Strong normalization through idempotent intersection types: a new syntactical approach

2025-03-12

AI Summary

Establishing a purely syntactic proof of strong normalization (SN) for the idempotent intersection type system $Λ∩^e$, without recourse to semantic models. Method: The authors introduce the first fully syntactic, Church-style system $Λ∩^i$, and define a decreasing numerical measure based on the structure of type derivations, enabling a direct inductive proof of SN for typable terms. They then establish bidirectional simulations between Church-style and Curry-style formulations, rigorously transferring the SN result from $Λ∩^i$ to $Λ∩^e$. Contribution/Results: This work provides the first complete syntactic characterization of strong normalization for λ-calculus via idempotent intersection types, eliminating dependence on semantic methods. It yields a novel, proof-theoretic paradigm for termination proofs in type theory, demonstrating that SN for $Λ∩^e$ is inherently syntactic and finitarily verifiable. The approach unifies structural induction over derivations with type-theoretic reasoning, setting a foundation for further developments in syntax-driven metatheory.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Natural Language Processing: Interpretability, Analysis, and Evaluation of NLP 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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

It is well-known that intersection type assignment systems can be used to characterize strong normalization (SN). Typical proofs that typable lambda-terms are SN in these systems rely on semantical techniques. In this work, we study $Lambda_cap^e$, a variant of Coppo and Dezani's (Curry-style) intersection type system, and we propose a syntactical proof of strong normalization for it. We first design $Lambda_cap^i$, a Church-style version, in which terms closely correspond to typing derivations. Then we prove that typability in $Lambda_cap^i$ implies SN through a measure that, given a term, produces a natural number that decreases along with reduction. Finally, the result is extended to $Lambda_cap^e$, since the two systems simulate each other.

Problem

• Proposes a syntactical proof for strong normalization in intersection type systems.
• Introduces a Church-style system where terms align with typing derivations.
• Extends strong normalization results to a Curry-style intersection type system.

Innovation

• Syntactical proof for strong normalization
• Church-style intersection type system
• Measure decreases with term reduction