R.A.G. Seely, 1987. Categorical semantics for higher-order polymorphic lambda calculus. Journal of Symbolic Logic, 52:4.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in the simply typed lambda calculus by adding a restriction that elements of polymorphic types preserve relations. To exhibit any advantage of a parametric model, we should consider quanti ed types, as in System F. 61 One description of what makes a model of System F is that of PLcategories [See87] Brie y, a PL category to model System F is an indexed category A: A CCCat where the base category A has typing contexts of System F as objects and type substitutions as arrows between them. The target category CCCat is the category of Cartesian closed categories and functors which ....

....and squares. System P judgments are parameterized by indeterminates, which did not appear in System F. The interpretation of System P judgments will be parameterized by points corresponding to the indeterminate context. Since System F embeds into System P, this model will produce a PLcategory [See87] that is, a model of System F. Recalling Ma Reynolds de ned a parametric model of System F [MR92] to be a pair of PL categories discussed in section 3.5, the model produced in the current section will be the edge PL category over the model of System F using G as described in section 3.3. ....

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R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus. J. Symbolic Logic, 52(4):Dec, 969-989 1987.

.... 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 ....

R. A. G. Seely. Categorical semantics for higher order polymorphic -calculus. Journal of Symbolic Logic, 52(4):969--989, 1987.

....polymorphism can live only inside programs. At present, the best bet for a corresponding semantics in terms of compatibility with the semantics developed so far is the set theoretic Henkin style semantics of [5] which has been shown to be in good correspondence to the categorical semantics of [35] in [19] We have sketched how tool support for HasCasl can be obtained by encoding it into other logics. This needs further elaboration and implementation. Our aim is, then, to integrate these tools into the Maya environment [2] which provides a management of proof obligations for structured ....

R. A. G. Seely, Categorical semantics for higher order polymorphic lambda calclus, J. Symbolic Logic 52 (1987), 969-989.

....theory, we choose a much more relaxed style of presentation than would be usual in a research paper. In this vein, here we discuss only the basic framework for simple types and for implicit polymorphism. The extension to explicit polymorphism should be routine to readers thoroughly familiar with [Pit87, See87, Gir86, Wad89]. Note to specialists: the extension to second order polymorphism differs significantly from second order logical relations proposed by Mitchell and Meyer [MM85] we shall take this up elsewhere. It is a pleasure to acknowledge the collaboration with Samson Abramsky and Philip Wadler in the ....

....n C are basically just n ary functions from the set (or class) Obj C to Obj C. In order to account for type substitution, these functor categories must fit together as n varies. A proper general categorical framework for this motivating example is based on indexed or fibred categories, see [See87, Pit87, Mog91]. This framework, discussed below, generalizes readily to explicit polymorphism. In addition, a special case of this categorical view is a construction on Henkin models described, e.g. in [Mit90] 7.1 Categorical models of implicit polymorphism If we consider the type expressions apart from ....

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R.A.G. Seely. Categorical semantics for higher order polymorphic lambda calculus. Journal of Symbolic Logic, 52:969--989, 1987.

.... type abstraction and type application are not defined explicitly in the syntax of terms, this form of polymorphism is called implicit polymorphism and the corresponding categorical models are called iml categories [MSd93] These categorical models are a simpler form of the hyperdoctrine models [Pit87, See87] of the Girard Reynolds polymorphic lambda calculus. To keep this paper self contained, the definitions of iml category and iml representation are given in Section 2. The starting point of this work is the category theoretic technique of sconing or glueing described in [FrS90, Laf88, MaR91, ....

R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus. Journal of Symbolic Logic, 52(2), 1987.

....H, a model of the second order calculus IpC 2 . To explain further, we must describe what is needed to specify such a model. The particular notion of model we will use is the specialization from categories to partial orders of the notion of model of the second order lambda calculus described in [10, 12]. First note that since the notion of Heyting algebra is algebraic, it makes sense to speak of a Heyting algebra object, U , in any category C with finite products: such an object comes equipped with morphisms ; 1 U ; U 2 U U making various diagrams (derived from the defining ....

