A Category-Theoretic Perspective on Approximation Fixpoint Theory

TL;DR

This work presents a program to lift Approximation Fixpoint Theory (AFT) from non-monotonic logics into a general algebraic framework for constructive knowledge, using an approximating operator $A: L^2 \to L^2$ monotone in $\le_p$ over a lattice $L$. It defines three research strands—general approximation spaces, recursively defined domains and higher-order semantics, and explanations for AFT (WP1–WP3)—to broaden AFT beyond bilattices and simple operators with a new, category-aware foundation. The approach leverages category and domain theory to unify non-monotone construction processes across domains such as databases, argumentation, and logic programming, enabling modularity, explainability, and higher-order reasoning. Early results include higher-order stable semantics for logic programming and a generalized notion of approximation spaces, with ongoing work on wADFs and justification-based explanations.

Abstract

Approximation Fixpoint Theory (AFT) was founded in the early 2000s by Denecker, Marek, and Truszczyński as an abstract algebraic framework to study the semantics of non-monotonic logics. Since its early successes, the potential of AFT as a unifying semantic framework has become widely recognised, and the interest in AFT has gradually increased, with applications now ranging from foundations of database theory to abstract argumentation. The non-monotonic constructive processes that occur in many more areas of computer science, together with their associated semantic structures, can be successfully studied using AFT, which greatly simplifies their characterizations. The goal of my research is to take a step towards the lifting of AFT into a more general framework for constructive knowledge.