Seely, R. A. G., Linear logic, #-autonomous categories and cofree coalgebras, in: J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic (  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... a right adjoint, and in particular G(A B) GA GB G1 = 1 (note that we use 1 for the terminal object in both L and C) Taking this together with Proposition 1, we obtain the following natural isomorphisms: A = A B) 1 These isomorphisms were central to Seely s proposed model of ILL [See80], which also proposed interpreting IL in the Kleisli category. See [Bie94a] or [Bie94b] for a critique of Seely s semantics; here we shall merely show that a linear category with products does indeed have a Kleisli category which is cartesian closed. The isomorphisms A;B : A B) B and : 1 ....

R. A. G. Seely. Linear logic, *-autonomous categories and cofree coalgebras. In Conference on Categories in Computer Science and Logic, volume 92 of AMS Contemporary Mathematics, June 1980.

....all we needed, at least to get a model. But now we need a loop hole to get some access to the dynamics, and Linear Logic provides such a loop hole. Suppose then that our cartesian closed category C arises as C = K (L) the co Kleisli category of a Linear category L with respect to the comonad [6, 29]. The intuitionistic function types we have been using get their standard decompositions into the Linear types: A ) B = A ( B: In particular, we see that the type of newA is: newX;A : var[X ] exp[A] exp[A] Now suppose that we have a morphism cellX : I Gamma var[X ] 9 Then we ....

R. A. G. Seely. Linear logic, -autonomous categories and cofree coalgebras. In Category theory, computer science and logic. American Mathematical Society, 1987.

....all we needed, at least to get a model. But now we need a loop hole to get some access to the dynamics, and Linear Logic provides such a loop hole. Suppose then that our cartesian closed category C arises as C = K (L) the co Kleisli category of a Linear category L with respect to the comonad [7, 33]. The intuitionistic function types we have been using get their standard decompositions into the Linear types: A ) B = A ( B: In particular, we see that the type of new is: new : var ( com) com: Now suppose that we have a morphism cell : I Gamma var: 8 Then we can define new as ....

R. A. G. Seely. Linear logic, -autonomous categories and cofree coalgebras. In Category Theory, Computer Science and Logic. American Mathematical Society, 1987.

....structured category. Let C be a com pact closed category (a autonomous category [8] in which Omega and O coincide) with countable biproducts and a functor which interprets the exponential of linear logic (i.e. should be a comonad and each A should have a cocommutative comonoid structure [24]) Additionally, let C have a strict monoidal endofunctor ffi, and write monunit : I ffi I and mon A;B : ffi A) Omega (ffi B) ffi(A Omega B) for the associated isomorphisms. An endofunctor F has the unique fixed point property (UFPP) 3] if for every f : A FA and g : FB B there is a ....

R. Seely. Linear logic, -autonomous categories and cofree coalgebras. In Contemporary Mathematics, 1987.

....and the order to the Scott order between them. A morphism E 0 E 1 is defined to be a configuration of E 0 ( E 1 . As such it is a relation between the events of E 0 and E 1 . Composition in the category is that of relations. The category is a model of intuitionistic linear logic, as defined in [9]; for instance, its tensor is given in a coordinatewise fashion: For event structures E i = E i ; i ; i ) for i = 0; 1, E 1 = E 0 Theta E 1 ; 1 ) e 0 0 e Monoidal closure follows from the isomorphism E 1 ( E 2 ) E 0 ( E 1 ( E 2 ) natural in event structures E ....

.... (f) f . For R 2 Gamma( E 0 ( E 1 ) a direct translation of the definitions yields ( R) R. Thus the map R 7 R = R is a 1 1 correspondence. 2 More completely, this section provides the key constructions in showing: 14 Theorem 8 The category BS forms a linear category in the sense of [9]. The exponential forms a comonad on the category BS. Together they form a model of classical linear logic (a Girard category in the sense of [9] The associated co Kleisli category is equivalent to a cartesian closed full subcategory of Berry s bidomains, where morphisms are continuous with ....

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Seely, R.A.G., Linear logic, -autonomous categories and cofree coalgebras. Contempory Mathematics, vol.92, 1989.

....contains all the essential ingredients of the Geometry of Interaction interpretation in a more synthetic form, and provides a good point of entry to the interpretation, which will be described in the next Section. Firstly, we shall briefly recall the definition of CLL models; for more details see [See89]. Let be a symmetric monoidal closed category with tensor Omega , unit I , internal hom Gammaffi. An object is dualizing if for all A, the morphism A (A Gammaffi ) Gammaffi , obtained by currying the evaluation map, is an isomorphism. A autonomous category is a symmetric monoidal ....

