scholarly journals Graded Modal Dependent Type Theory

2021 ◽  
pp. 462-490
Author(s):  
Benjamin Moon ◽  
Harley Eades III ◽  
Dominic Orchard
Keyword(s):  
Case Studies ◽  
Type Theory ◽  
Data Use ◽  
Type Checking ◽  
Fine Grained ◽  
Dependent Type Theory ◽  
Checking Procedure ◽  
Dependent Type

AbstractGraded 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 quantitative dataflow tracking, usually via a semiring-like structure which provides analysis on variables (often called ‘quantitative’ or ‘coeffect’ theories). We present Graded Modal Dependent Type Theory (Grtt for short), which equips a dependent type theory with a general, parameterizable analysis of the flow of data, both in and between computational terms and types. In this theory, it is possible to study, restrict, and reason about data use in programs and types, enabling, for example, parametric quantifiers and linearity to be captured in a dependent setting. We propose Grtt, study its metatheory, and explore various case studies of its use in reasoning about programs and studying other type theories. We have implemented the theory and highlight the interesting details, including showing an application of grading to optimising the type checking procedure itself.

scholarly journals Observational equality: now for good

2022 ◽  
Vol 6 (POPL) ◽  
pp. 1-27
Author(s):  
Loïc Pujet ◽  
Nicolas Tabareau
Keyword(s):  
Type Theory ◽  
Identity Relation ◽  
Logical Relation ◽  
Type Checking ◽  
Dependent Type Theory ◽  
Proper Extension ◽  
New Type ◽  
Recent Extension ◽  
Dependent Type

Building on the recent extension of dependent type theory with a universe of definitionally proof-irrelevant types, we introduce TTobs, a new type theory based on the setoidal interpretation of dependent type theory. TTobs equips every type with an identity relation that satisfies function extensionality, propositional extensionality, and definitional uniqueness of identity proofs (UIP). Compared to other existing proposals to enrich dependent type theory with these principles, our theory features a notion of reduction that is normalizing and provides an algorithmic canonicity result, which we formally prove in Agda using the logical relation framework of Abel et al. Our paper thoroughly develops the meta-theoretical properties of TTobs, such as the decidability of the conversion and of the type checking, as well as consistency. We also explain how to extend our theory with quotient types, and we introduce a setoidal version of Swan's Id types that turn it into a proper extension of MLTT with inductive equality.

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

2021 ◽  
Vol 31 ◽  
Author(s):  
ANDREA VEZZOSI ◽  
ANDERS MÖRTBERG ◽  
ANDREAS ABEL
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Type Checking ◽  
Dependent Type Theory ◽  
Wide Range ◽  
General Schema ◽  
Direct Definition ◽  
Inductive Types ◽  
Definition Of ◽  
Cubical Type Theory

Abstract Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types (HITs). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of HITs. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

scholarly journals A relationally parametric model of dependent type theory

2014 ◽  
Vol 49 (1) ◽  
pp. 503-515 ◽  
Author(s):  
Robert Atkey ◽  
Neil Ghani ◽  
Patricia Johann
Keyword(s):  
Type Theory ◽  
Parametric Model ◽  
Dependent Type Theory ◽  
Dependent Type

scholarly journals Implementing a modal dependent type theory

2019 ◽  
Vol 3 (ICFP) ◽  
pp. 1-29 ◽  
Author(s):  
Daniel Gratzer ◽  
Jonathan Sterling ◽  
Lars Birkedal
Keyword(s):  
Type Theory ◽  
Dependent Type Theory ◽  
Dependent Type

scholarly journals Guarded Dependent Type Theory with Coinductive Types

2016 ◽  
pp. 20-35 ◽  
Author(s):  
Aleš Bizjak ◽  
Hans Bugge Grathwohl ◽  
Ranald Clouston ◽  
Rasmus E. Møgelberg ◽  
Lars Birkedal
Keyword(s):  
Type Theory ◽  
Dependent Type Theory ◽  
Dependent Type

scholarly journals Parametric quantifiers for dependent type theory

2017 ◽  
Vol 1 (ICFP) ◽  
pp. 1-29 ◽  
Author(s):  
Andreas Nuyts ◽  
Andrea Vezzosi ◽  
Dominique Devriese
Keyword(s):  
Type Theory ◽  
Dependent Type Theory ◽  
Dependent Type

Introduction to the Special Issue on Dependent Type Theory Meets Practical Programming

2004 ◽  
Vol 14 (1) ◽  
pp. 1-2
Author(s):  
GILLES BARTHE ◽  
PETER DYBJEN ◽  
PETER THIEMANN
Keyword(s):  
Programming Languages ◽  
Type Theory ◽  
Type System ◽  
Type Systems ◽  
Special Issue ◽  
Dependent Type Theory ◽  
Dependent Type

Modern programming languages rely on advanced type systems that detect errors at compile-time. While the benefits of type systems have long been recognized, there are some areas where the standard systems in programming languages are not expressive enough. Language designers usually trade expressiveness for decidability of the type system. Some interesting programs will always be rejected (despite their semantical soundness) or be assigned uninformative types.

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