Nested Sequents for Intuitionistic Modal Logics via Structural Refinement

We employ a recently developed methodology – called"structural refinement"– to extract nested sequent systems for a sizable class of intuitionistic modal logics from their respective labelled sequent systems. This method can be seen as a means by which labelled sequent systems can be transformed into nested sequent systems through the introduction of propagation rules and the elimination of structural rules, followed by a notational translation. The nested systems we obtain incorporate propagation rules that are parameterized with formal grammars, and which encode certain frame conditions expressible as first-order Horn formulae that correspond to a subclass of the Scott-Lemmon axioms. We show that our nested systems are sound, cut-free complete, and admit hp-admissibility of typical structural rules.

💡 Research Summary

The paper presents a systematic method for deriving nested sequent calculi for a broad class of intuitionistic modal logics from their labelled sequent counterparts. The authors build on the recently introduced “structural refinement” technique, which consists of three main steps: (1) replace the structural rules of a labelled sequent system (such as the d‑rule and the Sₙ,ₖ‑rules) with propagation rules, (2) parameterise the applicability of these propagation rules by formal grammars that generate strings over the alphabet {♦, ◇}, and (3) translate the resulting labelled sequents into nested sequents, a more compact tree‑like representation of worlds.

Intuitionistic modal logic IK is first recalled together with its bi‑relational Kripke semantics (a preorder ≤ for intuitionistic entailment and an accessibility relation R for modal operators). The authors consider extensions of IK by the seriality axiom D and by a family of Horn‑Scott‑Lemmon (HSL) axioms of the form (♦ⁿ ◇ A ⊃ ◇ᵏ A) ∧ (♦ᵏ A ⊃ ◇ⁿ ♦ A). Each HSL axiom corresponds to a first‑order Horn frame condition, e.g. ∀w,u,v (wRⁿ u ∧ wRᵏ v → uRv).

The key technical innovation is the use of an A‑grammar g(A) associated with a set A of HSL axioms. For every axiom φ(n,k)∈A the grammar contains two production rules: ♦ⁿ ◇ → ◇ᵏ and ◇ᵏ → ♦ⁿ ◇. Strings over Σ={♦, ◇} encode paths in the underlying labelled graph; a propagation rule may be applied only when the string describing the path can be derived from the grammar. In this way, frame conditions are enforced syntactically, without resorting to explicit relational atoms.

The labelled sequent system L◇♦(A) (a variant of Simpson’s system) is presented with initial, logical, and structural rules. The structural refinement replaces the structural rules by the grammar‑controlled propagation rules, yielding a refined labelled system. The final step translates each labelled sequent into a nested sequent: worlds become nodes of a tree, and the hierarchical nesting of □‑ and ◇‑sequents mirrors the ≤‑relation. The resulting nested calculus N◇♦(A) retains the original logical rules, adds the propagation rules (now written as “propagation(σ)” where σ∈Σ*), and discards all explicit structural rules.

The authors prove three metatheoretic properties for N◇♦(A): (i) Soundness – every derivation corresponds to a valid argument in all bi‑relational A‑models; (ii) Cut‑free completeness – the calculus without the cut rule is sufficient to derive every theorem of IK(A), and the cut rule is admissible; (iii) Height‑preserving admissibility (hp‑admissibility) of weakening, contraction, and exchange, meaning these rules can be applied without increasing the height of proof trees. The proofs rely on a systematic simulation of labelled derivations by nested derivations, and on the closure properties of the grammars.

Compared with earlier work, which provided nested sequents only for the “intuitionistic modal cube” (logics obtained by adding subsets of T, B, 4, 5, D), this approach handles any combination of HSL axioms, thus covering a much larger landscape of intuitionistic modal logics. The grammar‑based propagation mechanism abstracts away from the concrete shape of each axiom, allowing new axioms to be incorporated simply by adding the corresponding production rules to the grammar. Consequently, the method offers a uniform, modular way to generate nested calculi for new logics.

The paper concludes by highlighting the broader applicability of structural refinement: the same three‑stage pipeline (labelled → propagation → nested) can be adapted to other non‑classical logics (e.g., tense logics, linear logics) where labelled calculi are available. Future work is suggested on automating the generation of propagation rules from arbitrary Horn frame conditions, developing decision procedures based on the nested systems, and implementing prototype theorem provers to assess practical performance.