Identify a succinct structured representation with minimal canonical forms

Construct or determine whether there exists a structured representation format whose canonical forms are minimal for a given vtree and Boolean function and that is exponentially more succinct than ordered binary decision diagrams.

Background

The paper contrasts TDDs and SDDs with respect to canonicality and succinctness. TDDs, like ordered binary decision diagrams (OBDDs), have canonical minimal forms for a fixed vtree or variable order, but TDDs are only quasipolynomially more succinct than OBDDs. SDDs can be exponentially more succinct than OBDDs, but their canonical form is not necessarily of minimal size. The authors explicitly ask whether both properties can coexist in a single structured representation format.

References

With that said, there are still several open research questions: first of all, can either restructuring in time polynomial in the input and output or the polynomial time equivalence test be lifted to SDDs or even all d-SDNNFs?

Note that for SDDs the canonical form is not of minimal size, whereas canonical forms of TDDs (and OBDDs) are minimal. On the other hand, SDDs are exponentially more succinct than OBDDs, whereas TDDs are not (as we now know). Can we have both? That is, does there exists a structured representation format with canonical minimal forms (for a given vtree and a Boolean function) that are exponentially more succinct than OBDDs?

— Restructuring Tree Decision Diagrams  (2609.21842 - Berkholz et al., 18 Sep 2026) in Section 5, Conclusion