Undifferentiated-block function class and its symmetry characterization

Determine whether there exists a class of list functions that operates on inputs as undifferentiated blocks—meaning the functions do not identify elements by their values or positions—and that includes the functions double (defined as identity + identity via list concatenation), identity, and reverse; and identify a specific collection of list operators such that this class is exactly the symmetry class of those operators (i.e., the set of list functions that commute with every operator in the collection).

Background

The paper introduces filter-equivariant (FE) and natural filter-equivariant (NFE) list functions, characterizing classes of functions that commute with value-based removal (filter) and, for NFEs, also with element-wise transformations (map). These symmetry constraints enable length-general extrapolation through a formal amalgamation procedure.

Beyond map and filter symmetries, the authors suggest exploring broader symmetry classes of list functions. They highlight a potential class whose actions treat the list as an undifferentiated block, without reference to element values or positions, and note that it should encompass well-known functions like identity, reverse, and a doubling function formed by concatenating the input list with itself.

The conjecture asks for a precise characterization of such a class and for the identification of operators whose shared symmetry (commutation) exactly defines this class, thereby extending the symmetry-based framework for extrapolation beyond map/filter equivariances.