Pitts and Intuitionistic Multi-Succedent: Uniform Interpolation for KM

Abstract: 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.

• The paper proves that KM has uniform interpolation by adapting Pitts’ proof-theoretic construction to the terminating multi-succedent calculus G4KM.
• The authors establish termination, contraction and cut admissibility, soundness, and completeness for G4KM, thereby also obtaining decidability and an effective procedure for computing interpolants.
• The result yields coherence of KM-algebras through algebraic interpolation theory, while the Rocq formalisation verifies the metatheory and exposes open problems concerning stronger quantifiers and related logics.

Background and motivation

Uniform interpolation is a strong form of Craig interpolation: for every formula $\varphi$ and propositional variable $p$, there exist $p$-free formulas $\forall p\,\varphi$ and $\exists p\,\varphi$ that act as strongest/weakest $p$-free approximations of $\varphi$, thereby internalising propositional quantifiers. Pitts' proof-theoretic method computes these interpolants from terminating sequent calculi via mutual recursion over finite proof-search trees [Pit92]. Until this work by Férée and Shillito (arXiv (2602.16445)), applications of Pitts' technique to logics with an intuitionistic basis were confined to single-succedent calculi; multi-succedent adaptations existed only on a classical basis (Bílková's work on K, T, GL, later extended by van der Giessen, Jalali and Kuznets to nested/labelled settings).

The target logic is KM, the intuitionistic modal logic extending iSL — itself extending iGL and IPC — with the Kuznetsov–Muravitsky axiom $\Box\varphi \to (\psi \lor (\psi \to \varphi))$. KM is mathematically significant because its lattice of extensions is isomorphic to that of GL, and it is connected in computer science to Nakano's "later" modality and guarded recursion. Prior to this paper, uniform interpolation for KM was neither proved nor refuted.

A terminating multi-succedent calculus

Existing calculi were unsuitable: Darjania's calculi contain explicit structural rules and do not terminate; Clouston and Goré's calculus terminates but splits implication into reflexive and irreflexive variants and admits only a semantic cut-free completeness proof. The authors instead design G4KM, obtained from G4iSLt by porting to a multi-succedent basis over Dyckhoff–Negri's G4iP', and modifying the implication-right rule. The key insight comes from Kripke semantics: KM models require $R = <$ exactly (the modal relation coincides with strict intuitionistic succession), so boxed formulas forced at a world become unboxed at any modal successor while remaining boxed at the world itself under reflexivity. The modified ImpR rule exploits this by keeping the succedent in one premise (the reflexive case) and replacing it with the antecedent's boxes in the other (the strict-successor case). A naive two-premise refinement distinguishing reflexive from strict jumps collapses under weakening for plain intuitionistic implication — the distinction is only available because KM can syntactically separate the two kinds of jumps.

Termination follows from a Dershowitz–Manna multiset ordering induced by formula weight (size plus number of conjunctions, so conjunctions count double), applied to the triple $(\Gamma,\Delta,\Delta)$; counting the succedent twice accommodates the BoxR rule. This yields decidability of KM as an immediate corollary.

The structural metatheory required genuine new work. Because ImpR duplicates material across premises, right contraction depends on left contraction, which in turn heads the standard lemma chain toward left contraction — a circular dependency absent from Dyckhoff–Negri's G4iP' argument. The authors resolve it by proving admissibility of right contraction, left contraction, ImpL, and ImpLDup simultaneously by well-founded induction on sequents. Cut elimination then follows via the now-standard methodology for provability logics, using the termination measure as induction measure. The calculus is shown sound and complete for the generalised Hilbert calculus KMH. A Rocq-specific subtlety arises here: since KMH lives in Prop and the calculus in Type, completeness cannot directly inspect axiomatic proofs; decidability of the calculus supplies constructive existence-to-proof conversion.

Notably, the Kripke semantics is not strongly complete for KM: the infinite set $p$0 entails $p$1 semantically but not derivationally, due to non-compactness of converse-well-founded frames — a limitation inherited from provability logics generally.

Adapting Pitts' technique to multi-succedent sequents

The core conceptual contribution is a careful rephrasing of the uniform interpolation property for multi-succedent sequents. The tempting formulation of clause 3(b) — allowing arbitrary $p$2-free formulas in the succedent context — would interpret a constant-domain universal propositional quantifier satisfying $p$3, which fails intuitionistically. The adopted definition restricts the property precisely to avoid capturing this stronger quantifier; the authors note they see no theoretical obstruction to interpreting stronger quantifier versions, leaving that open.

The interpolants are defined by recursive clauses mirroring the rules of G4KM, with $p$4 and $p$5. Two technical adjustments matter:

• Clause (Aₚ8) recurses on $p$6 rather than the actual premise containing $p$7 on the succedent side, since ImpLImp is the only left rule whose premise mixes antecedent and succedent formulas of the conclusion. This is justified by invertibility of ImpLImp with respect to its left premise (Proposition on ImpLImp-reversal), which lets the problematic premise be re-derived without $p$8.
• Invertibility structure: OrR is fully invertible thanks to multi-succedency, whereas ImpR is only partially invertible, unlike in single-succedent G4iP.

The uniformity proof proceeds by well-founded induction; the critical case is ImpLImp, where the leftmost premise cannot be handled by direct application of the induction hypothesis (clause 3(a) fails since $p$9 need not be $p$0-free), and the reversal proposition is used essentially. The result: KM has the uniform interpolation property, resolving a previously open question. The authors expect the same adaptation to transfer to Gödel–Dummett logic LC and KM_lin, though they do not prove this.

Algebraic consequence: coherence of KM-algebras

KM is finitely algebraisable over the class KMAlg of Heyting algebras equipped with a unary operator satisfying $p$1, $p$2, $p$3, $p$4... specifically $p$5 together with $p$6. Via the bridge theorem from abstract algebraic logic connecting uniform deductive interpolation to coherence (finitely generated subalgebras of finitely presented algebras being finitely presented), the paper deduces a novel corollary: KMAlg is coherent. This illustrates how proof-theoretic interpolation results feed directly into universal-algebraic properties.

Mechanisation

All results — syntax, Hilbert calculus, Kripke semantics, sequent calculus, termination, contraction and cut admissibility, equivalence, uniform interpolation, algebraisability, coherence — are formalised in Rocq, with hyperlinks throughout the paper to the machine-checked proofs. Beyond correctness assurance, the mechanisation enables extraction of executable programs computing interpolants concretely.

Limitations and open questions

Several caveats are stated plainly. The multiset ordering may be overly coarse: a weight of the form $p$7 suffices for termination and yields explicit depth bounds, but the more permissive ordering remains useful for admissibility proofs. Cut admissibility for the variant calculus using the alternative rule (ImpLImp-two, implicitly used in clause Aₚ8) has not been established — a straightforward adaptation of the argument does not obviously work. Whether the constant-domain universal quantifier admits interpretation in some strengthening of the framework is open, as is the transfer of the technique to LC (where interpolant size or production time might improve on procedures exploiting local finiteness plus Craig interpolation) and to KM_lin, whose uniform interpolation status is genuinely unresolved.

Conclusion

This paper establishes uniform interpolation for KM through three coordinated contributions: a novel terminating, cut-admissible multi-succedent calculus G4KM built on the semantic separation of reflexive and strict-successor jumps; a careful multi-succedent reformulation of Pitts' interpolant construction that avoids capturing the constant-domain quantifier; and the derivation of coherence for KM-algebras via algebraisability and a bridge theorem. The full mechanisation in Rocq certifies the results and supports effective interpolant computation.