Intermediate matching-logic regime analogous to H(@)

Determine whether a useful matching-logic analogue of the hybrid logic H(@) can be constructed in which definedness is present but inaccessible to the quantifier, and whether such a regime recovers any of the decidability available at the middle hybrid level.

Background

In the matching-logic setting studied here, definedness and the satisfaction operator are interdefinable because element variables can be quantified: definedness can be expressed as $\exists x.@_x\varphi$. Consequently, matching logic has two regimes where hybrid logic has three.

The proposed intermediate regime would block this collapse by allowing definedness while preventing the quantifier from reaching it. The paper leaves open whether this distinction is semantically possible or merely a syntactic artifact, and whether it would preserve some decidability properties of H(@).