M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science, 146(1--2):5--23, July 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Y with the compact open topology. We may equip CGHaus with a structure of strict CS4 category as follows: Definition 52 ( Comonad in C(Haus) For every topological space X, the path space [ X over X is the disjoint sum l[xoCX [ o X, where the [ o X is the space of all continuous functions from [0, 1] to X such that (0) Xo, with the compact open topology. For every continuous function f: X Y, let [ f : X [ Y be the function mapping each C [ X to f o [ Y. The counit d maps every C [X to c(1) C X. The comultiplication s maps every [X to the map t (t (tt ) in [2 X. This comonad is ....

....produce a value at time 1. The counit d is the operator that extracts the final value of the process c as argument. Proposition 53. The construction ( d, s) of Definition 52 is a strict monoidal comohad on C(Haus, making it a strict CSJ category. Proof. First show that [xo X is Kelley. Since [0, 1] is compact, it is locally compact Hausdorff; it is then well known that the space of continuous functions from [0, 1] to X is Kelley: this is HomcGsaus( 0, 1] X) In general HomcGsaus(Y,X) is the kelleyfication of the space of continuous functions from Y to X, not the space itself. Since x0 ....

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Moez Alimohamed, A characterization of lambda definability in categorical models of implicit polymorphism, Theoretical Computer Science 146 (1995), no. 1-2, 5-23.

....the definable elements in any Henkin model [4] Although not emphasised in [4] relations of varying arity are powerful enough to characterise relative definability with respect to any given set of elements considered as constants. The full generality of the approach is demonstrated in Alimohamed [1], where such relations are used to characterise relative definability in an arbitrary cartesian closed category. In general, results about the pure simply typed calculus extend easily to analogous results for systems containing finite product types. This is not the case for finite coproduct ....

....relative definability in any bicartesian closed category in which the finite coproducts are stable (as is the case in Set) We do not know if the characterisation extends also to the non stable case. From the categorical point of view our results are best explained in terms of glueing [12, 1]. However, for this conference version of the paper, we keep our exposition elementary, in the hope that it will be accessible to most type theorists with some background in categorical semantics. It should be said that the research in this paper originated as part of a strategy conceived by the ....

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M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science, 146:5--23, 1995.

....speaking the completeness proof sketched out here is a mild variant of theirs. The work presented here, however, is not based on Henkin models, but on cartesian closed categories, and therefore the relevant comparison is the Alimohamed s characterization of lambda definability for ccc s in [2] (which, however, also deals with ML style polymorphism) The second major source is the work of Claudio Hermida in his thesis [3] where he explores the connection between logical predicates and hyperdoctrines defined over the semantic category. This provides a formal link between the theory of ....

M. Alimohamed. A Characterization of Lambda Definability in Categorical Models of Implicit Polymorphism. Theoretical Computer Science, 146:5--23, 1995.

....by a minor change of notation. From the definitions. 2 The modification to Jung and Tiuryn s result hinted at in the proof, together with Theorem 3 in [JT93] can be used to generalize Jung and Tiuryn s lambdadefinability result (their Theorem 5) to signatures containing term constants, cf. [Ali95]. 7 Pre logical Relations via Composition of Logical Relations Our weakening of the definition of logical relations may appear to be ad hoc, but for extensional structures it turns out to be the minimal weakening that is closed under composition. There are variants of this result for several ....

....that a corresponding weakening of the definition would lead to analogues of the results above, cf. PPS98] It is worth noting that Prop. 6. 2 links pre logical relations to KLRwVAs, which have a formulation in terms of pre sheaf categories and have been extended to cartesian closed categories in [Ali95]. 11 Conclusions and Directions for Future Work Our feeling is that by introducing the notion of pre logical relation we have, metaphorically and a little immodestly, removed a blind spot in the existing intuition of the use and scope of logical relations and related techniques. This is not to ....

M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science 146:5--23 (1995).

....semantic results on intuitionistic type theories. In particular, since Plotkin s work [38] a substantial study of characterizing the definability on models of the simply typed lambda calculus (and related typed languages such as PCF) using logical predicates has been carried out, see for instance [26, 37, 3, 17]. From the category theoretic point of view, it is known that a setting for logical predicates for the simply typed lambda calculus can be derived from the categorical glueing construction (also known as sconing and Freyd covering) on cartesian closed categories [29, 36] In terms of categorical ....

....its variants can be derived systematically from the glueing techniques (Example 3.6, 3.9, 3.11, 3.18 and 3.23) Then we are ready to introduce a notion of logical predicates for models of linear logic. The predicates we introduce are parameterized, in the same way as the Kripke logical relations [3]; the role of parameterization is essential in dealing with connectives of linear logic, especially the multiplicatives and modalities, roughly by the following reason. Suppose that we have a predicate P b A b for each base type b, where A s is a set in which the closed terms of type s are ....

Alimohamed, M. (1995), A characterization of lambda definability in categorical models of implicit polymorphism, Theoret. Comp. Sci. 146, 5--23.

....the definable elements in any Henkin model [4] Although not emphasised in [4] relations of varying arity are powerful enough to characterise relative definability with respect to any given set of elements considered as constants. The full generality of the approach is demonstrated in Alimohamed [1], where such relations are used to characterise relative definability in an arbitrary cartesian closed category. In general, results about the pure simply typed calculus extend easily to analogous results for systems containing finite product types. This is not the case for finite coproduct (sum) ....

....characterise relative definability in any bicartesian closed category in which the finite coproducts are stable (as is the case in Set) We do not know if the characterisation extends also to the non stable case. From the categorical point of view our results are best explained in terms of glueing [12, 1]. However, for this conference version of the paper, we keep our exposition elementary, in the hope that it will be accessible to most type theorists with some background in categorical semantics. It should be said that the research in this paper originated as part of a strategy conceived by the ....

[Article contains additional citation context not shown here]

M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science, 146:5--23, 1995.

....D [2] and Girard translation amounts to the ccc functor from a free ccc (the term model of the simply typed lambda calculus) to C. Then full completeness of Girard translation is no other than the full faithfulness of this functor, which can be shown using the technique of categorical glueing [1, 6]. The benefit of this semantics oriented view is not only the conceptual simplicity (for those familiar with the correspondence between syntax and semantics) but also the elegant proofs which avoid to handle (often too complicated) syntax directly. See [5, 6] for further details and examples. ....

....In this note we had to consider parameterized logical predicates just for showing Lemma 5. 3 (the crucial lemma which says P oe 1 oe 2 = P oe 1 ( P oe 2 ) The role of parameters is similar to that of the parameters of Kriple logical predicates ( logical relations with varying arities ) of [1, 7]. On the other hand, in [5, 6] we introduced parameterized logical predicates for overcoming the difficulties arising from the linearity. Since the source language of Girard translation is not a linear type theory, we did not need the full expressive power of parameterized logical predicates in ....

Alimohamed, M. (1995) A characterization of lambda definability in categorical models of implicit polymorphism. Theoret. Comp. Sci. 146, 5--23.

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M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science, 146(1--2):5--23, July 1995.

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M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science, 146(1--2), 1995.

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M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science, 146(1--2):5--23, July 1995.

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M. Alimohamed. A characterization of lambda definability in categorical models of implicit polymorphism. Theoretical Computer Science 146:5--23 (1995).

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