Semantics of Type Theory: Correctness, Completeness and Independence Results [Paperback]

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.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.