Davy Van Nieuwenborgh

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.

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.

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.

Open answer set programming for the semantic web

Abstract We extend answer set programming (ASP) with, possibly infinite, open domains. Since this leads to undecidable reasoning, we restrict the syntax of programs, while carefully guarding knowledge representation mechanisms such as negation as failure and inequalities. Reasoning with the resulting extended forest logic programs (EFoLPs) can be reduced to finite answer set programming, for which reasoners are available. We argue that extended forest logic programming is a useful tool for uniformly representing and reasoning with both ontological and rule-based knowledge, as they can capture a large fragment of the OWL DL ontology language equipped with DL-safe rules. Furthermore, EFoLPs enable nonmonotonic reasoning, a desirable feature in locally closed subareas of the Semantic Web.

Preferred answer sets for ordered logic programs

We extend answer set semantics to deal with inconsistent programs (containing classical negation), by finding a “best” answer set. Within the context of inconsistent programs, it is natural to have a partial order on rules, representing a preference for satisfying certain rules, possibly at the cost of violating less important ones. We show that such a rule order induces a natural order on extended answer sets, the minimal elements of which we call preferred answer sets. We characterize the expressiveness of the resulting semantics and show that it can simulate negation as failure, disjunction and some other formalisms such as logic programs with ordered disjunction. The approach is shown to be useful in several application areas, e.g. repairing database, where minimal repairs correspond to preferred answer sets.

Conceptual logic programs

Open answer set programming (OASP) solves the lack of modularity in closed world answer set programming by allowing for the grounding of logic programs with an arbitrary non-empty countable superset of the program’s constants. However, OASP is, in general, undecidable: the undecidable domino problem can be reduced to it. In order to regain decidability, we restrict the shape of logic programs, yielding conceptual logic programs (CoLPs). CoLPs are logic programs with unary and binary predicates (possibly inverted) where rules have a tree shape. Decidability of satisfiability checking of predicates w.r.t. CoLPs is shown by a reduction to non-emptiness checking of two-way alternating tree automata. We illustrate the expressiveness of CoLPs by simulating the description logic

Conditional planning with external functions

\mathcal{SHIQ}

On Programs with Linearly Ordered Multiple Preferences

. CoLPs thus integrate, in one unifying framework, the best of both the logic programming paradigm (a flexible rule-based representation and nonmonotonicity by means of negation as failure) and the description logics paradigm (decidable open domain reasoning).