Related papers: Call-by-name Gradual Type Theory
We reformulate recent advances in directed type theory--a type theory where the types have the structure of synthetic (higher) categories--as a logical calculus with multiple context 'zones', following the example of Pfenning and Davies.…
Most categorical models for dependent types have traditionally been heavily set based: contexts form a category, and for each we have a set of types in said context -- and for each type a set of terms of said type. This is the case for…
Type theory plays an important role in foundations of mathematics as a framework for formalizing mathematics and a base for proof assistants providing semi-automatic proof checking and construction. Derivation of each theorem in type theory…
Dependently typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional functor laws,…
We present an approach to type theory in which the typing judgments do not have explicit contexts. Instead of judgments of shape "Gamma |- A : B", our systems just have judgments of shape "A : B". A key feature is that we distinguish free…
In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…
Information flow type systems enforce the security property of noninterference by detecting unauthorized data flows at compile-time. However, they require precise type annotations, making them difficult to use in practice as much of the…
Liquid typing provides a decidable refinement inference mechanism that is convenient but subject to two major issues: (1) inference is global and requires top-level annotations, making it unsuitable for inference of modular code components…
Modern engineering systems include many components of different types and functions. Verifying that these systems satisfy given specifications can be an arduous task, as most formal verification methods are limited to systems of moderate…
Graded Type Theory provides a mechanism to track and reason about resource usage in type systems. In this paper, we develop GraD, a novel version of such a graded dependent type system that includes functions, tensor products, additive…
Reasoning about the sensitivity of functions with respect to their inputs has interesting applications in various areas, such as differential privacy. In order to check and enforce sensitivity, several approaches have been developed,…
Type soundness is an important property of modern programming languages. In this paper we explore the idea that "well-typed languages are sound": the idea that the appropriate typing discipline over language specifications guarantees that…
Gradual typing combines static and dynamic typing in the same language, offering the benefits of both to programmers. Static typing provides error detection and strong guarantees while dynamic typing enables rapid prototyping and flexible…
Gradualizing the Calculus of Inductive Constructions (CIC) involves dealing with subtle tensions between normalization, graduality, and conservativity with respect to CIC. Recently, GCIC has been proposed as a parametrized gradual type…
Graded type theories are an emerging paradigm for augmenting the reasoning power of types with parameterizable, fine-grained analyses of program properties. There have been many such theories in recent years which equip a type theory with…
We present a graded modal type theory, a dependent type theory with grades that can be used to enforce various properties of the code. The theory has $\Pi$-types, weak and strong $\Sigma$-types, natural numbers, an empty type, and a…
Erlang's dynamic typing discipline can lead to runtime errors that persist even after process restarts. Some of these runtime errors could be prevented through static type checking. While Erlang provides a type specification language, the…
We investigate gradual variations on the Calculus of Inductive Construction (CIC) for swifter prototyping with imprecise types and terms. We observe, with a no-go theorem, a crucial tradeoff between graduality and the key properties of…
We demonstrate the possibility of classifying causal systems into kinds that share a common structure without first constructing an explicit dynamical model or using prior knowledge of the system dynamics. The algorithmic ability to…
A key challenge when statically typing so-called dynamic languages is the ubiquity of value-based overloading, where a given function can dynamically reflect upon and behave according to the types of its arguments. Thus, to establish basic…