E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. In G. Huet, editor, Logical Foundations of Functional Programming, pages 315--330. Addison-Wesley, Reading, MA, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....f ( x] A ) k (x) B . Such a partial recursive function is said to realize f . The re exive graph category PER has the category of PERs over IN and PER morphisms for the vertex category. The edge category of PER is motivated by the relational parametricity identi ed by Bainbridge et.al. BFSS90] and used in [BAC95, PA93] An edge R: A B is a relation between natural numbers satisfying the following. An nRm mBm = n Such a relation R is called a saturated relation. A square from an edge R: A B to R : A B is a pair hf; gi where f : A A g: B B are ....

.... are of some particular polymorphic types in his paper Theorems for Free [Wad89] When Reynolds introduced his parametric model [Rey83] he included descriptions of the interpretation of some polymorphic types, later generalized in [RP93] Such descriptions, later dubbed representation results [BFSS90, BAC95, OT95] e ectively state that the interpretation of these polymorphic types contain only the intuitively uniform operations. We show that models of polymorphism in well pointed parametricity settings satisfy similar representation results. We approach the issue of representation results ....

E. S. Bainbridge, P. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. Theoretical Comput. Sci., 70:35-64, 1990.

.... structure used to interpret them is particularly clear; ii) there are a large number of naturally occurring categorical models of considerable practical interest which have been studied extensively and, iii) categorical models may be generalised to provide semantics for more complex calculi [3,44,64,74] while this is not always the case for other approaches [69] Typically these categorical models consist of a category C together with a map interpreting types as objects of C and term judgements as morphisms of C. The various type constructors and their associated introduction and elimination ....

E. S. Bainbridge, A. Scedrov, P. Freyd, and P. Scott. Functorial polymorphism. In G. Huet, editor, Logical Foundations of Functional Programming, chapter 14, pages 315--327. Addison Wesley, 1990.

....in Span, edge morphisms (f 0 ; fw ; f 1 ) have additional witness information in fw . So, Span is not relational. Other structures used for modelling parametricity, such as the categories REL(K;B;F ) of Ma and Reynolds [10] and the category of pers with saturated relations of Bainbridge at al. [1] can also be described as reflexive graphs. The following property is satisfied by Rel, Span and most reflexive graphs of interest: Definition A reflexive graph C is said to satisfy the identity condition if the functor I is full. 8] Note that the functor I is already faithful for any reflexive ....

E. S. Bainbridge, P. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. Theoretical Comput. Sci., 70:35--64, 1990.

....to deal with polymorphism is not something which can easily be remedied, since giving a semantics to polymorphic languages is rather harder than for simply typed languages. In fact, the development of a good semantic model for polymorphism is still very much the subject of active research ( Fre89, BFSS90, AMSW90, AP90] For languages using Hindley Milner style polymorphism, there is one rather crude way of adapting analyses intended for simply typed languages. That is to resolve a polymorphic function definition into the set of its monomorphic instances which are actually used in a given ....

E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. In G. Huet, editor, Logical Foundations of Functional Programming. Addison Wesley, 1990.

.... Girard, Taylor, and Lafont [GLT89] and by Leivant [Lei90] The representation of algebraic data types in polymorphic lambda calculus was proposed by Bohm and Berarducci [BB85] Reynolds s parametricity has been further explored by Reynolds [Rey84, RP90, MR91] Bainbridge, Freyd, Scedrov and Scott [BFSS90], Hasegawa [Has91] Pitts [Pit87, Pit89, Pit98] and Wadler [Wad89, Wad91] among others. Formulations of the abstraction theorem in terms of logics have been examined by Mairson [Mai91] and (in various combinations) by Abadi, Cardelli, Curien, and Plotkin [ACC93, PA93, PAC94] A number of the ....

.... of the abstraction theorem in terms of logics have been examined by Mairson [Mai91] and (in various combinations) by Abadi, Cardelli, Curien, and Plotkin [ACC93, PA93, PAC94] A number of the works cited above have observed some connection between parametricity and algebraic data types [RP90, Has91, BFSS90, ACC93, PA93, PAC94]. What is new here is that we don t require a semantic characterization of parametricity, and we don t require a specialized logic. For instance, Reynolds and Plotkin [RP90] use a categorical model, where the extra constraint posed by parametricity is replaced by the existence of certain ....

E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott, Functorial polymorphism, in G. Huet, editor, Logical Foundations of Functional Programming, pp. 315--330, Addison-Wesley, 1990.

....semantics and category theory is now well established. Two of the strongest examples of this interaction are the representation of function types as exponential objects in a cartesian closed category (Lambek Scott, 1986) and the description of polymorphic terms as natural transformations (e.g. (Bainbridge et al. 1990)) For example, the operation of appending lists can be represented as a natural transformation L L ) L where L : D D is the list functor on some category D. Of course, these natural transformations must have associated functors for their domain and codomain. System F supports a notion of ....

Bainbridge, E.S., Freyd, P.J., Scedrov, A., & Scott, P.J. (1990). Functorial polymorphism.

