Uniform Interpolation

Uniform interpolation is a strengthening of interpolation that holds for certain propositional logics. The starting point of this chapter is a theorem of A. Pitts, which shows that uniform interpolation holds for intuitionistic propositional logic. We outline how this theorem may be proved semantically via the definability of bisimulation quantifiers, and how it generalizes to an open mapping theorem between Esakia spaces. We also discuss connections between uniform interpolation and research in categorical logic, algebra, and model theory.

💡 Research Summary

This chapter surveys the concept of uniform interpolation, a strengthening of the classic Craig interpolation property, and its existence in several propositional logics, with a focus on intuitionistic propositional logic (IPL). The starting point is Pitts’s theorem (1995) which asserts that every IPL formula admits both left and right uniform interpolants with respect to any set of propositional variables. The authors present a semantic proof of this theorem based on the definability of bisimulation quantifiers, then connect the result to algebraic and categorical perspectives via adjunctions between free Heyting algebras and Esakia duality, culminating in an open‑mapping theorem for Esakia spaces. The chapter also outlines relationships to quantifier elimination in first‑order theories of algebras associated with propositional logics, and surveys known results and open problems for a variety of logics.

The exposition begins by defining right and left uniform interpolants Eₚ(φ) and Aₚ(φ) for a formula φ and a variable p. A right uniform interpolant must (i) not contain p, (ii) be entailed by φ, and (iii) entail any p‑free consequence of φ; the left version is dual. The authors note that iterating the single‑variable construction yields uniform interpolants for arbitrary variable sets, and that each formula has at most one such interpolant up to logical equivalence.

The semantic framework uses finite Kripke models for IPL: a model M = (W, ≤, π) consists of a set of worlds, a reflexive‑transitive order, and an upward‑closed valuation π for each propositional variable. For a pointed model (M,w), the forcing relation ⊩ is defined in the usual intuitionistic way. Soundness and completeness guarantee that φ ⊢ ψ iff every finite pointed model satisfying φ also satisfies ψ.

A central technical notion is q‑bisimulation between two models, which ensures that the two worlds agree on the truth of all formulas built from the variable set q and that the order structure can be matched forward and backward. Lemma 4 shows that q‑bisimilar worlds force exactly the same q‑formulas, proved by induction on formula structure.

Bisimulation quantifiers are then introduced. For a class K of finite pointed (p,q)‑models, the existential bisimulation quantifier Eₚ(K) consists of those worlds that are q‑bisimilar to some world in K; the universal quantifier Aₚ(K) consists of those worlds all of whose q‑bisimilar companions belong to K. Proposition 5 proves that if the class of models of a formula φ is closed under the appropriate bisimulation quantifier, then the resulting formula ε (or α) is a right (or left) uniform interpolant for φ. The proof uses the standard Craig interpolation theorem for IPL only to bridge from φ ⊢ θ to ε ⊢ θ′, but the authors remark that a modified argument can avoid this appeal entirely.

With these tools, the authors give a clean proof of Pitts’s theorem: the existential and universal bisimulation quantifiers are definable in IPL, which yields uniform interpolants for any formula and any set of variables. The definability is established via bounded bisimulation games (or equivalently, finite‑depth bisimulations), showing that the quantifiers can be expressed by formulas of bounded modal depth.

Section 3 shifts to an algebraic viewpoint. The free Heyting algebra on a set of generators admits left and right adjoints to the inclusion of the subalgebra generated by a smaller set of variables; these adjoints correspond precisely to the bisimulation quantifiers Eₚ and Aₚ. Via Esakia duality, the adjoints translate into open maps between Esakia spaces, giving an “open‑mapping theorem”: the dual of a Heyting algebra homomorphism that forgets variables is an open continuous map. This provides a topological characterisation of uniform interpolation.

Section 4 connects uniform interpolation to quantifier elimination in first‑order theories of the algebraic semantics of propositional logics. If a logic has uniform interpolation, then the corresponding class of algebras admits model‑complete theories where existential quantifiers can be eliminated, leading to model‑theoretic consequences such as completeness of certain deductive systems.

Section 5 surveys further developments. The authors compare syntactic approaches (based on sequent calculi with cut‑elimination) to the semantic bisimulation method, noting that the latter extends more readily to modal logics such as K, GL, and S4Grz. Table 1 (referenced but not reproduced) summarises which logics are known to have uniform interpolation, which have only implication‑based versions, and which remain open. Notably, classical propositional logic lacks uniform interpolation, while intuitionistic logic, Gödel‑Löb logic, and several substructural logics do possess it. The chapter concludes with open questions, especially regarding uniform interpolation for intuitionistic first‑order logic and for various non‑classical logics.

Overall, the chapter presents uniform interpolation as a unifying theme linking proof theory, model theory, algebra, and topology. By demonstrating that bisimulation quantifiers are definable in IPL and that these quantifiers correspond to adjoints in the algebraic setting, the authors provide a robust semantic foundation that both explains Pitts’s original result and opens pathways for extending uniform interpolation to a broad spectrum of logical systems.