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Applications of Categories in Computer Science: Proceedings by M. P. Fourman, P. T. Johnstone, A. M. Pitts (Editors)

By M. P. Fourman, P. T. Johnstone, A. M. Pitts (Editors)

Purposes of type idea and comparable themes of arithmetic to laptop technology were a becoming zone in recent times. This e-book comprises chosen papers at the topic from the London Mathematical Society Symposium held on the collage of Durham in July 1991.

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We shall leave it as an exercise to show that this is equivalent to the other presentation. However, we must stress that cut elimination does not hold for the second presentation of two-tensor-poly categories; the amount of cut built into the rules ( R) and (0 L) is necessary to prove cut elimination. ) So a functor F:C —• D is a map Ob(C) —• Ob(T>) and a map M

The (appropriately adjusted) Imerge and split2 operations also form a computational pairing, and split2 o Imerge = id; similar properties hold for rmerge and split2. 4 Pairing, currying and uncurrying on algorithms Using the split operation of a computational pairing provides a way to combine a pair of algorithms into an algorithm on pairs. If split is taken to be cr, this is the standard way to form the product of two morphisms in the Kleisli category. , C) and C(—,B —> C). 4 Let C be a category with finite products, let (T, e,£, 7) be a computational comonad, and let split be a natural transformation from T ( - x - ) to T - x T - .

An element of D is computable iff the set of (indices of) its finite approximations is recursively enumerable. 40 BROOKES AND GEVA : COMPUTATIONAL COMONADS The functor T\ can be adapted to this category, by defining T\D for an effectively given domain D to be the computable increasing paths over D (equivalently, the computable continuous functions from VNat to Z), ordered pointwise). All of the auxiliary operations (e, 8, 7, and so on) are computable. Hence we obtain a category of effectively given domains and computable algorithms, and this category quotients onto the underlying category of effectively given domains and computable functions.

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