Native globally complete calculus for H(∀)

Develop a native, syntax-directed Hilbert, sequent, tableau, or labelled calculus that is globally complete for the hybrid logic H(∀), and determine a minimal rule or structural principle that necessarily violates localization.

Background

The paper establishes that global consequence for H(∀), the hybrid logic obtained by adding nominals to the state-variable and arbitrary-arity modality language, is not captured by any well-founded calculus whose leaves are hypotheses or valid patterns and whose rules respect localization. Nevertheless, global consequence remains recursively enumerable under effective countability assumptions, so some sound and complete recursive system exists.

The unresolved issue is to find a native, syntax-directed calculus rather than obtaining completeness indirectly by pulling back a first-order calculus through the standard translation. Such a calculus must contain a rule or structural mechanism capable of seeing beyond the component generated by the evaluation point.

References

The open problem is to find a native, syntax-directed calculus (Hilbert, sequent, tableau, or labelled) for global consequence in $\mathrm{H(\forall)}$ and to identify a minimal rule or structural principle that necessarily violates localization (\Cref{cor:hforall,rem:whatcomplete}).

— Completeness and incompleteness of basic matching logic  (2608.13306 - Chen et al., 13 Aug 2026) in Section 6, subsection “Open problems,” item 1; see also Corollary 6.?? and Remark “what a complete system would have to look like”