{ "item_title" : "Semantics of Type Theory", "item_author" : [" T. Streicher "], "item_description" : "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.", "item_img_path" : "https://covers2.booksamillion.com/covers/bam/1/46/126/757/1461267579_b.jpg", "price_data" : { "retail_price" : "99.99", "online_price" : "99.99", "our_price" : "99.99", "club_price" : "99.99", "savings_pct" : "0", "savings_amt" : "0.00", "club_savings_pct" : "0", "club_savings_amt" : "0.00", "discount_pct" : "10", "store_price" : "" } }

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