An Introduction to Dependent Type Theory

Barthe, Gilles; Coquand, Thierry

doi:10.1007/3-540-45699-6_1

Cited by 7 publications

(6 citation statements)

References 92 publications

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“…Pure Type Systems have a well-understood theory and enjoy good meta-theoretical properties, and hence offer an appealing setting in which to specify the kernel of functional programming languages, see e.g. [1,6,25,28], and proof assistants based on the CurryHoward isomorphism, see e.g. [4,5].…”

Section: Introductionmentioning

confidence: 99%

Pure patterns type systems

et al. 2003

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We introduce a new framework of algebraic pure type systems in which we consider rewrite rules as lambda terms with patterns and rewrite rule application as abstraction application with built-in matching facilities. This framework, that we call "Pure Pattern Type Systems", is particularly well-suited for the foundations of programming (meta)languages and proof assistants since it provides in a fully unified setting higher-order capabilities and pattern matching ability together with powerful type systems. We prove some standard properties like confluence and subject reduction for the case of a syntactic theory and under a syntactical restriction over the shape of patterns. We also conjecture the strong normalization of typable terms. This work should be seen as a contribution to a formal connection between logics and rewriting, and a step towards new proof engines based on the Curry-Howard isomorphism.

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Section: Introductionmentioning

confidence: 99%

Pure patterns type systems

et al. 2003

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“…Moreover, the use of type theory, on the one hand, provides a strong mathematical formalism, and on the other hand, it renders the construction more transparent, compared with logical equations, because only by considering the type restrictions of terms it is possible to deduce whether model components may be combined. That said, it is not a binary choice, but rather a spectrum of possibilities of varying expressiveness: dependent types, in their full generality, allow arbitrary logical predicates to be expressed at the type level, while work such as LiquidHaskell demonstrate one approach to support selected classes of predicates at the type level while retaining full automation through the integration of a satisfiability modulo theories solver (SMT solver). Finally, the idea of composing types is typical for many programming languages, particularly for functional programming languages such as Haskell .…”

Section: Conceptual Approaches To Mathematical Modellingmentioning

confidence: 99%

Conceptual modelling: Towards detecting modelling errors in engineering applications

Gürlebeck

Legatiuk

Nilsson

et al. 2019

Math Methods in App Sciences

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Rapid advancements of modern technologies put high demands on mathematical modelling of engineering systems. Typically, systems are no longer "simple" objects, but rather coupled systems involving multiphysics phenomena, the modelling of which involves coupling of models that describe different phenomena. After constructing a mathematical model, it is essential to analyse the correctness of the coupled models and to detect modelling errors compromising the final modelling result. Broadly, there are two classes of modelling errors: (a) errors related to abstract modelling, eg, conceptual errors concerning the coherence of a model as a whole and (b) errors related to concrete modelling or instance modelling, eg, questions of approximation quality and implementation. Instance modelling errors, on the one hand, are relatively well understood. Abstract modelling errors, on the other, are not appropriately addressed by modern modelling methodologies. The aim of this paper is to initiate a discussion on abstract approaches and their usability for mathematical modelling of engineering systems with the goal of making it possible to catch conceptual modelling errors early and automatically by computer assistant tools. To that end, we argue that it is necessary to identify and employ suitable mathematical abstractions to capture an accurate conceptual description of the process of modelling engineering systems.

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“…We use dependent types to represent not only element type on an array but also the array's rank and shape vector. In the presence of unbounded recursion, type checking of a programming language with dependent types is undecidable [7]. To make type checking decidable, our approach resembles an indexed type system [8,9] that only allows types to depend on compile time terms of a specific index language, on which constraints may be resolved statically.…”

Section: Typesmentioning

confidence: 99%

Descriptor-Free Representation of Arrays with Dependent Types

Trojahner

Grelck

2011

Implementation and Application of Functional Languages

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Abstract. Besides element type and values, a multidimensional array is characterized by the number of axes (rank) and their respective lengths (shape vector). Both properties are essential to do bound checking and to compute linear offsets into heap memory at run time. In order to have an array's rank and shape available at any time during program execution both are typically kept in an array descriptor that is maintained at run time in addition to the array itself. In this paper, we propose a different approach: we treat array rank and shape as first-class citizens themselves. Firstly, we use dependent types to reflect structural properties of arrays in the type system. Secondly, we annotate a program with the array explicit array properties wherever necessary. This choice not only renders implicit run time array descriptors obsolete, but exposing all rank and shape computations explicitly in intermediate code also allows us to perform extensive compile time optimisation on them. We have implemented the proposed approach in our experimental array language Qube; preliminary experimental results indicate the suitability of the proposed approach.

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An Introduction to Dependent Type Theory

Cited by 7 publications

References 92 publications

Pure patterns type systems

Pure patterns type systems

Conceptual modelling: Towards detecting modelling errors in engineering applications

Descriptor-Free Representation of Arrays with Dependent Types

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