Filter Equivariant Functions: A symmetric account of length-general extrapolation on lists

TL;DR

This paper introduces filter equivariant functions, a new class of list functions that behave predictably under element removal, providing a geometric perspective and an algorithm for perfect length-general extrapolation.

Contribution

It defines filter equivariant functions, relates them to map equivariance, and presents an algorithm for constructing length-general extrapolations based on sublist behavior.

Findings

Contains interesting examples of filter equivariant functions

Proves basic theorems relating to filter equivariance

Provides an algorithm for perfect length-general extrapolation

Abstract

What should a function that extrapolates beyond known input/output examples look like? This is a tricky question to answer in general, as any function matching the outputs on those examples can in principle be a correct extrapolant. We argue that a "good" extrapolant should follow certain kinds of rules, and here we study a particularly appealing criterion for rule-following in list functions: that the function should behave predictably even when certain elements are removed. In functional programming, a standard way to express such removal operations is by using a filter function. Accordingly, our paper introduces a new semantic class of functions -- the filter equivariant functions. We show that this class contains interesting examples, prove some basic theorems about it, and relate it to the well-known class of map equivariant functions. We also present a geometric account of filter equivariants, showing how they correspond naturally to certain simplicial structures. Our highlight result is the amalgamation algorithm, which constructs any filter-equivariant function's output by first studying how it behaves on sublists of the input, in a way that extrapolates perfectly.