Dirk Vermeir

Dialectic semantics for argumentation frameworks

We provide a formalism for the study of dialogues, where a dialogue is a two-person game, initiated by the proponent who defends a proposed thesis. We examine several different winning criteria and several different dialogue types, where a dialogue type is determined by a set of positions, an attack relation between positions and a legal-move function. We examine two proof theories, where a proof theory is determined by a dialogue type and a winning criterion. For each of the proof theories we supply a corresponding declarative semantics.

An introduction to fuzzy answer set programming

In this paper we show how the concepts of answer set programming and fuzzy logic can be successfully combined into the single framework of fuzzy answer set programming (FASP). The framework offers the best of both worlds: from the answer set semantics, it inherits the truly declarative non-monotonic reasoning capabilities while, on the other hand, the notions from fuzzy logic in the framework allow it to step away from the sharp principles used in classical logic, e.g., that something is either completely true or completely false. As fuzzy logic gives the user great flexibility regarding the choice for the interpretation of the notions of negation, conjunction, disjunction and implication, the FASP framework is highly configurable and can, e.g., be tailored to any specific area of application. Finally, the presented framework turns out to be a proper extension of classical answer set programming, as we show, in contrast to other proposals in the literature, that there are only minor restrictions one has to demand on the fuzzy operations used, in order to be able to retrieve the classical semantics using FASP.

Nonmonotonic ontological and rule-based reasoning with extended conceptual logic programs

We present extended conceptual logic programs (ECLPs), for which reasoning is decidable and, moreover, can be reduced to finite answer set programming. ECLPs are useful to reason with both ontological and rule-based knowledge, which is illustrated by simulating reasoning in an expressive description logic (DL) equipped with DL-safe rules. Furthermore, ECLPs are more expressive in the sense that they enable nonmonotonic reasoning, a desirable feature in locally closed subareas of the Semantic Web.

Extending logic programming

An extension of logic programming, called “ordered logic programming”, which includes some abstractions of the object-oriented paradigm, is presented. An ordered program consists of a number of modules (objects), where each module is composed by a number of rules possibly with negated head predicates. A sort of “isa” hierarchy can be defined among the modules in order to allow for rule inheritance. Therefore, every module sees its own rules as local rules and the rules of the other modules to which it is connected by the “isa” hierarchy as global rules. In this way, as local rules may hide global rules, it is possible to deal with default properties and exceptions. This new approach represents a novel attempt to combine the logic paradigm with the object-oriented one in knowledge base systems. Moreover, this approach provides a new ground for explaining some recent proposals of semantics for classical logic programs with negation in the rule bodies and gives an interesting semantics to logic programs with negated rule heads.

Choice Logic Programs and Nash Equilibria in Strategic Games

We define choice logic programs as negation-free datalog programs that allow rules to have exclusive-only disjunctions in the head. We show that choice programs are equivalent to semi-negative datalog programs, at least as far as stable models are concerned. We also discuss an application where strategic games can be naturally formulated as choice programs; it turns out that the stable models of such programs capture exactly the set of Nash equilibria.

A Defeasible Ontology Language

We extend the description logic SHOQ(D) with a preference order on the axioms. With this strict partial order certain axioms can be overruled, if defeated with more preferred ones. Furthermore, we impose a preferred model semantics, thus effectively introducing nonmonotonicity into SHOQ(D). Since a description logic can be viewed as an ontology language, or a proper translation of one, we obtain a defeasible ontology language. Finally, we argue that such a defeasible language may be usefully applied for learning and integrating ontologies.

Open answer set programming with guarded programs

Open answer set programming (OASP) is an extension of answer set programming where one may ground a program with an arbitrary superset of the programs constants. We define a fixed-point logic (FPL) extension of Clarks completion such that open answer sets correspond to models of FPL formulas and identify a syntactic subclass of programs, called (loosely) guarded programs. Whereas reasoning with general programs in OASP is undecidable, the FPL translation of (loosely) guarded programs falls in the decidable (loosely) guarded fixed-point logic (μ(L)GF). Moreover, we reduce normal closed ASP to loosely guarded OASP, enabling, for the first time, a characterization of an answer set semantics by μLGF formulas. We further extend the open answer set semantics for programs with generalized literals. Such generalized programs (gPs) have interesting properties, for example, the ability to express infinity axioms. We restrict the syntax of gPs such that both rules and generalized literals are guarded. Via a translation to guarded fixed-point logic, we deduce 2-EXPTIME-completeness of satisfiability checking in such guarded gPs (GgPs). Bound GgPs are restricted GgPs with EXPTIME-complete satisfiability checking, but still sufficiently expressive to optimally simulate computation tree logic (CTL). We translate Datalog lite programs to GgPs, establishing equivalence of GgPs under an open answer set semantics, alternation-free μGF, and Datalog LITE.

On ETOL systems of finite index

The classical concept of finite index is investigated within the framework of ETOL systems.

Guarded open answer set programming

Open answer set programming (OASP) is an extension of answer set programming where one may ground a program with an arbitrary superset of the programs constants. We define a fixed point logic (FPL) extension of Clarks completion such that open answer sets correspond to models of FPL formulas and identify a syntactic subclass of programs, called (loosely) guarded programs. Whereas reasoning with general programs in OASP is undecidable, the FPL translation of (loosely) guarded programs falls in the decidable (loosely) guarded fixed point logic (μ(L)GF). Moreover, we reduce normal closed ASP to loosely guarded OASP, enabling a characterization of an answer set semantics by μLGF formulas. Finally, we relate guarded OASP to Datalog LITE, thus linking an answer set semantics to a semantics based on fixed point models of extended stratified Datalog programs. From this correspondence, we deduce 2-EXPTIME-completeness of satisfiability checking w.r.t. (loosely) guarded programs.

Semantic Web reasoning with Conceptual Logic Programs

We extend Answer Set Programming with, possibly infinite, open domains. Since this leads, in general, to undecidable reasoning, we restrict the syntax of programs, while carefully guarding useful knowledge representation mechanisms such as negation as failure and inequalities. Reasoning with the resulting Conceptual Logic Programs can be reduced to finite, normal Answer Set Programming, for which reasoners are available.