.... Girard, Taylor, and Lafont [GLT89] and by Leivant [Lei90] The representation of algebraic data types in polymorphic lambda calculus was proposed by Bohm and Berarducci [BB85] Reynolds s parametricity has been further explored by Reynolds [Rey84, RP90, MR91] Bainbridge, Freyd, Scedrov and Scott [BFSS90], Hasegawa [Has91] Pitts [Pit87, Pit89, Pit98] and Wadler [Wad89, Wad91] among others. Formulations of the abstraction theorem in terms of logics have been examined by Mairson [Mai91] and (in various cominations) by Abadi, Cardelli, Curien, and Plotkin [ACC93, PA93, PAC94] A number of the ....

.... Formulations of the abstraction theorem in terms of logics have been examined by Mairson [Mai91] and (in various cominations) by Abadi, Cardelli, Curien, and Plotkin [ACC93, PA93, PAC94] A number of the works cited above have observed some connection between parametricity and algebraic data types [RP90, Has91, BFSS90, ACC93, PA93, PAC94]. What is new here is that we don t require a semantic characterization of parametricity, and we don t require a specialized logic. For instance, Reynolds and Plotkin [RP90] use a categorical model, where the extra constraint posed by parametricity is replaced by the existence of certain ....

E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott, Functorial polymorphism, in G. Huet, editor, Logical Foundations of Functional Programming, pp. 315--330, AddisonWesley, 1990.

....the representation theorem, and a version similar to that used here appears in [Rey83] where it is called the abstraction theorem. Other versions include the logical relations of Mitchell and Meyer [MM85, Mit86] and the dinatural transformations of Bainbridge, Freyd, Girard, Scedrov, and Scott [BFSS87, FGSS88], from whom I have taken the name parametricity . So far as I am aware, all uses of parametricity to date have been general : they say something about possible implementations of the polymorphic lambda calculus (e.g. that the implementation is correct independent of the representation used) or ....

E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott, Functorial polymorphism. In G. Huet, editor, Logical Foundations of Functional Programming, Austin, Texas, 1987. Addison-Wesley, to appear.

....issue, discussed in Section 3.1, is that the equality relation of specifications must be general enough to be refined by implementations. To allow for equality relations to be refined in implementations, we define a parametric per semantics for IA . The basic ideas are from Bainbridge et al. [7]. See also [8] We adapt them to a predicative polymorphic context. A per E over a set X is a symmetric and transitive relation. It differs from an equivalence relation in that it need not be reflexive. The domain of E is defined by x 2 dom(E) x E x. Note that E reduces to a (total) ....

Bainbridge, E. S., Freyd, P., Scedrov, A., and Scott, P. J. Functorial polymorphism. Theoretical Comput. Sci. 70 (1990), 35--64.

....the type contructors has precluded a neat development of models. In particular, it is not known if there is a categorical parametric model. To our knowledge, apart from possibly a syntactic model in [8] the only mention of a parametric model was made by Plotkin in [14] as he says) adapted from [2] where they state that the standard interpretation of a polymorphic type satisfies [ 1X: T ] 8 : n] 2 P2PER [ T ] X=P) fi fi fi fi fi fi n [ T ] X=R) n; all relations R 9 = where T (X=R) is defined inductively for every relation R ae P 2Q as usual: i) the product of two ....

....1 is (the global sections of) C, so it is equivalent to the fibre on 1 of the standard indexed category P (C) on C in L. 3. 2 REMARK Applying this construction to the category M of modest sets, one gets a PL category for which the interpretation of the quantification over types is the one given in [2]. The argument requires an application of the uniformity principle. Note that in each fibre of P (P (M) the objects are pairs, so some adjustment is needed in order to connect the two presentations. 3.3 THEOREM If there are right Kan extensions for functors out of the categories D 0 and D into ....

S. Bainbridge, P. Freyd, A. Scedrov, and P. Scott. Functorial polymorphism. Theo. Comp. Sci., 70:35--64, 1990.

....are all continuous functions cannot be made complete. Since a continuous function need not be strict, its graph is not a complete relation in general. ffl The reflexive graph Per has per s over a partial combinatory algebra D as vertices, permorphisms as arrows, and saturated relations as edges (Bainbridge et al. 1990; Berlucci et al. 1995) A saturated relation R: A B is a relation R D Theta D such that A; R; B = R. Clearly, the graph of a morphism f : A B is a saturated relation. This gives a crg. ffl This example is suggested by Tennent. Let Pfn be the reflexive graph whose vertices are sets, ....

Bainbridge, E. S., Freyd, P., Scedrov, A., and Scott, P. J. (1990). Functorial polymorphism.

....of Computing, Imperial College, 180 Queen s Gate, London SW7 2BZ, U.K. Email: fsa,tpjg doc.ic. ac.uk in its first argument: Delta Delta] op Theta (we assume throughout that is cartesian closed) Several solutions to this problem have been suggested involving dinatural transformations [5], structors [9] section retraction pairs on domains etc. In this paper we pursue Reynolds idea of viewing a type as a relation [14] We give this a categorical formulation introducing the concepts of relators and transformations and arrive at characterising polymorphic functions as ....

