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

2025-10-30

AI Summary

Real-world modeling demands abstraction that simultaneously simplifies representations and preserves essential properties under incomplete information—a longstanding challenge in formal methods. Method: This paper introduces a novel logical abstraction framework grounded in both necessity and sufficiency conditions—departing from conventional necessity-only approaches. It formally defines approximate abstractions and their tightest forms, enabling multi-level, composable hierarchical abstraction structures. Built upon classical logic, the framework unifies abstraction mappings, logical reasoning, and computational complexity analysis within a single formal system. Results: The framework significantly improves abstraction fidelity while providing tight complexity characterizations for core reasoning tasks. It establishes a unified theoretical foundation for both exact and approximate abstraction, advancing the state of the art in abstraction-based verification and synthesis.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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 computational frameworks. Formally, abstraction can be understood as the transformation of a source representation into an abstract representation that discards certain details while retaining desirable features. In real-world modeling and reasoning, abstraction is crucial, particularly when managing imperfect or incomplete information that calls for approximate representations. This paper introduces a novel logic-based framework for modeling abstraction processes that goes beyond the traditional entailment of necessary conditions to encompass sufficient conditions 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.

Problem

• Modeling abstraction processes using a logic-based framework
• Extending abstraction beyond necessary to sufficient conditions
• Studying exact and layered abstractions for hierarchical simplification

Innovation

• Logic-based framework for modeling abstraction processes
• Approximate abstractions with tightest and exact forms
• Layered abstractions enabling hierarchical system simplification