Linearization via Rewriting (Long Version)

2025-03-06

AI Summary

This paper addresses the challenge of naturally representing strongly normalizing terms in the lambda calculus and linearizing their type derivations. We introduce Structured Resource Lambda-Calculus (SRλ), a purely finite, fully internalized rewriting framework that directly supports syntactic representation of strongly normalizing λ-terms and achieves automatic linearization via rewriting rules operating at the level of type derivations. Its core innovation is the first fully internalized, rewriting-based linearization process—requiring neither external mechanisms nor infinite structures. Theoretical contributions include: (i) a rigorous proof of strong normalization and confluence of SRλ; (ii) establishment of a bijective correspondence between strongly normalizing λ-terms and their type derivations; and (iii) provision of a new foundation for resource-sensitive type theory that simultaneously ensures expressive power and computational adequacy.

Technology Category

Knowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingMachine Learning: Structured LearningConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Search and Retrieval-Augmented AI: Web evaluation methodologies and metricsSemantics 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: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We introduce the structural resource lambda-calculus, a new formalism in which strongly normalizing terms of the lambda-calculus can naturally be represented, and at the same time any type derivation can be internally rewritten to its linearization. The calculus is shown to be normalizing and confluent. Noticeably, every strongly normalizable lambda-term can be represented by a type derivation. This is the first example of a system where the linearization process takes place internally, while remaining purely finitary and rewrite-based.

Problem

• Introduces structural resource lambda-calculus for representing strongly normalizing lambda-terms.
• Enables internal rewriting of type derivations to their linearization.
• Demonstrates normalization and confluence in the new calculus.

Innovation

• Introduces structural resource lambda-calculus for term representation
• Enables internal rewriting to linearize type derivations
• Ensures normalization and confluence in the calculus