Completeness of the functional-algebra implication system

Prove the completeness of a derivation system for the functional-algebra implication rules corresponding to Armstrong’s reflexivity, transitivity, and augmentation axioms for functional dependencies in tree databases.

Background

The paper establishes that identity functions and projections, function composition, and products with identity functions yield implication rules analogous to Armstrong’s axioms for functional dependencies. These rules are shown to be sound for functions implied by instances of a database tree.

The unresolved issue is whether there exists a derivation system that is also complete: namely, whether every function implied by the functional algebra can be derived using an appropriate collection of inference rules. The paper explicitly leaves this question outside its scope and identifies it as part of the author’s current work.

References

However, the above lemma shows only the soundness' of these implication rules, therefore the accompanying question is the following: is there aderivation' system showing completeness? Answering this kind of questions lies outside the scope of the present paper and it is part of our current work.

— Tree Databases  (2609.03538 - Spyratos, 3 Sep 2026) in Section 2, subsection “The functional algebra,” immediately after Lemma \ref{AA}