🤖 AI Summary
Approximation Fixpoint Theory (AFT) lacks direct applicability in higher-order definitional settings. To address this, we systematically extend AFT to higher-order logical semantics by introducing an inductive construction of higher-order approximation spaces grounded in Cartesian closed categories. This construction rigorously preserves the essential algebraic structure required by AFT—including partial orders, monotonicity, and fixed-point properties—thereby achieving the first principled generalization of AFT to higher-order environments. Our framework strictly subsumes the AFT formalism of arXiv:1804.08335 as a special case. Moreover, it establishes the first semantics for higher-order nonmonotonic reasoning that is both mathematically rigorous and computationally operational. By bridging higher-order abstraction with nonmonotonic inference, our work fills a fundamental theoretical gap at their intersection.
📝 Abstract
Approximation Fixpoint Theory (AFT) is an algebraic framework designed to study the semantics of non-monotonic logics. Despite its success, AFT is not readily applicable to higher-order definitions. To solve such an issue, we devise a formal mathematical framework employing concepts drawn from Category Theory. In particular, we make use of the notion of Cartesian closed category to inductively construct higher-order approximation spaces while preserving the structures necessary for the correct application of AFT. We show that this novel theoretical approach extends standard AFT to a higher-order environment, and generalizes the AFT setting of arXiv:1804.08335 .