Kripke Semantics for Martin-L"ofs Extensional Type Theory

It is well-known that simple type theory is complete with respect to non-standard set-valued models. Completeness for standard models only holds with respect to certain extended classes of models, e.g., the class of cartesian closed categories. Similarly, dependent type theory is complete for locally cartesian closed categories. However, it is usually difficult to establish the coherence of interpretations of dependent type theory, i.e., to show that the interpretations of equal expressions are indeed equal. Several classes of models have been used to remedy this problem. We contribute to this investigation by giving a semantics that is standard, coherent, and sufficiently general for completeness while remaining relatively easy to compute with. Our models interpret types of Martin-L"of’s extensional dependent type theory as sets indexed over posets or, equivalently, as fibrations over posets. This semantics can be seen as a generalization to dependent type theory of the interpretation of intuitionistic first-order logic in Kripke models. This yields a simple coherent model theory, with respect to which simple and dependent type theory are sound and complete.

💡 Research Summary

The paper addresses a long‑standing gap in the model theory of dependent type theory: providing a standard, coherent, and complete semantics that is also computationally tractable. While simple type theory enjoys completeness with respect to non‑standard set‑valued models, its standard models require additional categorical structure such as Cartesian closed categories (CCC). Dependent type theory (specifically Martin‑Löf’s extensional type theory, MLTT) is known to be complete for locally Cartesian closed (LCC) categories, but the “coherence problem” – ensuring that syntactically equal expressions are interpreted as equal objects – makes such models difficult to work with. Existing solutions (categories with attributes, fibrations, etc.) are abstract and cumbersome.

The authors propose a Kripke‑style semantics that generalises the classic Kripke models for intuitionistic first‑order logic to the setting of dependent types. The key idea is to interpret a context Γ as a partially ordered set (poset) P, and a type S in that context as a P‑indexed set, i.e. a functor P → SET. Terms become elements of the corresponding indexed sets. This viewpoint identifies a type‑valued function on a context with a fibration over a poset, and the category of elements ∫ₚA provides a concrete representation of such fibrations.

To obtain coherence, the authors construct a canonical choice of pullback functors in the categories SETᴾ. Using a theorem from topology (essentially a choice principle for pullbacks over posets), they show that for every monotone map p : Q → P there exists a unique pullback functor p* : SETᴾ → SETᴼ that satisfies the usual universal property. Moreover, they exhibit left and right adjoints Σₚ ⊣ p* ⊣ Πₚ, which serve as the categorical semantics of dependent sum (Σ‑type) and dependent product (Π‑type). These adjoints are built from simple set‑theoretic constructions (pre‑image and function space) and are shown to be compatible with the chosen pullbacks, thereby solving the coherence problem without recourse to sophisticated categorical machinery.

With these ingredients, the interpretation of the syntax is defined inductively: contexts become posets, types become indexed sets, Σ‑types are interpreted by the left adjoint Σₚ, Π‑types by the right adjoint Πₚ, and function types by exponentials in SETᴾ. Substitution is interpreted as functorial reindexing, β‑reduction and η‑expansion correspond to ordinary function application and λ‑abstraction at the set level, and the identity type Id(s, t) is interpreted as the set of proofs that the underlying elements are equal.

The paper proves soundness by a straightforward induction on derivations, showing that every typing and equality rule is respected by the model. Completeness is more subtle: the authors demonstrate that any valid judgment in all Kripke models of the described form can be derived in MLTT. The proof proceeds by constructing a canonical term for each semantic object (a “realizer”) and then using the adjunctions to reflect the semantic structure back into syntax. Importantly, the completeness result holds for full MLTT with Σ‑, Π‑, Id‑, and unit types, not merely for the base fragment.

The authors compare their approach with earlier work: Henkin‑style non‑standard models, categories with attributes, and previous Kripke‑style models for simple type theory. Their contribution is distinguished by using only standard set‑theoretic objects, preserving the usual exponential interpretation of function types, and achieving completeness for theories with non‑trivial type constructors. They also note that the same framework yields a Kripke semantics for simple type theory as a special case.

In conclusion, the paper presents a clean, concrete class of models—poset‑indexed sets—that simultaneously satisfies the three desiderata for a model of MLTT: standardness (no exotic set‑theoretic constructions), coherence (equal syntactic objects have equal interpretations), and completeness (every semantically valid statement is provable). This work simplifies the meta‑theoretic study of dependent type theory and opens the door to more practical implementations and further extensions, such as adding universes or inductive families, within the same Kripke‑style setting.