Conservativity of currying

Prove or refute that the currying translation from polyadic definedness-free matching logic into applicative matching logic reflects derivability.

Background

The paper proves global completeness directly for arbitrary finitary one-sorted signatures and separately observes that applicative matching logic is the binary-application instance. A natural alternative would curry each polyadic symbol into a constant and repeated applications, thereby reducing the general case to the applicative case.

Semantic preservation in one direction is established, but an applicative derivation may use partial applications or the curried symbol alone, which have no direct polyadic counterparts. A conservativity or derivability-reflection theorem is therefore required and remains unresolved.

References

The converse transfer of derivability remains unproved. An applicative derivation of $\varphi\dagger$ may pass through patterns with no polyadic counterpart, such as $c_\sigma$ alone or a partial application $c_\sigma\varphi_1$.

— Completeness and incompleteness of basic matching logic  (2608.13306 - Chen et al., 13 Aug 2026) in Remark “currying is not known to be conservative”; Section 6, subsection “Open problems,” item 3

Applicative matching logic. Currying $\mathsf{s},\oplus,\otimes$ into constants under a single application is expected to go through, since \Cref{lem:muinj} applies to each argument position of application, but the constraints then quantify over partial applications and we have not written them.

— Completeness and incompleteness of basic matching logic  (2608.13306 - Chen et al., 13 Aug 2026) in Remark “what is not minimized”; Section 6, subsection “Open problems,” item 5