T. Streicher

Semantics of Type Theory (eBook, PDF)

Correctness, Completeness and Independence Results

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Birkhäuser Boston
• Seitenzahl: 299
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461204336
• Artikelnr.: 44199148

Inhaltsangabe

1 Contextual Categories and Categorical Semantics of Dependent Types.- 2 Models for the Calculus of Constructions and Its Extensions.- 3 Correctness of the Interpretation of the Calculus of Constructions in Doctrines of Constructions.- 4 The Term Model of the Calculus of Constructions and Its Metamathematical Applications.- 5 Related Work, Extensions and Directions of Future Investigations.- Appendix Independence Proofs by Realizability Models.- References.