ASPERTI, A. and LONGO, G. Categories, Types and Structures: An Introduction to Category Theory for the Working Computer Scientist. Cambridge: MIT Press, 1991. 306p.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from category theory. Category theory ( 3] 21] 28] is a branch of mathematics that has been widely applied to specification in computer science. Examples of this application include: abstract data types [13] 14] semantics of programming languages [24] 25] and functional programming [1]. The techniques of category theory are graphical and based on a simple axiom set. They are highly valued for powerful semantic expressiveness. We make extensive use of the idea of logical data independence (see, for example, 27] Logical data independence supports views of systems data that ....

A. Asperti and G. Longo. Categories, Types and Structures: An introduction to category theory for the working computer scientist. MIT Press, 1991.

....all this structure is preserved by the faithful functor C oe C S . The above example of structure on C is illustrative. Exactly similar definitions can be given for a range of structures, including: ffl models of Classical (or Intuitionistic) Linear Logic including the additives and exponentials [10] ffl cartesian closed categories [15] ffl models of polymorphism [15] 2.1 Examples of Specification Structures In each case we specify the category C , the assignment of properties P to objects and the Hoare triple relation. 1. C = Set; PX = X; affgb j f(a) b: In this case, C S is the ....

....of coherence spaces and linear maps [20] 4. C = Set ; PX = fs : X j 8x 2 X:9n 2 :s(n) xg; sffgt j 9n 2 w:f ffi s t ffi n where n is the nth partial recursive function in some acceptable numbering [34] Then C S is the category of modest sets, seen as a full subcategory of Set [10]. 5. C = the category of SFP domains; PD = K Omega Gamma D) the compact open subsets of D) UffgV j U f Gamma1 (V ) This yields (part of) Domain Theory in Logical Form [2] the other part arising from the local lattice theoretic structure of the sets PD and its interaction with the ....

A. Asperti and G. Longo. Categories, Types and Structures : An introduction to category theory for the working computer scientist. Foundations of Computing Series. MIT Press, 1991.

....that C(A; B) fp(oe) j oe 2 C(A; B)g; The compact elements of C(A; B) correspond to the finite consistent sets of complete plays, where s consistent with t means that s u t has even length. It is then clear that, if we start from basic data types which are given as enumerated sets [4], then for each IA type T , C(1; T ] is an effectively presentable domain. Theorem 27 (Effective Presentability) The fully abstract model of Idealized Algol is effectively presentable. Example Consider the type com ) com. The interpretation of this type in C is isomorphic to the lazy ....

A. Asperti and G. Longo. Categories, Types and Structures : An introduction to category theory for the working computer scientist. Foundations of Computing Series. MIT Press, 1991. 37

....branch of mathematics renowned for its semantic power, its simple axiom set, and its use of graphical techniques. It has been widely used for specification in computer science in, for example, abstract data types [14] 15] semantics of programming languages [23] 24] and functional programming [1]. The authors and their coworkers have, over a number of years, been developing a semantic data modelling paradigm based on category theory. The categorical specification of information systems (see for example [16] 17] has been motivated by categorical universal algebra (a categorical ....

A. Asperti and G. Longon. Categories, Types and Structures: An introduction to category theory for the working computer scientist. MIT Press, 1991.

....just nds here retraction pairs where one would nd pairs of inverse isomorphisms in the extensional case. Very general de nitions of models of F have already been proposed in the literature 10 , mainly: the de nition of Bruce Meyer Mitchell [11] and various categorical de nitions (see e.g. [3], and [23] Because of the strong reactions of one referee it seems we have to be still more explicit about the reasons why we do not start from one of these de nitions. 9 In fact it only covers the universal retraction models which are based on closures or projections. A slight generalization ....

....Is also true of the categorical de nitions, of course, that they do produce models, and that they embody the deep structural reasons why this is true. 12 It is far less evident to check whether our models t a given well established categorical de nition of models. For example, in the case of [3] the answer might be yes since it seems that our object T erms could play the r ole of the object c 1 of [3] which is intended to be an object of all internal morphisms) using that each X ) Y is a substructure of T erms . 13 On the other hand it is plausible that our de nition can be turned ....

