Kripke Semantics for Martin-L\"of's Extensional Type Theory

TL;DR

This paper introduces a Kripke semantics for Martin-Löf's extensional dependent type theory, providing a standard, coherent, and computationally manageable model that ensures soundness and completeness.

Contribution

It presents a novel Kripke-style semantics for dependent type theory that is both standard and coherent, facilitating completeness proofs and easier interpretation.

Findings

Models interpret types as sets indexed over posets

The semantics generalize Kripke models for first-order logic

The approach ensures soundness and completeness for the type theory

Abstract

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.