Refining Properties of Filter Models: Sensibility, Approximability and Reducibility

TL;DR

This paper investigates the relationship between sensibility and approximability in filter models of untyped lambda calculus, revealing that all known sensible models are also approximable and analyzing the limitations of current proof techniques.

Contribution

It demonstrates that sensibility implies approximability for filter models via an extension of lambda calculus with D-tests, highlighting limitations of existing proof methods.

Findings

All sensible filter models are approximable.

Traditional sensibility proofs extend to lambda calculus with D-tests.

The approach reveals weaknesses in current proof techniques.

Abstract

In this paper, we study the tedious link between the properties of sensibility and approximability of models of untyped {\lambda}-calculus. Approximability is known to be a slightly, but strictly stronger property that sensibility. However, we will see that so far, each and every (filter) model that have been proven sensible are in fact approximable. We explain this result as a weakness of the sole known approach of sensibility: the Tait reducibility candidates and its realizability variants. In fact, we will reduce the approximability of a filter model D for the {\lambda}-calculus to the sensibility of D but for an extension of the {\lambda}-calculus that we call {\lambda}-calculus with D-tests. Then we show that traditional proofs of sensibility of D for the {\lambda}-calculus are smoothly extendable for this {\lambda}-calculus with D-tests.