Named Models in Coalgebraic Hybrid Logic

Hybrid logic extends modal logic with support for reasoning about individual states, designated by so-called nominals. We study hybrid logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specifically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding.

💡 Research Summary

This paper investigates hybrid logics—modal logics enriched with nominals that name individual states—within the broad and uniform framework of coalgebraic semantics. In coalgebraic modal logic, the traditional Kripke frames are replaced by T‑coalgebras for a chosen endofunctor T, and modal operators are interpreted via predicate liftings. By fixing a similarity type Λ (the set of modal operators together with their arities) and a Λ‑structure T, the authors obtain a parametrised family of hybrid logics that covers a wide spectrum of reasoning paradigms, including probabilistic, graded, conditional, neighbourhood, and coalition logics.

The central technical contribution is the development of generic criteria guaranteeing the existence of named canonical models. A named model is a canonical model in which every nominal denotes a distinct world; such models are crucial for proving strong completeness of hybrid logics, especially when the language contains the satisfaction operator “@i” and the local binding operator “↓x.φ”. The authors present two complementary existence criteria:

1. Strong one‑step completeness – If the set of one‑step rules R (rules of the form φ/ψ where φ is a propositional premise and ψ a modal conclusion) is strongly complete for the functor T, then a named canonical model can be built using the standard “Name” rule (i → @i i). This condition is satisfied by many well‑studied coalgebraic logics, such as the basic modal logic K and its hybrid extension.

2. Finite boundedness – If each modal operator in Λ is k‑bounded for some finite k (i.e., the truth of an operator at a state depends only on at most k successors), then a named model can be constructed via a finite “unraveling” process. Graded modalities ♦k, which assert the existence of more than k successors, fall into this category.

Both criteria are proved to be sufficient for the existence of named models, and they are largely independent: a logic may satisfy one without the other. Once a named model is available, the authors show that the Hilbert‑style proof system LR (which includes propositional tautologies, the standard hybrid axioms for @, the rule (Name), and the one‑step rules R) can be extended with pure axioms (axioms containing no propositional variables) while preserving strong completeness. Pure axioms correspond to first‑order frame conditions (e.g., transitivity, reflexivity) expressed purely in terms of nominals.

The paper then tackles the more demanding extension with the local binding operator ↓. The operator ↓x.φ reads “the current state, named x, satisfies φ”. To handle ↓, the authors augment LR with two additional schemata: (i) a ↓‑introduction rule that equates ↓x.φ with @x φ, and (ii) a rule that allows any nominal i to be bound by ↓ (i → ↓i ⊤). Using the named canonical model, they prove that LR + pure axioms + ↓ is strongly complete for both global and local consequence, even in the presence of arbitrary T‑boxes (global assumptions).

To demonstrate the breadth of their framework, the authors instantiate the general results for several concrete logics:

• Hybrid K: The classic case where T is the powerset functor and the modal operator □ is interpreted via the usual subset lifting. The named model construction recovers the well‑known completeness of hybrid K.
• Graded hybrid logic: Here T is the multiset functor B, and ♦k asserts “more than k successors”. The logic is k‑bounded, so the boundedness criterion applies. The authors also show that adding the pure axiom ¬♦1 i (which forbids a single successor named i) aligns the multigraph semantics with the standard Kripke semantics, yielding completeness for graded hybrid logic with ↓.
• Conditional logic CK: Interpreted over selection‑function frames via the functor Cf. The one‑step completeness of the corresponding rule set ensures the existence of named models.
• Classical and monotone neighbourhood logics: Using the double‑powerset functor N (or its upward‑closed subfunctor M). Again, the required one‑step rules are known to be strongly complete.
• Coalition logic: Modeled by the game‑frame functor G, with operators