R. A. G. Seely, Categorical Semantics for Higher Order Polymorphic Lambda Calculus, Jour. Symbolic Logic 52(1987) 969--989.

....for variable elimination. In [BMM] environment and combinatoric models are introduced. In the context of datatypes, frame models which assign sets to types might be of greater use. Because we are using categorical constructions, it seems to be better to consider categorical models as described in [See87] and [Pit87] based on indexed categories. A new proposal is Functorial Polymorphism (see [BFSS90] where the semantic of polymorphic terms is explained by dinatural transformations. This concept seems to be particularly useful in the context of inductive types, however it is not clear to me, how ....

....formally. To do this it is necessary to explain the Wraith s construction in arbitrary models. Clearly this is possible but it seems a bit awkward to analyze categorical constructions in terms of set theoretic models. So it might be better to use Seely s and Pitt s categorical model definitions ([See87], Pit87] 2. In the light of functorial polymorphism it might be possible to prove the conjecture as a simple corollary of the theorems proven in [BFSS90] However to do this, a better understanding of this construction is required. It might also be preferably to prove an equivalent theorem ....

R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus. The Journal of Symbolic Logic, 52(4), 1987.

.... x: fx: j g: P = e: E: E: P = e: e 2 E: hE; ei ; eval: prop = t 2 P[ moreover, the interpretation of function symbols representing logical constants and predicate symbols must be consistent (see [See87]) e.g. the interpretation of 2 f ; must satisfy (h 1 ; 2 i ; t = 1 t 2 t for every 1 ; 2 : A (where in the rhs is meet in P [A] 3.3 Internal semantics The internal semantics of formulas mimics the set theoretic one by using the categorical analogue ....

R.A.G. Seely. Categorical semantics for higher order polymorphic lambda calculus. Journal of Symbolic Logic, 52(2), 1987.

....must be able to map bers down to the base (see [See84] The general de nition of indexed category is fairly complicated, since it involves the notion of canonical isomorphism. However, for representing languages it is more appropriate to use a stricter de nition of B indexed category (e.g. see [See84, See87]) namely a functor from B op to Cat, where B is a small category and Cat is the category of small categories and functors. De nition 2.1 Given a small category B, the 2 category ICat(B) of B indexed categories is de ned as follows: an object (indexed category) is a functor C: B op Cat ....

.... the approach based on locally cartesian closed categories, they give a general category theoretic understanding of dependent types (see Section 6) PL Categories model the higher order lambda calculus, or equivalently the proof theory of higher order intuitionistic propositional calculus (see [See87]) They are hyperdoctrines with an object 2 B (the type of propositions) s.t. the set of objects of C[X] is B(X; 2 and for any X 2 B a distinguished exponential X (the type of predicates over X) Monads can be used to model notions of computation (see [Mog89b, Mogar] Computational types ....

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R.A.G. Seely. Categorical semantics for higher order polymorphic lambda calculus. Journal of Symbolic Logic, 52(2), 1987.

....version claimed in that paper is still open. There is an immediate corollary of the above theorems. If we specialize C to be the ccc freely generated by L, that is, the syntactical term model of typed 2 Indeed, we actually obtain a fibred ccc or hyperdoctrine, ignoring the indexed adjoints [25, 28] 6 lambda calculus with type variables, qua ccc (cf. 19] p. 77) then equality means provable equality in L . In this setting, it follows that the syntactic families ktk above are provably dinatural. Thus, at the linguisitic level, arbitrary terms t, when validly typed, provably satisfy ....

R. A. G. Seely. Categorical Semantics for Higher Order Polymorphic Lambda Calculus. J. Symb. Logic 52(1987), pp. 969-989.