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A. Asperti and G. Longo, Categories, Types and Structures : an Introduction to Category Theory for the working Computer Scientist, Cambridge, Mass., MIT Press, 1991. 47

....representation can have operations from various classes defined for them. This provides the same functionality which is described as multiple inheritance in object oriented programming(Lochovsky 1986; Meyer 1988; Rumbaugh et al. 1991) but in the mathematically rigorous context of category theory(Asperti and Longo 1991; Barr and Wells 1990) and denotational semantics (Peyton Jones 1987; Stoy 1977) Algebras for specification are in our practice very small, similar to the traits in Larch (Guttag, Horning, and Wing 1985) They are combined, as one algebra can use other algebras previously defined. The methods ....

Asperti, A., and G. Longo. 1991. Categories, Types and Structures - An Introduction to Category Theory for the Working Computer Scientist. Edited by G. M. a. M. Albert, Foundations of Computing.

....with the impressive application to the full abstraction problem for PCF [1] However, if we restrict ourselves to intuitionistic linear logic, it is not difficult to give semantics by familiar category theoretic means. In the following, we briefly review the categorical semantics for linear logic [2]. In fact, the logic without contraction and weakening has been studied since the late 1960 s exactly because of its neat correspondence with closed monoidal categories. Roughly speaking, a category is a collection of a certain kind of structures and structure preserving maps between them. For ....

....monoidal category was first noticed by Lambek. The idea of a category with a self dual endofunctor, or autonomous category, was formulated by Barr. Finally the observation that the exponential is just a comonad is due to Seely. Our presentation is based on the exposition by Asperti and Longo [2]. We now give the precise definitions. Definition 6.1 Let C be a symmetric monoidal closed category. A functor F : C C is closed if for every pair of objects A; B 2 C there exists an arrow B A f AB FB FA CHAPTER 6. PHASE VALUED MODELS OF LINEAR SET THEORY 111 such that for every arrow A ....

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A. Asperti and G. Longo. Categories, Types and Structures: An Introduction to Category Theory for the Working Computer Scientist, The MIT Press, Cambridge, Massachusetts, 1991.

....to c and d are homomorphisms of F algebras. Somewhat confusingly, F substitutive relations are called F congruences in [Man76, RT94] 12. Much remains to be done 22 It can be easily shown that Sigma algebras are the F algebras for a particular functor F on the category of sets (see, e.g. [AL90, RT94]) Although the notion of F algebra is dual to that of F coalgebra, the category C F of F algebras is not dual to the category C F of F coalgebras. Informally speaking, this can be explained by the following two diagrams (of a homomorphism of F algebras and a homomorphism of F ....

A. Asperti and G. Longo. Categories, types and structures: An introduction to category theory for the working computer scientist. The MIT Press, 1990.

.... values sorts (types) use call instantiation formal parameters formal parameters for values formal parameters for sorts actual parameters values representable data types combination call of procedure within another abstraction use as a sub algebra within another algebra Category theory [2, 3, 14, 23] abstracts from individual values to sets of values (types, domains) Algebras group operations which are applied to the same data types. Axioms in the algebra define the properties (behavior) of these operations. Algebras are naturally parameterized in the types of the arguments the operations in ....

Asperti, A. and G. Longo, Categories, Types and Structures - An Introduction to Category Theory for the Working Computer Scientist. The MIT Press, Cambridge, MA (1991).

....since they are used in a signicant way during the completeness proof. We also make a rather detailed presentation of the interpretation of terms in our models, since it is not standard, even if it looks familiar, and since the existing presentations, in general much more complex (for example [3] or [10] are not directly usable. Finally, we would like to stress here that the lemma which justies that the interpretation is correct (Lemma 66) is very natural and has a straightforward proof, but that the fact that such a simple statement is possible deeply uses the fact that we do not ....

