Approximation Fixpoint Theory with Refined Approximation Spaces
This addresses a theoretical bottleneck in knowledge representation for AI, though it appears incremental as it builds on existing AFT.
The paper tackles limitations in Approximation Fixpoint Theory (AFT) for non-monotonic reasoning by extending it with refined approximation spaces, resulting in improved expressiveness and the ability to handle previously problematic examples.
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.