Bridge and Bound: A Logic-Based Framework for Abstracting (Extended Report)

TL;DR

This paper introduces a logic-based framework for modeling abstraction processes, emphasizing the treatment of sufficient conditions alongside necessary ones, and exploring hierarchical and approximate abstractions within classical logic.

Contribution

It presents a novel, logic-centric approach to abstraction, extending traditional models to include sufficient conditions and layered abstractions, with analysis of computational complexity.

Findings

Defined approximate and exact abstractions

Extended framework to layered, hierarchical abstractions

Discussed computational complexity of reasoning tasks

Abstract

At its core, abstraction is the process of generalizing from specific instances to broader concepts or models, with the primary objective of reducing complexity while preserving properties essential to the intended purpose. It is a~fundamental, often implicit, principle that structures the understanding, communication, and development of both scientific knowledge and everyday beliefs. Studies on abstraction have evolved from its origins in Ancient Greek philosophy through methodological approaches in psychological and philosophical theories to modern computational frameworks. This paper presents a novel logic-based framework for modeling abstraction processes in which all components are expressed within logic. The framework extends beyond the traditional focus on the entailment of necessary conditions by making sufficient conditions first-class citizens as well. We define approximate abstractions, study their tightest and exact forms, and extend the approach to layered abstractions, enabling hierarchical simplification of complex systems and models. The computational complexity of the related reasoning tasks is also discussed. For clarity, our framework is developed within classical logic, chosen for its simplicity, expressiveness, and computational friendliness.