Scope of categorical methodology for efficient and practical AD

Determine how far categorical reasoning for correctness-guaranteed, structure-preserving automatic differentiation can be extended to provide a systematic roadmap for efficiency, optimisation, and implementation.

Background

The thesis develops categorical semantics and homomorphic transformations for forward- and reverse-mode automatic differentiation, with correctness established through universal properties and categorical logical relations. The concluding discussion identifies efficiency, optimisation, and implementation as areas not fully addressed by the semantic foundation.

The unresolved issue is whether the same categorical reasoning that establishes derivative correctness can also systematically guide the practical development of efficient implementations and optimisation techniques. This question concerns the broader applicability and limits of the thesis methodology rather than a specific theorem proved elsewhere in the work.

References

Ultimately, the open question is how far this methodology can be taken. This thesis has shown that categorical semantics, when held accountable to practice, is not only explanatory but generative: it can deliver transformations that are provably correct and practically relevant. The challenge now is to push this dialogue further: to see whether the same categorical reasoning that has secured correctness can also chart a systematic roadmap to efficiency, optimisation, and implementation.

— Freely Generated Categorical Structures and Automatic Differentiation, PhD Thesis (Introduction and Conclusion)  (2609.40046 - Nunes, 30 Sep 2026) in Chapter 8, Section "Limitations"