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Semantics of Type Theory

Correctness, Completeness and Independence Results

  • Book
  • © 1991

Overview

Authors:
  1. Thomas Streicher

    1. Fakultat für Mathematik und Informatik, Universität Passau, Passau, Germany

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Part of the book series: Progress in Theoretical Computer Science (PTCS)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xii
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  2. Introduction

    • Thomas Streicher
    Pages 1-42

  3. Contextual Categories and Categorical Semantics of Dependent Types

    • Thomas Streicher
    Pages 43-111

  4. Models for the Calculus of Constructions and Its Extensions

    • Thomas Streicher
    Pages 112-155

  5. Correctness of the Interpretation of the Calculus of Constructions in Doctrines of Constructions

    • Thomas Streicher
    Pages 156-220

  6. The Term Model of the Calculus of Constructions and Its Metamathematical Applications

    • Thomas Streicher
    Pages 221-264

  7. Related Work, Extensions and Directions of Further Investigations

    • Thomas Streicher
    Pages 265-280

  8. Back Matter

    Pages 281-299
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Keywords

About this book

Typing plays an important role in software development. Types can be consid­ ered as weak specifications of programs and checking that a program is of a certain type provides a verification that a program satisfies such a weak speci­ fication. By translating a problem specification into a proposition in constructive logic, one can go one step further: the effectiveness and unifonnity of a con­ structive proof allows us to extract a program from a proof of this proposition. Thus by the "proposition-as-types" paradigm one obtains types whose elements are considered as proofs. Each of these proofs contains a program correct w.r.t. the given problem specification. This opens the way for a coherent approach to the derivation of provably correct programs. These features have led to a "typeful" programming style where the classi­ cal typing concepts such as records or (static) arrays are enhanced by polymor­ phic and dependent types in such a way that the types themselves get a complex mathematical structure. Systems such as Coquand and Huet's Calculus of Con­ structions are calculi for computing within extended type systems and provide a basis for a deduction oriented mathematical foundation of programming. On the other hand, the computational power and the expressive (impred­ icativity !) of these systems makes it difficult to define appropriate semantics.

Authors and Affiliations

  • Fakultat für Mathematik und Informatik, Universität Passau, Passau, Germany

    Thomas Streicher

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