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Abstract:

In the field of knowledge representation and reasoning, many different logics are developed. Often, these logics exhibit striking similarities, either because they emerged from related ideas, or because they use similar underlying fundamental principles. Approximation fixpoint theory (AFT) is an abstract algebraical unifying framework that aims at exposing these principles by formalising them in lattice theory. It has been successfully applied to unify all common semantics of logic programs, autoepistemic logic, default logic, and more recently Dung's argumentation frameworks and abstract dialectical frameworks. In this dissertation, we extend approximation fixpoint theory to expose more underlying principles common to the aforementioned logics. In these domains, researchers have made use of a similar intuition: that facts (or models) can be derived from the ground up. They typically phrase this intuition by saying, e.g., that the facts should be grounded, or that they should not be unfounded, or that they should be supported by cycle-free arguments. In different domains, semantics that allow ungrounded models have received a lot of criticism. In logic programming for example, this was the case for Clark's completion semantics, which was later improved by perfect model semantics, stable semantics and well-founded semantics. In autoepistemic logic, a similar evolution happened: Moore's expansion semantics turned out to allow self-supporting models; this resulted in the development of many different semantics in attempts to get rid of this erroneous behaviour. In the first part of this dissertation, we formalise groundedness in approximation fixpoint theory. We study how groundedness relates to other concepts and fixpoints studied in AFT. We apply our abstract theory to the aforementioned domains: we show that our notion of groundedness indeed captures the intuitions that existed in these domains and study complexity of reasoning with grounded models. We study which existing semantics are grounded and which are not. For example, for logic programming, we find that Clark's completion semantics (indeed) is not grounded, while stable and well-founded semantics are grounded. We show that the well-founded model is not just any grounded model: it is the least precise partial grounded model. In the second part of this thesis we define a class of autoepistemic theories for which it is informally clear how to construct the intended model. Unfortunately, existing constructive semantics for autoepistemic logic, in particular, the well-founded semantics, fail to identify this model. In order to overcome this limitation, we propose, algebraically, a new constructive semantics based on the notion of groundedness. Our new construction refines the well-founded model construction and succeeds in identifying the intended model for the class of motivating examples. Furthermore, we show that for this class of examples, our novel construction constructs the unique grounded fixpoint. Summarised, in this dissertation, we continue the work on approximation fixpoint theory by identifying novel concepts occurring in all of the application domains and by refining existing semantics to better capture the intended meaning of a class of theories.