R.A.G. Seely. Linear logic, *-autonomous categories, and cofree coalgebras. In: Categories in Computer Science and Logic, June 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... A n ) f B] 13 Note that in case n = 0 then fl(f) m I ; f . In case n = 1 this definition is consistent with the usual cokleisli operator. It should be mentioned that the definition of fl is due to [BBdPH92] Note that the definition of fl is unrelated to the product structure. In [See89], another generalised cokleisli operator is used which is related to the product structure. Proposition 5.5 If f : A A n B, then fl(f ) B = f . Proof. Straightforward calculation. 2 Proposition 5.6 If f i : I A i for i 2 f1; ng, and h : A A n B, then B] ....

....objects in C , I is isomorphic to 1, and analogously, since both ffi) m A; B ) and ( A Theta B) ffi) are products of ( A; ffi) and ( B; ffi) in C , B is isomorphic to (A Theta B) such that the isomorphism is natural in A and B. Thus we can define a model of ILL as described in [See89]. Calculations show that the way the isomorphisms are defined and the universal property of (I; m I ) and ffi) m A; B ) forces to take the cocommutative comonoid structure w.r.t. the finite products to the cocommutative comonoid structure w.r.t. the symmetric monoidal structure, that is: e A ....

R. A. G. Seely. Linear logic, -autonomous categories, and cofree coalgebras. In Contemporary Mathematics, Categories in Computer Science and Logic, volume 92. American Mathematical Society, 89.

.... operations on infinite games [25] and by Barr in so called autonomous categories [17] Both settings, in addition to coherent domains, 3 are now understood to yield mathematical models for linear logic proofs, more precisely, for the relation t is a linear logic proof of a formula A [26, 88, 18, 19]. Other versions of game semantics are given by Abramsky and Jagadeesan [2] and by Lafont and Streicher [65] Event spaces, which come about from Pratt s work in semantics of concurrency, also provide models for certain linear logic proofs [82] Models investigated by de Paiva [39] are motivated ....

R.A.G. Seely. Linear logic, *-autonomous categories, and cofree coalgebras. In Categories in Computer Science and Logic, pages 371--382. Contemporary Math., vol. 92, American Math. Soc., Providence, RI, 1989.

.... (a; b) A ffi B (a 0 ; b 0 ) iff a 0 A a b B b 0 ; A Theta B = f0g Theta A [ f1g Theta B; A ThetaB ) where (i; a) A ThetaB (j; a 0 ) iff (i = j = 0 a A a 0 ) or (i = j = 1 a B a 0 ) ffflg; f(ffl; ffl)g) We follow the notation and terminology introduced in [1,15] except , 0, and 1 for which we follow Girard [6] A linear category is a symmetric monoidal closed category with all finite products and a dualizing object. Theorem 30 PIS.l is a linear category. PIS.a is a cartesian closed category. PROOF. We only give a proof of the first conclusion here. ....

....we have A B : A Theta B ) A Theta B; For tensorsum fi, we have A fi B : A Omega B ) A Omega B: 2 Note that gluing the bottoms and the tops of two prime algebraic domains does not give back a prime algebraic domain. PIS.l is, moreover, a Girard category [15]. Let A B = Fin(A) Theta B; A B ) with (u; a) A B (v; b) iff v A u a B b; A = Fin(A) A ) with u A v iff u A v; where Fin(A) stands for the set of finite subsets of A. The function space can be decomposed into two more primitive constructions mentioned above. This is the ....

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R. Seely, Linear logic, *-autonomous categories, and cofree coalgebras, in: Categories in Computer Science and Logic, 371--382 (Amer. Math. Soc., Providence, RI, 1989).

....the basis of a programming language. Some elaboration of the above points can be found in our previous work [41,42] Every term of our language is also a term of the language in [41] from which it is easy to see how to give a semantics to this language in a categorical model in the style of Seely [35] as amended by Bierman [11,12] 3 The call by name lambda calculus Figure 2 reviews the call by name lambda calculus Name and presents its translation into the linear lambda calculus. Types, terms and values are standard: a type is the base type or a function type, a term is a value or a ....

R. A. G. Seely, Linear logic, -autonomous categories, and cofree coalgebras, in: Categories in Computer Science and Logic (AMS Contemporary Mathematics 92, June 1989).

.... A n ffi A 1 Omega : Omega ffi An A 1 Omega : Omega A n mA 1 ; A n ( A 1 Omega : Omega A n ) f B In case n = 1 this definition is consistent with the usual coKleisli operator. Note that the definition of fl does not assume the presence of finite products. In [See89] another notion of a categorical model for ILL is defined where the induced generalised coKleisli operator relies on the existence of finite products. Given a category C equipped with a comonad ( ffi) the coEilenberg Moore category, C , is the category of coalgebras. We have an ....

.... n 1 : I 1, and a natural isomorphism n A;B : A Omega B (A Theta B) It can be shown that the map n 1 and the natural isomorphism n makes a symmetric monoidal functor from (C; 1; Theta) to (C; I; Omega ) Given these isomorphisms, we can define a model for ILL as described in [See89]. Calculations show that the way the isomorphisms are defined and the universal properties of I and ( A; ffi) Omega ( B; ffi) in C forces to take the cocommutative comonoid structure with respect to the finite products to the cocommutative comonoid structure with respect to the symmetric ....

