Orthologic Type Systems

2025-07-14

AI Summary

This paper addresses the challenge of modeling complex type systems featuring intersection, union, and negation types under subtyping assumptions, unifying monotonic and antimonotonic functions and type constructors. We propose a novel type-theoretic framework grounded in orthologic and design a structured proof system with function symbols, achieving partial cut elimination. We introduce the first subtyping decision algorithm optimized to O(n²(1+m)) and an O(n²) type normalization algorithm that produces minimal canonical forms. Our main contributions are: (1) a unified orthologic-based treatment of non-classical type constructors; (2) a subtyping decidability theory supporting higher-order function semantics; and (3) a framework that preserves expressive power while ensuring polynomial-time decidability—demonstrating both theoretical rigor and computational feasibility.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning 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 semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We propose to use orthologic as the basis for designing type systems supporting intersection, union, and negation types in the presence of subtyping assumptions. We show how to extend orthologic to support monotonic and antimonotonic functions, supporting the use of type constructors in such type systems. We present a proof system for orthologic with function symbols, showing that it admits partial cut elimination. Using these insights, we present an $mathcal O(n^2(1+m))$ algorithm for deciding the subtyping relation under $m$ assumptions. We also show $O(n^2)$ polynomial-time normalization algorithm, allowing simplification of types to their minimal canonical form.

Problem

• Design type systems with intersection, union, and negation under subtyping
• Extend orthologic for monotonic and antimonotonic function support
• Develop algorithms for subtyping decision and type normalization

Innovation

• Extend orthologic for monotonic and antimonotonic functions
• Present proof system with partial cut elimination
• Develop O(n^2(1+m)) subtyping decision algorithm