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

2025-07-11

AI Summary

This work addresses the problem of reliable extrapolation of list functions under unknown input perturbations—e.g., arbitrary element deletions. Existing methods lack structural constraints, leading to uncontrolled and unreliable extrapolation behavior. To resolve this, we introduce the novel class of *filter-equivariant functions*, which enforce behavioral consistency across all sublists (i.e., outputs on any sublist must align with the restriction of the full-list output). Theoretically, we establish an equivalence between filter-equivariant functions and mapping-equivariant functions, revealing their natural correspondence to a specific class of simplicial complexes—a geometric characterization. Algorithmically, we design a *complete extrapolation algorithm* that exactly reconstructs the output of any filter-equivariant function under arbitrary sublist inputs. Our framework unifies functional programming (via filtering), equivariance theory, combinatorial semantics, and geometric representation, yielding the first formal and computationally tractable theory for structured function extrapolation.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Defining criteria for good extrapolant functions on lists
• Introducing filter equivariant functions for predictable behavior
• Developing an algorithm to construct filter-equivariant extrapolants

Innovation

• Introduces filter equivariant functions
• Relates to map equivariant functions
• Develops amalgamation algorithm