A dependent nominal type theory

Abstract

Abstract. Nominal abstract syntax is an approach to representing names and binding pioneered by Gabbay and Pitts. So far nominal techniques have mostly been studied using classical logic or model theory, not type theory. Nominal extensions to simple, dependent and ML-like polymorphic languages have been studied, but decidability and normalization results have only been established for simple nominal type theories. We present a LF-style dependent type theory extended with name-abstraction types, prove soundness and decidability of βη-equivalence checking, discuss adequacy and canonical forms via an example, and discuss extensions such as dependently-typed recursion and induction principles.

Nominal Coalgebraic Data Types with Applications to Lambda Calculus

“…Nominal extensions of typed λ-calculus have been proposed in [Pit10,Che09] for system T, in [Che12] for LF and in [WSA09] for the Calculus of Inductive Constructions. Further research could include a study of typing that combines nominal syntax with coinductive data types.…”

Nominal essential intersection types

“…In Cheney (2009), a simple type system is presented for nominal abstract syntax, where the nominal semantics is added to the λ-calculus, with βηreduction shown as a primitive notion. Using the same approach, Cheney (2012) and Pitts et al (2015) presented dependent type systems, where a dependent name-abstraction type constructor is used in the syntax of types.…”

Validating Brouwer's continuity principle for numbers using named exceptions

“…It is also designed with decidable typechecking in mind. Cheney extended SNTT to a dependent type theory, called λ Π N [23], with dependent products (Π) and dependent name-abstraction types ( N). As for SNTT, one of his main focus was to provide a strongly normalizing theory with decidable type checking.…”