Semantics of Type Theory: Correctness, Completeness and Independence Results
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 specification. By translating a problem specification into a proposition in constructive logic, one can go one step further: the effectiveness and unifonnity of a constructive 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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Semantics of Type Theory: Correctness, Completeness and Independence Results
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 specification. By translating a problem specification into a proposition in constructive logic, one can go one step further: the effectiveness and unifonnity of a constructive 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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Semantics of Type Theory: Correctness, Completeness and Independence Results
299
Semantics of Type Theory: Correctness, Completeness and Independence Results
299Paperback(Softcover reprint of the original 1st ed. 1991)
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Product Details
| ISBN-13: | 9781461267577 |
|---|---|
| Publisher: | Birkh�user Boston |
| Publication date: | 10/29/2012 |
| Series: | Progress in Theoretical Computer Science |
| Edition description: | Softcover reprint of the original 1st ed. 1991 |
| Pages: | 299 |
| Product dimensions: | 6.10(w) x 9.25(h) x 0.03(d) |
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