....F and its models. In this section, we brieAEy introduce system F ( 16] and its term model, the only fij complete model known up to now. Then we introduce a notion of models of system F , which is less general but much easier than those which can be found in the literature (for example [10] or [3]) and we build a class of complete models of F out of it. Our denition requires no more category theory than knowing what is a c.c.c. functors are not explicitly used here) It is abstract enough to t several dioeerent settings (for example all universal retraction models 8 , the class in ....

A. Asperti and G. Longo, Categories, Types and Structures : an Introduction to Category Theory for the working Computer Scientist, Cambridge, Mass., MIT Press, 1991.

....of this structure is preserved by the faithful functor C oe C S . The above example of structure on C is illustrative. Exactly similar definitions can be given for a range of structures, including: ffl models of classical (or intuitionistic) linear logic including the additives and exponentials [13] ffl cartesian closed categories [20] ffl models of polymorphism [20] 5 2.1 Examples of Specification Structures In each case we specify the category C , the assignment of properties P S to objects and the Hoare triple relation. 1) C = Set , P S X = X, affgb def , f(a) b. In this ....

.... spaces and linear maps [25] 5) C = Set ; P S X = fs : X j 8x 2 X:9n 2 :s(n) xg; sffgt def ,9n 2 w: f ffi s t ffi OE n where OE n is the nth partial recursive function in some acceptable numbering [45] Then C S is the category of modest sets, seen as a full subcategory of Set [13]. 6) C = the category of SFP domains; P S D = K D) the compact open subsets of D) UffgV def , U f Gamma1 (V ) This yields (part of) Domain Theory in Logical Form [3] the other part arising from the local lattice theoretic structure of the sets P S D and its interaction with the ....

A. Asperti and G. Longo. Categories, Types and Structures : An introduction to category theory for the working computer scientist. Foundations of Computing Series. MIT Press, 1991.

....relations which are amenable to this treatment. The extension of relation calculus to a function calculus is discussed here, linking two previously unconnected tools. The two tools are not as different and their conceptual merging is in category theory (Barr and Wells 1990, Herring et al. 1990, Asperti and Longo 1991, Walters 1991) Function 27 composition tables can be used similarly to relation composition tables; they show patterns which can then be succinctly formulated as rules. In this paper we applied a linguistic method based on prepositions to describe image schemata. A rich set of relations for 4 ....

A. Asperti and G. Longo (1991) Categories, Types and Structures - An Introduction to Category Theory for the Working Computer Scientist. The MIT Press, Cambridge, Mass.

....theory of early may be worth pursuing as an interesting development on the grounds of the concrete model below. Indeed, by some general categorical tools, one may even avoid to start with a model of type free lambda calculus, but this may require some technicalities from Category Theory (see [AL91] Thus we use here a model (D; Delta) of type free lambda calculus and a fundamental type structure out of it. We survey first the basic ideas for the construction. Later we specialize the general construction by starting out of a specific type free model which will yield a semantic for our ....

....(notation: n j A B f) Thus PER is a category where the identity map, in each type, is computed by (at least) the interpretation of the term x:x, i.e. the identity function on D. Theorem 4.3 PER is a CCC s. Proof. Hint, the proof is in several papers since [Sco76] in particular, in [AL91] The exponent object A B is defined by 8m; n: m(A B)n , 8p; q(pAq ) m Delta p)B(n Delta q) Products are defined by taking a coding of pairs of D into D, as given for example by the fact that D is a model of type free lambda calculus. 2 To clarify the construction, let us look more ....

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A. Asperti and G. Longo. Categories, Types and Structures: An Introduction to Category Theory for the Working Computer Scientist. MIT-Press, 1991.

....relations which are amenable to this treatment. The extension of relation calculus to a function calculus is discussed here, linking two previously unconnected tools. The two tools are not as different and their conceptual merging is in category theory (Barr and Wells 1990, Herring et al. 1990, Asperti and Longo 1991, Walters 1991) Function composition tables can be used similarly to relation composition tables; they show patterns which can then be succinctly formulated as rules. In this paper we applied a linguistic method based on prepositions to describe image schemata. We showed examples for large scale ....

