Continuation-passing Style Models Complete for Intuitionistic Logic

A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic. The proofs of soundness and completeness are constructive and the computational content of their composition is, in particular, a $\beta$-normalisation-by-evaluation program for simply typed lambda calculus with sum types. Although the inspiration comes from Danvy’s type-directed partial evaluator for the same lambda calculus, the there essential use of delimited control operators (i.e. computational effects) is avoided. The role of polymorphism is crucial – dropping it allows one to obtain a notion of model complete for classical predicate logic. The connection between ours and Kripke models is made through a strengthening of the Double-negation Shift schema.

💡 Research Summary

The paper introduces a new class of models for minimal intuitionistic predicate logic based on continuation‑passing style (CPS) monads that are polymorphic over first‑order individuals. Traditional Kripke semantics, while sound and complete for intuitionistic logic, become cumbersome and non‑constructive when all logical connectives—including falsum, disjunction, and the existential quantifier—are considered. To overcome these difficulties, the author defines “Intuitionistic Kripke‑CPS” (IK‑CPS) models, which enrich the usual Kripke frame (a preorder of worlds) with two forcing relations: a strong forcing ⊩ₛ that captures constructive evidence for atomic formulas, and a weaker forcing ⊩ that corresponds to the usual Kripke forcing. An additional exploding‑node predicate ⊩⊥ marks worlds where falsum holds.

Strong forcing is defined inductively on formula structure. For atomic formulas it coincides with the existence of a normal‑form derivation in the underlying proof system. For composite formulas it uses a higher‑order condition: for any formula C, if every extension world that strongly forces A also forces C⊥, then the current world forces C⊥. This definition mirrors a strengthened version of the Double‑Negation Shift (DNS) schema, which the paper calls D‑DNS⁺. The weak forcing ⊩ is then derived from strong forcing via a monadic “unit” operation, and the usual Kripke monotonicity properties are proved (Lemma 3.3). The monadic bind operation (Lemma 3.4) allows one to compose continuations, reflecting the intuitionistic implication rule.

A universal IK‑CPS model U is constructed where worlds are exactly the proof contexts Γ of the minimal intuitionistic natural deduction system (MQC). In U, strong forcing of an atomic formula X means there is a normal‑form derivation Γ ⊢ₙ𝚏 X, while the exploding predicate ⊩⊥ means there is a normal‑form derivation of any formula C from Γ. The domain of individuals is constant across worlds. Within this universal model, the two key operations “reify” (↓) and “reflect” (↑) are defined: ↓ maps a strong forcing witness to a normal λ‑term (a proof term in β‑normal η‑long form), while ↑ maps a neutral λ‑term back to a strong forcing witness. These are precisely the components of a β‑normalisation‑by‑evaluation (NBE) algorithm for simply‑typed λ‑calculus with sum types (λ→∨). The paper shows that the composition of soundness (from proof terms to forcing) and completeness (from forcing back to proof terms) yields a constructive NBE procedure, and the algorithm is formalised in Coq.

A crucial technical contribution is handling disjunction and existential quantification without resorting to delimited control operators (shift/reset) that Danvy’s original NBE algorithm employed. By embedding continuations directly into the Kripke structure, the IK‑CPS model can treat neutral terms of type A∨B or ∃x.A(x) constructively: the strong forcing condition forces a case analysis on the neutral term, and the monadic bind supplies the necessary continuation to complete the proof. Thus the model achieves full intuitionistic completeness in CPS style while staying within a pure, effect‑free meta‑language.

The role of polymorphism is highlighted. When the continuation monad is polymorphic over first‑order individuals, the model captures intuitionistic logic; if polymorphism is dropped, the forcing relations collapse to those of a classical completeness model (as shown in related work