Relationship between μ-downward self-reducibility and traditional downward self-reducibility

Determine whether every problem that admits a μ-downward self-reduction also admits a traditional downward self-reduction in which recursive calls are on strictly smaller input sizes.

Background

The paper generalizes traditional downward self-reducibility to μ-d.s.r., where “smaller” is measured by a problem-specific potential μ rather than input size. This enables broader applicability of the framework across TFΣ_i^P problems.

Clarifying whether μ-d.s.r implies traditional d.s.r would delineate the boundaries between the generalized and classical notions of self-reduction and their consequences for class collapses.