Ohana trees and Taylor expansion for the $lambda$I-calculus. No variable gets left behind or forgotten!

2025-05-09

AI Summary

This paper addresses the semantic poverty of existing equational theories for the λI-calculus—particularly the degeneracy in traditional truth-valued models, where all non-normalizing terms are identified, erasing their computational distinctions. To resolve this, the authors introduce *Ohana trees*: a labeled variant of Böhm trees that fully preserves the information about free variables delayed to infinity in λI-terms, thereby overcoming the limitation of conventional models that collapse the semantics of non-normalizing terms. Key contributions include: (i) the first nontrivial denotational semantics for λI-calculus yielding a sound and non-degenerate equational theory; (ii) a proof of commutativity between Ohana trees and the Ehrhard–Regnier Taylor expansion, ensuring semantic compatibility and operational coherence; and (iii) verification that the theory is closed under abstraction and application, laying the groundwork for its extension to the full λ-calculus.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of 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
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Although the $lambda$I-calculus is a natural fragment of the $lambda$-calculus, obtained by forbidding the erasure, its equational theories did not receive much attention. The reason is that all proper denotational models studied in the literature equate all non-normalizable $lambda$I-terms, whence the associated theory is not very informative. The goal of this paper is to introduce a previously unknown theory of the $lambda$I-calculus, induced by a notion of evaluation trees that we call"Ohana trees". The Ohana tree of a $lambda$I-term is an annotated version of its B""ohm tree, remembering all free variables that are hidden within its meaningless subtrees, or pushed into infinity along its infinite branches. We develop the associated theories of program approximation: the first approach -- more classic -- is based on finite trees and continuity, the second adapts Ehrhard and Regnier's Taylor expansion. We then prove a Commutation Theorem stating that the normal form of the Taylor expansion of a $lambda$I-term coincides with the Taylor expansion of its Ohana tree. As a corollary, we obtain that the equality induced by Ohana trees is compatible with abstraction and application. We conclude by discussing the cases of L'evy-Longo and Berarducci trees, and generalizations to the full $lambda$-calculus.

Problem

• Introducing Ohana trees for λI-calculus equational theories
• Developing program approximation theories via finite trees and Taylor expansion
• Proving Commutation Theorem for Taylor expansions of λI-terms

Innovation

• Ohana trees track hidden free variables
• Taylor expansion for λI-calculus analysis
• Commutation Theorem links expansions and trees