning techniques that allow for probabilistic uncertainty in the Rules, Logic, and Proof layers of the Semantic Web, we present probabilistic description logic programs (or pdl-programs), which are a combination of description logic programs (or dl-programs) under the answer set semantics and the well-founded semantics with Poole's independent choice logic. We show that query processing in such pdl-programs can be reduced to computing all answer sets
## more »

... of dl-programs and solving linear optimization problems, and to computing the well-founded model of dl-programs, respectively. Moreover, we show that the answer set semantics of pdl-programs is a refinement of the well-founded semantics of pdl-programs. Furthermore, we also present an algorithm for query processing in the special case of stratified pdl-programs, which is based on a reduction to computing the canonical model of stratified dl-programs. Introduction The Semantic Web [5, 14] aims at an extension of the current World Wide Web by standards and technologies that help machines to understand the information on the Web so that they can support richer discovery, data integration, navigation, and automation of tasks. The main ideas behind it are to add a machine-readable meaning to Web pages, to use ontologies for a precise definition of shared terms in Web resources, to use KR technology for automated reasoning from Web resources, and to apply cooperative agent technology for processing the information of the Web. The Semantic Web consists of several hierarchical layers, where the Ontology layer, in form of the OWL Web Ontology Language [46, 23] (recommended by the W3C), is currently the highest layer of sufficient maturity. OWL consists of three increasingly expressive sublanguages, namely OWL Lite, OWL DL, and OWL Full. OWL Lite and OWL DL are essentially very expressive description logics with an RDF syntax [23] . As shown in [21] , ontology entailment in OWL Lite (resp., OWL DL) reduces to knowledge base (un)satisfiability in the description logic SHIF(D) (resp., SHOIN (D)). On top of the Ontology layer, the Rules, Logic, and Proof layers of the Semantic Web will be developed next, which should offer sophisticated representation and reasoning capabilities. As a first effort in this direction, RuleML (Rule Markup Language) [6] is an XML-based markup language for rules and rule-based systems, whereas the OWL Rules Language [22] is a first proposal for extending OWL by Horn clause rules. A key requirement of the layered architecture of the Semantic Web is to integrate the Rules and the Ontology layer. In particular, it is crucial to allow for building rules on top of ontologies, that is, for rulebased systems that use vocabulary from ontology knowledge bases. Another type of combination is to build ontologies on top of rules, which means that ontological definitions are supplemented by rules or imported from rules. Towards this goal, the works [12, 13] have proposed description logic programs (or simply dlprograms), which are of the form KB = (L, P ), where L is a knowledge base in a description logic and P is a finite set of description logic rules (or simply dl-rules). Such dl-rules are similar to usual rules in logic programs with negation as failure, but may also contain queries to L in their bodies, which are given by special atoms (on which possibly default negation may apply). Another important feature of dl-rules is that queries to L also allow for specifying an input from P , and thus for a flow of information from P to L, besides the flow of information from L to P , given by any query to L. Hence, description logic programs allow for building rules on top of ontologies, but also (to some extent) building ontologies on top of rules. In this way, additional knowledge (gained in the program) can be supplied to L before querying. The semantics of dl-programs was defined in [12] and [13] as an extension of the answer set semantics by Gelfond and Lifschitz [17] and the well-founded semantics by Van Gelder, Ross, and Schlipf [45], respectively, which are the two most widely used semantics for nonmonotonic logic programs. The description logic knowledge bases in dl-programs are specified in the well-known description logics SHIF(D) and SHOIN (D). In this paper, we continue this line of research. Towards sophisticated representation and reasoning techniques that also allow for modeling probabilistic uncertainty in the Rules, Logic, and Proof layers of the Semantic Web, we present probabilistic description logic programs (or simply pdl-programs), which generalize dl-programs under the answer set and the well-founded semantics by probabilistic uncertainty. This probabilistic generalization of dl-programs is developed as a combination of dl-programs with Poole's independent choice logic (ICL) [35] . It is important to point out that Poole's ICL is a powerful representation and reasoning formalism for single-and also multi-agent systems, which combines logic and probability, and which can represent a number of important uncertainty formalisms, in particular, influence diagrams, Bayesian networks, Markov decision processes, and normal form games [35] . Furthermore, Poole's ICL also allows for natural notions Proof (sketch). Let α ∈ {ψ, ψ ∧ φ}. It is sufficient to show that Pr wf KB (α) is equal to Pr (α) for all answer set models Pr of KB . Observe that Pr wf KB (α) is the sum of all µ(B) such that (i) B is a total choice of C, (ii) µ(B) > 0, and (iii) I B (α) = true, where I B denotes the well-founded model of (L, P ∪{p ← | p ∈ B}). By induction on the structure of classical formulas, it is not difficult to see that I B (α) = true iff I |= α for all answer sets I of (L, P ∪ {p ← | p ∈ B}). This already shows that Pr wf KB (α) is equal to Pr (α) for all answer set models Pr of KB . 2 Query Processing in Stratified PDL-Programs The canonical model of an ordinary positive (resp., stratified) normal program P has a fixpoint characterization in terms of an immediate consequence operator T P , which generalizes to positive (resp., stratified)

doi:10.1016/j.ijar.2006.06.012
fatcat:uwt4uvjzszhula5jh5tzmewz4u