Approximation Fixpoint Theory with Refined Approximation Spaces

TL;DR

This paper extends Approximation Fixpoint Theory by introducing refined approximation spaces, enhancing its expressiveness and ability to handle a broader range of non-monotonic reasoning semantics.

Contribution

It proposes a generalization of approximation spaces in AFT, overcoming previous limitations and enabling more precise approximations for non-monotonic semantics.

Findings

Enhanced expressiveness of AFT with refined approximation spaces

Better handling of simple non-monotonic reasoning examples

Established relations between different approximation spaces

Abstract

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.