A. Asperti and G. Longo (1991) Categories, Types and Structures - An Introduction to Category Theory for the Working Computer Scientist. The MIT Press, Cambridge, Mass.

....is just an instance of the rule (z) of extensionality. In summary: z b ) is compatible with CL, but not admissible; z b ) is admissible for CLb, but not derivable; z b ) is derivable in lbh. In [LM90] a categorical classification is given of the models of CL, CLb, lbh (see also [AL91] However, in spite of the clear categorical notions that characterize the three classes of models, a surprising historical remark can still be made: an example of a proper mathematical model of Combinatory Logic, CL, has been given only recently [DiGiHon93] much later than the models of ....

.... Eff, w Set and PER, can be built out of any (partial) combinatory algebra or (partial) model of Combinatory Logic or l calculus, even a very concrete one, with no reference to the syntax of l calculus, such as Scott s Pw model, say, or any reflexive object in a Cartesian Closed Category, AL91] 5.3 Definition: The category w Set has as objects: A , w Set iff A is a set and w A , i.e. is a relation in w A , s.t. A A n (n,A) write n A) morphisms : f w Set [ A , B ] iff f : A B and n A A p A A , n . p B f(A) notation : ....

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A. Asperti, G. Longo Categories, Types and Structures: an introduction to Category Theory for the working computer scientist. M.I.T.- Press, 1991

....relations which are amenable to this treatment. The extension of relation calculus to a function calculus is discussed here, linking two previously unconnected tools. The two tools are not as different and their conceptual merging is in category theory (Barr and Wells 1990; Herring et al. 1990; Asperti and Longo 1991; Walters 1991) Function composition tables can be used similarly to relation composition tables; they show patterns which can then be succinctly formulated as rules. Acknowledgments Numerous discussions with Werner Kuhn, David Mark and Andrea Rodriguez have contributed to our understanding of ....

A. Asperti and G. Longo (1991) Categories, Types and Structures - An Introduction to Category Theory for the Working Computer Scientist. The MIT Press, Cambridge, Mass.

....in the basic proof theoretic property of these calculi: the normalization (cut elimination) theorem. In the semantic interpretations, this essential type independence of computations is understood by the fact that the meaning of polymorphic functions is given by essentially constant functions (see [AL91]) It is clear, instead, that overloaded functions express computations which depend on types, as different codes may be applied on the basis of input types. This is so in various imperative as well as functional languages; our motivation, though, comes from considering overloading as a key ....

A. Asperti and G. Longo. Categories, Types and Structures: An Introduction to Category Theory for the Working Computer Scientist. MIT-Press, 1991.

....to suggest consistent extensions. This is one of the reasons for which we construct a specific (class of) model(s) instead of suggesting general definitions. These may be obtained by slight modifications of the work in [Bruce Longo 88] or, even better, by following the categorical approach in [Asperti Longo 91] Indeed, in the latter case, the invention of a general categorical meaning for subtyping and subkinds would be a relevant contribution. In this part, we first try to give the structural (and partly informal) meaning of kinds, types, and terms, as well as their crucial properties. The reader ....

.... framework the results in [Hyland 87] Pitts 87] Hyland Pitts 87] Carboni Freyd Scedrov 87] and [Bainbridge Freyd Scedrov Scott 87] The general treatment of models, as internal categories of categories with finite limits, which was suggested by Moggi, is given in [Asperti Martini 89] and [Asperti Longo 90]. The elegant presentation in [Meseguer 88] compares various approaches. We use here the fact that w Set is closed under products indexed over itself and, in particular, we use the completeness of PER as an internal category. The categorical products are exactly those naively defined below (to ....

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A.Asperti, G.Longo: Categories, types and structures: an introduction to category theory for the working computer scientist, M.I.T. Press, to appear.

