Interaction Improvement

2026-01-04

arXiv.org

AI Summary

Although relational semantics can model resource-sensitive λ-calculi, the induced preorder and equational theory offer only qualitative characterizations, lacking the ability to quantify the number of interactions between terms and contexts. This work introduces, for the first time, the inspector calculus into the relational semantics framework, proposing a quantitative and context-sensitive interpretation mechanism that refines program behavior by bounding interaction counts. By imposing such quantitative constraints, the approach overcomes the limitations of traditional qualitative models and demonstrates that relational semantics can effectively refine the contextual preorder, thereby enhancing the discriminative power over λ-term behaviors.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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 heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The relational semantics of linear logic is a powerful framework for defining resource-aware models of the $\lambda$-calculus. However, its quantitative aspects are not reflected in the preorders and equational theories induced by these models. Indeed, they can be characterized in terms of (in)equalities between B\"ohm trees up to extensionality, which are qualitative in nature. We employ the recently introduced checkers calculus to provide a quantitative and contextual interpretation of the preorder associated to the relational semantics. This way, we show that the relational semantics refines the contextual preorder constraining the number of interactions between the related terms and the context.

Problem

• relational semantics
• quantitative interpretation
• contextual preorder
• linear logic
• lambda calculus

Innovation

• relational semantics
• checkers calculus
• quantitative interpretation
• contextual preorder
• resource-aware