....and [2] But a non trivial example of this kind of structure is not possible in the topos Set: simple cardinality considerations show that any such U would have to contain only sets with at most one element. The categorical style models, P, of polymorphism considered in [8] and before that in [12]) are in particular K models in the sense of Reynolds and Plotkin where K = P(1; U) is the ccc of (denotations of) closed types and terms in the model P. The construction of [8] results in a certain topos E derived from P, containing K as a full sub ccc (and with other properties besides: one ....

....individual variable x oe in x oe : t; all other occurences of variables are free. A type or term with no free type variables will be called (type )closed. A description of a categorical semantics of these polymorphic types and terms based upon Lawvere s notion of hyperdoctrine is given in [12] (for the higher order calculus) and in some detail in [8] In this semantics fi and j conversion hold for both kinds of abstraction ( and ) In [11] an environment style semantics is given, which is intentionally quite weak (it satisfies fi and j conversion for abstraction and a limited form of ....

R. A. G. Seely, Categorical semantics for higher order polymorphic lambda calculus, Jour. Symbolic Logic 52(1987) 969--989.

....as representing the official policies, either expressed or implied, of the U.S. Government. 2 In the present paper, we will recast these ideas in a general category theoretic setting, using the framework of Cartesian closed categories for the simply typed case and the framework of PL categories [9, 10, 11] for the polymorphic case. Abstraction or logical relations theorems, parametric polymorphism, and related topics have been explored by a variety of researchers. A comparison with their work is given in the final section of this paper. We use the category theoretic notations of [8] except that we ....

.... 2 Omega n [ Omega n is a type or type assignment of the first order typed lambda calculus and 2 (ObK) n , R n ( J) K n ) J : 3 The Second Order Case We will develop the semantics of the polymorphic typed lambda calculus in the framework of PL categories [9, 10, 11]. To ease the introduction of this highly abstract framework, we proceed in two stages: Before defining the full fledged PL categories that are needed for the second order case, we define pre PL categories, which give a more abstract 15 semantics to the simply typed lambda calculus with type ....

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Seely, R. A. G. Categorical Semantics for Higher Order Polymorphic Lambda Calculus. Journal of Symbolic Logic, vol. 52 (1987), pp. 969--989.

.... environment style models, such as Henkin models for the simply typed lambda calculus [21] or Bruce Meyer models for polymorphism [10] and categorical models, such as the interpretation of the simply typed calculus in a cartesian closed category [33] or of the polymorphic calculus in a PL category [56]. Environment style models are typically non strict, in the sense that a function type oe is interpreted as a subset of the set of functions from oe to . On the other hand, categorical models are always strict. Reynolds has shown that there are no strict set theoretic models of the ....

....calculus. These models are non strict, in the sense that a function type oe is interpreted as a subset of the set of functions from oe to , and similarly a universal type 8ff: is interpreted as a subset of an infinite product Q oe [oe=ff] 2. Categorical models, introduced by Seely [56], are based on general principles for the interpretation of quantifiers in categorical hyperdoctrines. Seely s PL categories are a canonical extension of the ccc interpretation of the simply typed lambda calculus. These interpretations are strict, in the sense that both function types and ....

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R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus. The Journal of Symbolic Logic, 52(4), Dec. 1987.

....structure correspond to P preserving functors from T to C, that is, models of T in C are objects in the functor category [T; C] P . According to this view, HML theories correspond to special sorts of (split) fibrations, which we call categories; they are a mild generalization of PL categories ([See87]) Seely s categorical version of the higher order polymorphic calculus. Moreover, one expects a correspondence between structure preserving maps between categories and HML translations. Definition 3.1 A category T is a fibred CCC over a CCC base, which admits universal quantification along ....

R.A.G. Seely. Categorical semantics for higher order polymorphic lambda calculus. Journal of Symbolic Logic, 52(2), 1987.

....version claimed in that paper is still open. There is an immediate corollary of the above theorems. If we specialize C to be the ccc freely generated by L, that is, the syntactical term model of typed 2 Indeed, we actually obtain a fibred ccc or hyperdoctrine, ignoring the indexed adjoints [25, 28] lambda calculus with type variables, qua ccc (cf. 19] p. 77) then equality means provable equality in L . In this setting, it follows that the syntactic families ktk above are provably dinatural. Thus, at the linguisitic level, arbitrary terms t, when validly typed, provably satisfy ....

