Comparing Semantic Frameworks for Dependently-Sorted Algebraic Theories

2024-12-27

Asian Symposium on Programming Languages and Systems

AI Summary

Existing work lacks a systematic characterization and comparative analysis of the relationships among various categorical semantic models—such as fibrations, sheaves, and extensions of homotopy type theory—for dependently sorted algebraic theories. Method: This paper establishes, for the first time, a semantic taxonomy for dependently sorted algebraic theories, introducing cross-framework expressivity hierarchies and structure-preserving translation mechanisms. Leveraging category theory, semantics of dependent types, and fibration-based methods, we rigorously determine the relative expressivity ordering among three mainstream semantic frameworks. Contribution/Results: We precisely characterize equivalence boundaries and prove fundamental intranslatability results between certain models. Our framework unifies the treatment of type dependency and algebraic structure, yielding a comparable and transferable semantic foundation for higher-order formal verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Relational Probabilistic Models

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 semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Problem

• Algebraic Theory
• Dependent Sorting
• Category Models

Innovation

• Unified Framework
• Algebraic Theory Models
• Comparative Understanding