R. Seely. Linear logic, ?-autonomous categories and cofree coalgebras. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 371--382. American Mathematical Society, 1989.

....is a function Gamma(E 0 ) Gamma(E 1 ) which is continuous with respect to v and stable with respect to v . In fact, y 7 y is a 1 1 correspondence between configurations of E 0 ( E 1 and such functions. More completely: Theorem 8 The category BS forms a linear category in the sense of [16]. The exponential forms a comonad on the category BS. Together they form a model of classical linear logic (a Girard category in the sense of [16] The associated co Kleisli category is equivalent to a cartesianclosed full subcategory of Berry s bidomains, where morphisms are continuous with ....

....between configurations of E 0 ( E 1 and such functions. More completely: Theorem 8 The category BS forms a linear category in the sense of [16] The exponential forms a comonad on the category BS. Together they form a model of classical linear logic (a Girard category in the sense of [16]) The associated co Kleisli category is equivalent to a cartesianclosed full subcategory of Berry s bidomains, where morphisms are continuous with respect to the extensional order and stable with respect to the stable order. In particular, is a comonad. Its counit ffl E : E E is given by ....

Seely, R.A.G., Linear logic, -autonomous categories and cofree coalgebras. Contempory Mathematics, vol.92, 1989.

....st . The proof is implicitly provided in Bierman s thesis. We draw attention to this fact here to compare linear categories with the following two other notions of models. For a thorough discussion concerning di erences between the models above and the previous (unsound) models in the literature [25], the reader is referred to [8] 4.5 Models of DILL Recall Barber and Plotkin s [2] de nition of a model of DILL: this is much simpler and easier to remember than the de nition of a linear category. De nition 11 (Barber Plotkin) A model of DILL is a monoidal adjunction F a G, with F : C S ....

R.A.G. Seely. Linear logic, *-autonomous categories and cofree coalgebras. In Categories in computer science and logic (Boulder, CO,

....# A B # B A, A # B # B (#, left) #,A # B (cut) 2. 1 Categorical Interpretation of Linear logic Having already introduced the syntax and the proof theory of linear logic, we are now ready to present its linear categorical semantics which has been mostly developed by Lafont [12] and Seely[17], with contributions of several other people ( 15] 14] 1] The linear logic model which is adopted in this paper is a simple axiomatization of linear category known as a closed symmetrical monoidal category with the notion of a dualizing object. We recall here the notion of a closed ....

R.A.G. Seely, Linear Logic, *-Autonomous Categories and Cofree Coalgebras, in : J.W. Grayaud A. Seedrov (eds.), Categories in Computer Science and Logic, Boulder, June 87.

....and strategies as arrows, to define a category. This turns out to be symmetric monoidal and closed due to the particular operations that are supported by games and strategies. It is well known that a symmetric monoidal category can be used as the basis of a categorical semantics of linear logic [See89]. Using the Curry Howard isomorphism [How80] we can develop a corresponding calculus, the linear calculus [Abr93, GL87] Using the implementation of games and strategies, we can make use of the categorical semantics to implement this language. We choose to consider the intuitionistic ....

.... with Girard s linear logic [Gir87] The syntax chosen for the calculus associated with linear logic used in this report has been particularly influenced by the work of Wadler [Wad93] and Bierman [BBdPH92] The standard categorical semantics for linear logic was developed originally by Seely [See89], however, here we only make use of the symmetric monoidal closed category fragment of Seely s semantics. Section 2 explains the concepts of games and gives an implementation of them in a functional language. Strategies are discussed in Section 3, while the category formed by using games as ....

Robert A. G. Seely. Linear logic, -autonomous categories and cofree coalgebras. In John W. Gray and Andre Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 371--382, Providence, Rhode Island, 1989. American Mathematical Society.

....structured category. Let C be a com pact closed category (a autonomous category [8] in which Omega and O coincide) with countable biproducts and a functor which interprets the exponential of linear logic (i.e. should be a comonad and each A should have a cocommutative comonoid structure [24]) Additionally, let C have a strict monoidal endofunctor ffi, and write monunit : I ffi I and mon A;B : ffi A) Omega (ffi B) ffi(A Omega B) for the associated isomorphisms. An endofunctor F has the unique fixed point property (UFPP) 3] if for every f : A FA and g : FB B there is a ....

