Sensible Intersection Type Theories

2026-03-04

AI Summary

This work investigates the construction of filter lambda-models endowed with the sensibility property—namely, that all unsolvable terms are interpreted as the least element in the model. By viewing intersection type theories as specific meet-semilattices and leveraging morphisms in their dual categories together with the Tait–Girard computability method, the paper establishes, for the first time, a connection between sensibility and morphisms of meet-semilattices. The main contribution consists of two classes of intersection type theories that induce sensible filter models: the effective class not only supports concrete model constructions but also generalizes Mendler’s criterion to intersection types and head-normalizable terms. Nevertheless, a complete characterization of sensible filter models remains an open problem.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Relational Probabilistic 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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Finitary/static semantics in the form of intersection type assignments have become a paradigm for analysing the fine structure of all sorts of lambda-models. The key step is the construction of a filter model isomorphic to a given lambda-model. A property of great interest of filter lambda-models is sensibility, i.e. the interpretation of all unsolvable terms is the least element. The flexibility of intersection type assignments derives from their parametrisation on intersection type theories. We construe intersection type theories as special meet-semilattices and show that appropriate morphisms, in the opposite category of meet-semilattices, preserve sensibility of the induced lambda-models. Interestingly the set of saturated sets together with the set of lambda-terms is such a meet-semilattice, thus showing that arguments based on Tait-Girards's computability amount to the construction of a morphism. We characterise two classes of intersection type theories which induce sensible filter models. The first is non-effective while the second is effective and it amounts to the generalisation of Mendler's criterion to intersection types and head normalising terms. The complete characterisation of sensible filter models however still escapes.

Problem

• intersection type theories
• sensible lambda-models
• filter models
• unsolvable terms
• meet-semilattices

Innovation

• intersection type theories
• sensible lambda-models
• filter models
• meet-semilattices
• computability