Linde Vanbesien

Đorđe Marković, Simon Vandevelde, Linde Vanbesien et al.

Substantial efforts have been made in developing various Decision Modeling formalisms, both from industry and academia. A challenging problem is that of expressing decision knowledge in the context of incomplete knowledge. In such contexts, decisions depend on what is known or not known. We argue that none of the existing formalisms for modeling decisions are capable of correctly capturing the epistemic nature of such decisions, inevitably causing issues in situations of uncertainty. This paper presents a new language for modeling decisions with incomplete knowledge. It combines three principles: stratification, autoepistemic logic, and definitions. A knowledge base in this language is a hierarchy of epistemic theories, where each component theory may epistemically reason on the knowledge in lower theories, and decisions are made using definitions with epistemic conditions.

Linde Vanbesien, Bart Bogaerts, Marc Denecker

Approximation Fixpoint Theory (AFT) is a powerful theory covering various semantics of non-monotonic reasoning formalisms in knowledge representation such as Logic Programming and Answer Set Programming. Many semantics of such non-monotonic formalisms can be characterized as suitable fixpoints of a non-monotonic operator on a suitable lattice. Instead of working on the original lattice, AFT operates on intervals in such lattice to approximate or construct the fixpoints of interest. While AFT has been applied successfully across a broad range of non-monotonic reasoning formalisms, it is confronted by its limitations in other, relatively simple, examples. In this paper, we overcome those limitations by extending consistent AFT to deal with approximations that are more refined than intervals. Therefore, we introduce a more general notion of approximation spaces, showcase the improved expressiveness and investigate relations between different approximation spaces.

Linde Vanbesien, Maurice Bruynooghe, Marc Denecker

Aggregates provide a concise way to express complex knowledge. The problem of selecting an appropriate formalisation of aggregates for answer set programming (ASP) remains unsettled. This paper revisits it from the viewpoint of Approximation Fixpoint Theory (AFT). We introduce an AFT formalisation equivalent with the Gelfond-Lifschitz reduct for basic ASP programs and we extend it to handle aggregates. We analyse how existing approaches relate to our framework. We hope this work sheds some new light on the issue of a proper formalisation of aggregates. This paper is under consideration for acceptance in TPLP.