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Closure of d‑DNNF under complementation

Prove or refute that the class of deterministic decomposable negation normal form (d‑DNNF) circuits is closed under Boolean complementation.

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Background

The paper’s circuit approach connects to knowledge compilation. Understanding whether d‑DNNFs are closed under negation is a central unresolved question in that area and influences the expressive power of corresponding tractable counting frameworks.

The authors explicitly refer to this as an open problem when discussing whether their negation‑extended circuits are more concise than negation‑free ones.

References

One obvious question is whether $\times,\uplus,$-circuits are more concise than $\times,\uplus$-circuits, which relates to the open problem of whether d-DNNFs are closed under complementation (see).

— On the Complexity of Language Membership for Probabilistic Words (2510.08127 - Amarilli et al., 9 Oct 2025) in Section 7, Conclusion and Future Work