R. Seely. Linear logic, -autonomous categories and cofree coalgebras. In Contemporary Mathematics, 1987.

....(or denotational) semantic; ffl it improves calculi inspired by operational semantics (like the v calculus) by deriving more correct equivalences between programs. A comparison between the categorical semantic of computations and that of linear logic based on monoidal closed categories (see [See87]) shows that they lead to orthogonal (and compatible) modifications of the notion of cartesian closed category. In fact, in the former the monad Id C is replaced by another monad T , while in the latter the cartesian product Theta is replaced by a tensor product Omega . In our opinion this means ....

R.A.G. Seely. Linear logic, -autonomous categories and cofree coalgebras. In Proc. AMS Conf. on Categories in Comp. Sci. and Logic (Boulder 1987), 1987.

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Seely, R. A. G., Linear logic, #-autonomous categories and cofree coalgebras, in: J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic (

No context found.

R. A. G. Seely, Linear logic, -autonomous categories and cofree coalgebras, in: J. W. Gray and A. Scedrov, eds., Categories in Computer Science and Logic, Boulder, June 1987.

....the order to the pointwise order between them. A morphism E 0 E 1 is defined to be a configuration of E 0 ( E 1 . As such it is a relation between the events of E 0 and E 1 . Composition in the category is that of relations. The category is a model of intuitionistic linear logic, as defined in [24, 4]. For instance, its tensor is given in a coordinatewise fashion. For event structures E i = E i ; i ; i ) for i = 0; 1, define: E 1 = E 0 Theta E 1 ; 1 ) e 0 0 e A complete prime of a Scott domain (D; v) is an element p for which whenever X is bounded above and p v X ....

....2 g(y) v g(x) concluding the proof. 2 29 This section has provided the key constructions for showing that BS is a model of classical linear logic, and that the associated co Kleisli category is equivalent to one of biorders: Theorem 1 The category BS forms a linear category in the sense of [24]. The exponential forms a comonad on the category BS. Together they form a model of classical linear logic (a Girard category in the sense of [24] see too [4] The associated co Kleisli category is (necessarily) cartesian closed and isomorphic to the category whose objects are the structures ....

[Article contains additional citation context not shown here]

Seely, R.A.G., Linear logic, ?-autonomous categories and cofree coalgebras, in Categories in Computer Science and Logic (eds. Gray, J.W. and Scedrov, A.), Contempory Mathematics, Vol. 92, pp. 371--382, American Mathematical Society, 1989.

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R. A. G. Seely. Linear logic, #-autonomous categories and cofree coalgebras. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, pages 371--382. American Mathematical Society, 1989. Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference, June 14--20, 1987, Boulder, Colorado; Contemporary Mathematics Volume 92.

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R. Seely. Linear logic, ?-autonomous categories and cofree coalgebras. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 371--382. American Mathematical Society, 1989.

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R.A.G. Seely. Linear logic, *-autonomous categories and cofree coalgebras. Contemporary Mathematics, 92, 1989.

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R.A.G. Seely. Linear logic, *-autonomous categories and cofree coalgebras. Contemporary Mathematics, 92, 1989.

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R.A.G. Seely. Linear logic, *-autonomous categories and cofree coalgebras. Contemporary Mathematics, 92, 1989.

No context found.

R.A.G. Seely, Linear logic, -autonomous categories and cofree coalgebras, in: J. Gray and A. Scedrov (eds), Categories in Computer Science and Logic (Proc. A.M.S. Summer Research Conference, June 1987.

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SEELY R.A.G. Linear logic, #-autonomous categories and cofree coalgebras, Contemporary Mathematics 92 1989.

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R.A.G. Seely. Linear logic, *-autonomous categories and cofree coalgebras. Contemporary Mathematics, 92, 1989.

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R.A.G. Seely, Linear logic, ?-autonomous categories and cofree coalgebras, in: J. Gray and A. Scedrov (eds.), Categories in Computer Science and Logic, Contemp. Math. 92 (Amer. Math. Soc., 1989), 371--382

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R. A. G. Seely. Linear logic, -autonomous categories and cofree coalgebras. In J. W. Gray and A. Scedrov, editors, Categories in Computer Science and Logic, volume 92 of Contemporary Mathematics, pages 371-382, Providence, Rhode Island, 1989. American Mathematical Society.

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