M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... m : m m 1 maps i 1 to (i) 1 for any 0 i m 1. We leave the definition of , and all other calculations, as a long(ish) Exercise. 4.5 Where to now There are a number of books which cover basic category theory. For a short and gentle introduction, see [29] For a longer first text see [19]. Both of these books are intended for computer scientists. The original and recommended general reference for category theory is [25] which was written for mathematicians. A very concise and fast paced introduction can be found in [21] which also covers the theory of allegories (which, roughly, ....

M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.

....serif font while multiple letter functors are written in normal italic font. just to the extent that is su#cient for our purposes. Then we will show how these basic building blocks help us to model the concept of datatypes. For a more complete account of category theory, the reader is directed to [17, 8]. The second task is to generalise from functions to relations. The inverse of a function is not necessarily a function. One of the ways to formally talk about inverses is to generalise to relations, of which functions are a special case. While the semantics of Haskell is built upon functions ....

....with Minimum Height Given is a list of trees. The task is to combine them into a single tree, retaining the left to right order of the subtrees. How can we make the height of the resulting tree as small as possible Figure 5. 1 illustrates one such tree, of height 11, for given subtrees of heights [2, 9, 8, 3, 6, 9]. As the actual content of the subtrees is not important, we can think of them simply as numbers representing the heights. The problem is therefore again one of turning a list of numbers to a tree. A linear time algorithm to this problem has been proposed in [14] Here we will demonstrate how a ....

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M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1985.

....order functions, and we ought to address these at least to the extent that the above derivation can be fully justi ed. Our starting point is the standard semantics of expressions in cartesian closed categories. It will be useful if the reader has some knowledge of elementary category theory [6]. For the purpose of this paper, one can think of a category as a universe of typed speci cations: it consists of objects (types) and arrows (speci cations or programs) Each arrow h has a source type A and a target type B : we write h : A B to indicate this type information. For each object A ....

M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.

....order functions, and we ought to address these at least to the extent that the above derivation can be fully justified. Our starting point is the standard semantics of expressions in cartesian closed categories. It will be useful if the reader has some knowledge of elementary category theory [6]. For the purpose of this paper, one can think of a category as a universe of typed specifications: it consists of objects (types) and arrows (specifications or programs) Each arrow h has a source type A and a target type B : we write h : A B to indicate this type information. For each object ....

M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.

....from the context, or irrelevant. Similar notation will be used for other sequences below, of types, etc. If F : m then D n hGi D m F D is their composite. If F : m 1 is a functor then m F : m represents the functor whose action on the tuple X yields the initial F (X; algebra (e.g. [BW90]) F (X ; m F (X) m F (X) used to nd minimal solutions of recursive domain equations. These constructions motivate the choice of functors in the following description of the raw syntax for functors, types and type schema. F; G : X j C j m i j F hGi n j m F : X j F ....

M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.

.... determine which is the most suitable for programming the Product Model [Rossiter95, Nelson94] The product model is a framework for object based databases, intended to provide a formal basis for the object relational database model [Stonebraker94] It uses standard constructs from category theory [Barr90], based on multi level mappings and products. The languages we are going to look at are: DAPLEX, Gofer (as one of many candidate functional programming languages) Prolog, Lisp (as a lambda calculus) C and Pascal. These languages form a relatively wide cross section of the programming paradigms ....

....object oriented features, and Scheme [Hanson91a, 91b] The expressiveness of the lambda calculus should enable any language such as Lisp to support categorical constructs, although this would usually involve the rewriting of any formalism into unnatural lambda calculus concepts. In Barr and Wells [Barr90], they briefly show a formalism for mapping between cartesian closed categories and the typed lambda calculus, although our categorical model would require more than a lambda calculus representation for cartesian closed categories. Most dialects of Lisp, including Common Lisp, support ....

Barr M, Wells C. Category Theory for Computing Science. Prentice Hall. International Series in Computer Science, 1990.

....context, or irrelevant. Similar notation will be used for other sequences below, of types, etc. If F : m then D n hGi D m F D is their composite. If F : m 1 is a functor then m F : m represents the functor whose action on the tuple X yields the initial F (X; Gamma) algebra (e.g. [BW90]) F (X; m F (X) m F (X) used to find minimal solutions of recursive domain equations. These constructions motivate the choice of functors in the following description of the raw syntax for functors, types and type schema. F; G : X j C j Pi m i j F hGi n j m F : X j ....

M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.

.... supplied by the programmer, e.g. Jon95] Perhaps surprisingly, shape theory yields a type free implementation based on a completely uniform method for locating data within a shape [BJM96] This parametric shape polymorphism is captured in a type system based on (categorical) functors, see, e.g. [BW90]) which extends the standard Hindley Milner type system. The shape polymorphic operations are examples of shapely operations, i.e. the shape of the output is a function of the shape of the input alone, without regard for the data. When a program is built from shapely operations then static shape ....

M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.

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M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.

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M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.

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M. Barr and C. Wells. Category Theory for Computing Science (second edition). International series in computer science. Prentice Hall, 1996.

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M. Barr and C. Wells. Category Theory for Computing Science (second edition). International series in computer science. Prentice Hall, 1996.

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M. Barr and C. Wells. Category Theory for Computing Science. International Series in Computer Science. Prentice Hall, 1990.

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