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

The paper investigates hybrid logic – a modal logic enriched with nominals that name individual states – within the broad setting of coalgebraic semantics. In coalgebraic semantics, the usual Kripke frames are replaced by coalgebras for an arbitrary functor F, which allows one to capture a wide variety of modal operators such as probabilistic, graded, default, or coalition operators. The central technical contribution is a pair of generic criteria guaranteeing the existence of named canonical models for any coalgebraic hybrid logic that satisfies them. The first criterion requires the underlying F‑coalgebraic category to be ω‑complete (closed under countable coproducts) and to admit a “named‑canonical” construction, i.e., a way of interpreting nominals densely enough to separate states. The second criterion applies to bounded functors and demands strong one‑step completeness, meaning that every one‑step satisfiable set of formulas can be realized in a single transition step. When either of these conditions holds, one can build a named model in which every state is denoted by some nominal, and the usual truth lemmas go through unchanged.

Using these model existence results, the authors prove two families of completeness theorems. The first concerns pure extensions of the hybrid language, i.e., additional axioms that involve only nominals and the box modality. Because pure axioms are interpreted purely in terms of the underlying frame structure, the named canonical model provides a sound and complete semantics for any such extension that respects the above criteria. The second family deals with an extended hybrid language that includes a local binding operator ↓ (read “bind the current state to a fresh nominal”). The paper shows how to give a coalgebraic semantics to ↓ by interpreting it as a dynamic renaming of the current state, and then demonstrates that the same named‑model construction yields completeness for logics that combine ↓ with the usual hybrid operators.

To illustrate the general theory, the authors instantiate the framework for several concrete functors. For the probability functor P, they obtain completeness of probabilistic hybrid logic with ↓. For the graded functor N (counting successors), they prove completeness of graded hybrid logic with local binding – a result not previously available. They also treat default functors D (capturing “if‑then‑else” style reasoning) and coalition functors C (modeling multi‑agent abilities), showing that each satisfies the required criteria and therefore admits named canonical models and pure‑extension completeness.

Overall, the paper establishes a robust, uniform method for constructing named canonical models in coalgebraic hybrid logics, thereby extending the reach of hybrid reasoning to a large class of non‑standard modal operators. The results unify and generalize earlier completeness proofs for Kripke‑based hybrid logics, and open the door to automated reasoning tools that can handle hybrid specifications over probabilistic, graded, default, or cooperative systems. Future work suggested includes algorithmic generation of named models, integration with model‑checking frameworks, and exploration of richer binding mechanisms.