An Unconventional View on Beta-Reduction in Namefree Lambda-Calculus

2026-03-04

AI Summary

Traditional beta-reduction in lambda calculus centers on the global structure of terms, making it ill-suited to capture the branching characteristics inherent in tree representations. This work models lambda terms as labeled planar trees and introduces a branch-oriented, expansive beta-reduction mechanism: after reduction, the resulting term tree always contains the original term tree as a subtree, thereby overcoming the limitations of conventional contractive reduction. By employing de Bruijn notation and symbolically annotated abstraction and application nodes, the approach not only reconstructs several classical reduction strategies but also naturally yields novel reduction forms that satisfy the tree-inclusion property. This provides a fresh, tree-structure-based perspective on lambda calculus.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Terms in the lambda-calculus can be represented as planar trees decorated with symbols for abstraction and application, and having variables as leaves. In this paper, we concentrate on the branches of such trees, rather than on the trees themselves. We reformulate several well-known notions of beta-reduction in this view. In a natural manner, this reconsideration eventually leads to a new form of beta-reduction, being expanding in the sense that the reduction of term t1 to term t2 entails that the tree of t1 is a subtree of the tree of t2.

Problem

• beta-reduction
• namefree lambda-calculus
• planar trees
• subtree expansion

Innovation

• beta-reduction
• nameless lambda-calculus
• planar trees
• expanding reduction
• branch-based representation