....to suggest consistent extensions. This is one of the reasons for which we construct a specific (class of) model(s) instead of suggesting general definitions. These may be obtained by slight modifications of the work in [Bruce Longo 88] or, even better, by following the categorical approach in [Asperti Longo 91] Indeed, in the latter case, the invention of a general categorical meaning for subtyping and subkinds would be a relevant contribution. In this part, we first try to give the structural (and partly informal) meaning of kinds, types, and terms, as well as their crucial properties. The reader ....

.... framework the results in [Hyland 87] Pitts 87] Hyland Pitts 87] Carboni Freyd Scedrov 87] and [Bainbridge Freyd Scedrov Scott 87] The general treatment of models, as internal categories of categories with finite limits, which was suggested by Moggi, is given in [Asperti Martini 89] and [Asperti Longo 90]. The elegant presentation in [Meseguer 88] compares various approaches. We use here the fact that w Set is closed under products indexed over itself and, in particular, we use the completeness of PER as an internal category. The categorical products are exactly those naively defined below (to ....

[Article contains additional citation context not shown here]

A.Asperti, G.Longo: Categories, types and structures: an introduction to category theory for the working computer scientist, M.I.T. Press, to appear.

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Asperti A., Longo G. (1990) Categories, types and structures: an introduction to category theory for the working computer scientist, M.I.T. Press, to appear.

.... of the higher order types by quotient sets and internal categories, which we develop here, has been first suggested by Moggi and widely developped by several authors in Category Theory (Rosolini[1986] Hyland[1987] Hyland al[1987] Carboni al[1987] Bainbridge al[1987] Robinson[1989] Asperti Longo[1990]. The objects of the category PER below are equivalence relations on subsets of the natural numbers or partial equivalence relations (p.e.r. s) Morphisms are defined by Kleene s application: n . p is the result of the application of the n th partial recursive function to the number p . By n . ....

.... of the completeness properties of the internal category M , necessary to turn it into a model of construction, in Hyland[1987] Ehrhard[1988] and Robinson[1989] The definition of model for system F , by internal categories, originally due to Moggi, is described and related to other approaches in Asperti Longo[1990]. 8.10 Remark: There is another property of the right adjoint that has to be verified in order to guaranty that substitution is well behaved, namely the Beck Chevalley condition (see Hyland[1987] Hyland Pitts[1987] However, this condition is automatically true when quantification is restricted ....

Asperti A., Longo G. [1990] Categories, Types and Structures: an introduction to Category Theory for the working computer scientist, M.I.T. Press, to appear.

....be closed under products indexed over itself. All this gives a new structure for the variation and a strong closure property. The circularity of the impredicative definitions becomes then a theorem, the closure of certain categories under generalised products, whose origin is geometrical (see [Asperti Longo, 1991]) G. Longo 15 The only difficulty is that the construction cannot be done inside a classical Set Theory ( Reynolds, 1984] instead one needs an intuitionistic environment ( Pitts, 1987] Hyland, 1988] Longo, Moggi, 1991] Once again, but this is complicated, the geometrical symmetry ....

....environment) 12. The few technical notions in this section will not be used in the sequel: they are just examples of elementary connections between principles of proof and principles of construction. For more details on intuitionistic systems of types and Category Theory, see [Lambek Scott,1986] [Asperti Longo,1991]. 13. We could say the same about Girard s Linear Logic as its nature makes even Classical Linear Logic . constructive [Girard, 1991] 14. The research on the unshakeable certainties of Hilbert and Brouwer (see [Brouwer, 1927] has given us this century a very solid notion of mathematical ....

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Asperti A. & Longo G. (1991) Categories, Types and Structures: an introduction to Category Theory for the working computer scientist. M.I.T.- Press. (pp. 1--300).

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ASPERTI, A. and LONGO, G. Categories, Types and Structures: An Introduction to Category Theory for the Working Computer Scientist. Cambridge: MIT Press, 1991. 306p.

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Asperti, A. and Longo, L. Categories, Types and Categories, Types and Structures: an introduction to Category Theory for the working computer scientist. M.I.T.- Press, 1991.

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Andrea Asperti and Giuseppe Longo. Categories, Types and Structures : An Introduction to Category Theory for the Working Computer Scientist. MIT Press, 1991.

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