Pitts' proof-theoretic technique for uniform interpolation, which generates uniform interpolants from terminating sequent calculi, has only been applied to logics on an intuitionistic basis through single-succedent sequent calculi. We adapt the technique to the intuitionistic multi-succedent setting by focusing on the intuitionistic modal logic KM. To do this, we design a novel multi-succedent sequent calculus for this logic which terminates, eliminates cut, and provides a decidability argument for KM. Then, we adapt Pitts' technique to our calculus to construct uniform interpolants for KM, while highlighting the hurdles we overcame. Finally, by (re)proving the algebraisability of KM, we deduce the coherence of the class of KM-algebras. All our results are fully mechanised in the Rocq proof assistant, ensuring correctness and enabling effective computation of interpolants.
Uniform interpolation is a mighty form of interpolation, ensuring for any formula φ and variable p the existence of left and right uniform interpolants, respectively denoted ∀pφ and ∃pφ. Intuitively, ∀pφ is the strongest formula without p that implies φ, and ∃pφ the weakest p-free formula that is implied by φ. Technically, these interpolants satisfy the following, where ψ is a p-free formula:
Uniform interpolants are propositional formulas, but their notation is suggestive: they provide an interpretation of propositional quantifiers inside the logic. Because of its strength, uniform interpolation is a notoriously difficult property to prove. Still, a variety of proof techniques are presented in the literature: model-theoretic [53], universal-algebraic [26,33,40], and proof-theoretic [48]. The latter kind of technique, developed in 1992 for intuitionistic logic IPC by Pitts, requires a terminating sequent calculus: a calculus whose naive backward proof search, i.e. the process of repetitively applying backward rules of the calculus in no specific order, necessarily comes to a halt. The proof search tree of a sequent in a terminating calculus is finite, and hence becomes data from which uniform interpolants are computed via mutual recursion.
In his proof, Pitts used the single-succedent terminating calculus G4iP for IPC, which was invented several times through the decades by Vorob’ev [54], Dyckhoff [19] and Hudelmaier [37]. By exploiting the existence of such calculi, which extend G4iP through Iemhoff’s methodology [39], Pitts’ original technique was recently applied to a variety of intuitionistic modal logics [38,27], including the intuitionistic provability logic iSL [21] for which a single-succedent terminating calculus G4iSLt was defined [49]. In parallel, Bílková ported the technique to multi -succedent calculi for classical modal logics K, T and provability logic GL [2]. The shift to a classical basis allowed for technical simplifications, e.g. the recursive definition stops being mutual: it focuses on the left uniform interpolant without needing the right one. Following her footsteps, van der Giessen, Jalali and Kuznets extended her approach to additional classical modal logics using multi-succedent but richer, i.e. nested or labelled, sequents [30,31,32]. As it stands, the applicability of Pitts’ technique to multi -succedent calculi for logics on an intuitionistic basis remains unclear.
By scanning the literature, one easily finds logics with an intuitionistic basis requiring multi-succedent calculi. Most famous is the Gödel-Dummett logic LC [17], an intermediate logic extending IPC with the linearity axiom (φ → ψ) ∨ (ψ → φ). This logic is given a crucially multi-succedent terminating calculus in Dyckhoff’s work [18]. While uniform interpolation is known for LC, as it is locally finite and has Craig interpolation [25,45], it could at least serve as a good example for methodological purposes. Of yet better interest is the intuitionistic modal logic KM [46], which extends iSL [28] (and hence iK [5] and IPC) with the Kuznetsov-Muravitsky axiom φ → (ψ ∨ (ψ → φ)). This logic possesses several multi-succedent calculi [11,8], and lacks a (dis)proof for uniform interpolation [41]. From a mathematical viewpoint, the interest in KM lies in its deep ties to GL and IPC: its lattice of extensions is isomorphic to the lattice of extensions of GL, and its extension with any non-modal axiom is conservative over the extension of IPC with the same axiom. Additionally, KM received some attention in computer science [8] in light of its connection to Nakano’s “later” modality [47], capturing the notion of guarded recursion [9]. Given its proximity to iSL and the recent application of Pitts’ technique to the latter logic [21], KM presents itself as a natural candidate for an investigation on the applicability of this technique to a combination of multi-succedent sequents and intuitionism.
Unfortunately, existing calculi for KM are not adequate for this investigation: the first sequent calculi for KM given by Darjania [11] clearly do not terminate, while Clouston and Goré’s calculus [8, Section 4] has complex rules and uses an alternative syntax for KM. So, we provide in Section 3 a novel multi-succedent terminating calculus for KM. Our calculus G4KM can be obtained from G4iSLt in two steps: first, port G4iSLt to a multi-succedent setting, making it an extension of the terminating multi-succedent calculus G4iP ′ for IPC [20,Section 7] ; second, modify the implication right rule, following (Kripke) semantic intuitions, to capture the characteristic axiom of KM. We show that naive backward proof search in G4KM terminates, which, together with cut elimination, provides a decidability procedure for KM. Our syntactic proof of cut elimination uses the termination measure as induction measure, a now standard methodology for provability logics [34,35,49], but requires a non-trivial refactoring of Dyckhoff and Negri’s argument
This content is AI-processed based on open access ArXiv data.