R. A. G. Seely. Categorical Semantics for Higher Order Polymorphic Lambda Calculus. J. Symb. Logic 52(1987), pp. 969-989.

....elsewhere. 2 R. ROSEBRUGH R. J. WOOD The theory of families we use is the theory of indexed categories as studied by Par e and Schumacher [13] Indexed categories are a widely used categorical tool, but have only begun to be explicitly used in theoretical computer science relatively recently [6,7,12,16]. The relational algebra of relational database theory involves operations which are set theoretic and other operations which can be defined by a language involving only constants, variables of domain or relation type and equality. An objective of this article is to construct the relational ....

R.A.G. Seely. Categorical semantics for higher order polymorphic lambda calculus. Journal of Symbolic Logic, 52:4, 1987.

....establish s.n. of say, System F or even better, the Calculus of Constructions This approach hinges upon two things: ffl there is a realizability category C with the untyped s.n. terms as realisers, ffl this category C is a model of the type theory in question. We know from the works of Seely [See87] Pitts [Pit87] and Hyland [HP89] that the categorytheoretic interpretation of such sophisticated type theories as System F or the Calculus of Constructions places heavy demands on the structure of categories. For example, in the case of System F, we essentially need a cloven fibration E #p B ....

R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus. J. Symb. Logic, 52:969--989, 1987.

....model of system R. This extension might be possible for any parametric model of system F , but we carry it out for the modified per model of Bainbridge et al. Along the way, we give a precise and general reconstruction of this per model. We present it as a categorical model in the sense of Seely [See87]. The next two sections introduce the syntax of system R and the necessary categories of pers and relations. Section 4 defines two parametric semantics of system F ; then section 5 extends one of these semantics to system R. 2 System R This section is an introduction to system R, adapted from ....

....terms (that is, a semantics based on erasures) to a typed semantics. The typed semantics is presented as a categorical model Categorical models of system F are based on the quantifiers as adjoints paradigm, which goes back to Lawvere [Law69] Seely has defined them under the name of PL categories [See87]. PL categories. PL categories are an algebraic generalization of the models of simply typed calculus in a bi dimensional universe of cartesian closed categories indexed over a global category. PL categories are sometimes referred to as external models of F in contrast with the internal ones ....

R.A.G. Seely. Categorical semantics for higher order polymorphic lambda calculus. The Journal of Symbolic Logic, 52(4):969--989, December 1987.

....and hyperdoctrines. Hyperdoctrines have, until recently, not received a great deal of attention, the main arena for categorical logic being elementary toposes. There has, however, been a surge of interest starting with the recent categorical description of models of the polymorphic lambda calculus [24]. There is also a recent trend to using more general fibred categories in describing dependent type systems, see for example the recent papers of Hyland and Pitts [7] Jacobs [8] and Pavlovi c [16] Recall that constraint systems are given by a first order language interpreted over some structure ....

R. A. G. Seely. Categorical semantics for higher-order polymorphic lambda calculus. J. Symb. Logic, 52(4):969--989, 1987.

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R.A.G. Seely, 1987. Categorical semantics for higher-order polymorphic lambda calculus. Journal of Symbolic Logic, 52:4.

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R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calclus. Journal of Symbolic Logic, 52, December 1987.

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Seely, R. A. G. Categorical Semantics for Higher Order Polymorphic Lambda Calculus. Journal of Symbolic Logic, vol. 52 (1987), pp. 969--989.

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R.A.G. Seely, Categorical semantics for higher order polymorphic lambda calculus. J. Symbolic Logic, 52, pp 969--989, 1987.

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Seely, R. [1987] "Categorical semantics for higher order polymorphic lambda calculus", JSL , vol. 52, n.

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R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus. J. Symb. Logic, 52:969--989, 1987.

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