E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. In G. Huet, editor, Logical Foundations of Functional Programming. Addison Wesley, 1990.

.... structure used to interpret them is particularly clear; ii) there are a large number of naturally occurring categorical models of considerable practical interest which have been studied extensively and, iii) categorical models may be generalised to provide semantics for more complex calculi [3,44,64,74] while this is not always the case for other approaches [69] Typically these categorical models consist of a category C together with a map interpreting types as objects of C and term judgements as morphisms of C. The various type constructors and their associated introduction and elimination ....

E. S. Bainbridge, A. Scedrov, P. Freyd, and P. Scott. Functorial polymorphism. In G. Huet, editor, Logical Foundations of Functional Programming, chapter 14, pages 315--327. Addison Wesley, 1990.

....Troelstra and Scott. The current understanding 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 ....

Bainbridge E., Freyd P., Scedrov A., P.J. Scott [1987] "Functorial Polymorphism" U. Texas Institute on Logical foundations of Functional Programming, Austin.

....relation is given on type parameters oe and , then (the interpretation of) X:M , applied to (the meaning of) oe and , should send related elements of oe and to related elements in the types of the outputs. Another meaning of the proper polymorphism of system F was given by Bainbridge et al. [BFSS90]. Consider x : X:N . Is it the case that x : X:N depends naturally on X , in the sense of natural transformations of Category Theory Indeed, natural transformations provide the core way to express uniformity on objects (as interpretation of types) in categories. Unfortunately, natural ....

.... dinatural transformations, yet another elegant categorical notion derived from tensor algebra and algebraic topology. The rub is that, in general, dinatural transformations do not compose, while terms do; however, the interpretation works fine (i.e. it is compositional) on relevant models (see [BFSS90, FGSS88, GSS]) On essentially similar lines, Freyd suggested the novel notion of structor in order to understand, categorically, the notion of uniformity inherent in second order terms. These attempts suggested brand new constructions and relevant mathematics, but seem still insufficient to fill the ....

E.S. Bainbridge, P.J. Freyd, A. Scedrov, P.J. Scott. "Functorial Polymorphism. " Theoretical Computer Science 70, pages 35--64, January 1990. Corrigendum in 71, page 431, April 1990.

....produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R N ThetaN ....

S. Bainbridge, P.J. Freyd, A. Scedrov, and P. Scott. Functorial polymorphism. Theo. Comp. Sci., 70:35--64, 1990.

....as A B = 8X( A)X) B)X) X) where X ranges over propositions. Notably Prawitz proved [32] that, in intuitionistic second order logic, implication and universal quantification can define all other connectives. For example, 9X:F (X) 8Y (8X(FX)Y ) Y ) A B = 8X( A)B)X) X) In [13] also see [2]) we proved Reynolds parametricity is a necessary and sufficient condition for these logical representations to have natural categorical meanings. For example, AB above is a coproduct of A and B if it is parametric and the converse is true if the relations are restricted to the graphs of ....

....the graphs of functions. Impredicativity of second order logic induces also inductive and coinductive types as initial and terminal fixed points. Although the first attempt of Reynolds to construct a parametric model in Set failed [35] now two models are known, that is, the parametric per model [2, 13] and the second order minimum model [29, 14] The aim of this paper lies on the same line. In order to explain it, let us consider 9X:F (X) 8Y (8X(F (X) Y ) Y ) and ask the question what is the idea behind this. Since 8X(A)X) is regarded to be :A intuitionistically, the representation of 9 ....

[Article contains additional citation context not shown here]

E. S. Bainbridge, P. J. Freyd, A. Scedrov and P. J. Scott, Functorial Polymorphism, Theoret. Comput. Sci. 70 (1990) 35--64; Corrigendum, 71 (1990) 431.

....all reasonable rules. For example, we might expect that a function in Root Nat be constant, but it need not be in the ideal semantics. A stronger semantics may be based on per models (e.g. Amadio, 1991; Cardone, 1989; Abadi and Plotkin, 1990) or, perhaps better, on parametric per models (e.g. (Bainbridge et al. 1990)) 32 For the sake of simplicity, we do not use pers in the body of this paper. Here we sketch the modifications necessary for obtaining a per semantics, and then discuss the result. As Amadio and Cardone, we take a metric approach. Finding a per semantics along the lines of (Abadi and Plotkin, ....

Bainbridge, E. S., Freyd, P. J., Scedrov, A. and Scott, P. J. 1990. Functorial polymorphism.

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Bainbridge E., Freyd P., Scedrov A., Scott P.J. [1990] "Functorial Polymorphism" Theoretical Comp. Sci. 70, 35-64.

No context found.

E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. In G. Huet, editor, Logical Foundations of Functional Programming, pages 315--330. Addison-Wesley, Reading, MA, 1990.

No context found.

E. S. Bainbridge, P. J. Freyd, A. Scedrov, and P. J. Scott. Functorial polymorphism. In G. Huet, editor, Logical Foundations of Functional Programming, pages 315--330. Addison-Wesley, Reading, MA, 1990.

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