R. A. G. Seely. Linear logic, -autonomous categories, and cofree coalgebras. In Contemporary Mathematics, Categories in Computer Science and Logic, volume 92. American Mathematical Society, 1989.

....in mall, then M(A) 1. Also, one may generalize these conditions somewhat, replacing all instances of 1 with any arbitrary constant c, and allowing propositions to have different (although fixed) values, where p has value v p , and p has value c Gamma v p [3] Other related work is given in [17] and [4] Since the above is only a necessary condition, there has been a question as to whether some form of simple truth table or numerical evaluation function like the above could yield a necessary and sufficient condition for provability of constant multiplicative (comll) expressions. The ....

R.A.G. Seely. Linear logic, *-autonomous categories, and cofree coalgebras. In: Categories in Computer Science and Logic, June 1989, 1989.

....MLL Sequents Functorial polymorphism can be extended to handle Barr s autonomous categories [7] i.e. smc categories C equipped with an involution functor ( C op C given by a dualising object. Such categories interpret the multiplicative fragment of classical linear logic [42, 9]. We modify the functorial interpretation of formulas mentioned earlier to the Omega ; fragment of classical linear logic by modifying the interpretation of atomic clauses. Thus associated to each formula OE(ff 1 ; ff n ) in Omega ; Gammaffi; with type variables (or ....

....Hopf algebra is a vector space, H, equipped with an algebra structure, a compatible coalgebra structure and an antipode. These must satisfy equations as outlined in [45] The following chart summarizes the necessary structure. 2 In fact, we obtain an indexed autonomous category in the sense of [42]. Structure k[G] Algebra m: H Omega H H m(g 1 Omega g 2 ) g 1 g 2 j: k H j(1 k ) 1 G Coalgebra Delta: H H Omega H Delta(g) g Omega g ffl: H k ffl(g) 1 k Antipode S: H H S(g) g Gamma1 One obtains a Hopf algebra from a group G by taking the vector space generated by the ....

R.A.G. Seely, Linear Logic, -Autonomous Categories, and Cofree Coalgebras, Contemporary Mathematics 92, (1989), pp. 371-382.

....the basis of a programming language. Some elaboration of the above points can be found in our previous work [36,37] Every term of our language is also a term of the language in [36] from which it is easy to see how to give a semantics to this language in a categorical model in the style of Seely [30] as ammended by Bierman [8,9] 3 Call by name Figure 2 reviews the call by name lambda calculus name and presents its translation into the linear lambda calculus. Types, terms, and values are standard: a type is a base type or function type, a term is a value or a function application, and a ....

R. A. G. Seely, Linear logic, -autonomous categories, and cofree coalgebras. In Categories in Computer Science and Logic, June 1989. AMS Contemporary Mathematics 92.

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R. A. G. Seely, Linear logic, -autonomous categories, and cofree coalgebras. In Categories in Computer Science and Logic, June 1989. AMS Contemporary Mathematics 92.

....following equation: lambda calculus intuitionistic logic = linear logic What does this language look like One would think the answer should be straightforward by now. There is the linear logic of Girard [Gir87] there is the syntax of Abramsky [Abr90] and there is the semantics of Seely [See89]. Each of these has become a standard. Abramsky was inspired by the earlier work of Lafont [Laf88] and Holmstrom [Hol88] and in turn inspired related systems by Chirimar, Gunter, and Riecke [CGR92] Lincoln and Mitchell [LM92] Mackie [Mac91] Troelstra [Tro92] and Wadler [Wad90, Wad91] Seely ....

R. A. G. Seely, Linear logic, -autonomous categories, and cofree coalgebras. In Categories in Computer Science and Logic, June 1989. AMS Contemporary Mathematics 92.

....on lambda terms, following in the footsteps of Abramsky [Abr90] the four rules associated with the of course type, Weakening, Contraction, Dereliction, and Promotion, are each represented by a separate term form. The semantics is based on category theory, following in the footsteps of Seely [See89]: Weakening and Contraction are modelled by a comonoid, while Dereliction and Promotion are modelled by a comonad. Surprisingly, when you combine a syntax like Abramsky s with a semantics like Seely s, various problems arise. For one thing, there is a term that looks as if it denotes the ....

.... 1 I ; A duplicate Gamma Gamma Gamma Gamma A Omega A = A diagonal Gamma Gamma Gamma Gamma Gamma (A Theta A) A) Omega ( A) where terminal : A 1 is the unique map to the terminal, and diagonal : A A Theta A is the diagonal map. The above definition is derived from Seely [See89]. Usually, Seely s definition is thought of as extending Barr s notion of a autonomous category [Bar79] but we have no need of a dualising object since we do not deal with negation. A categorical model is obtained by associating with each type variable an object in C, inducing a map from types ....

R. A. G. Seely, Linear logic, -autonomous categories, and cofree coalgebras. In Categories in Computer Science and Logic, June 1989. AMS Contemporary Mathematics 92.

.... Unity, a refinement of linear logic [7] This overcomes some technical problems with other presentations of linear logic, some of which are discussed by Benton, Bierman, de Paiva, and Hyland [2] and Wadler [23, 24] Much of the insight for this work comes from categorical models of linear logic [19, 15]. The particular system presented here was suggested to the author by Girard, and a similar system has been suggested by Plotkin. For further background on traditional logic see the wonderful introduction by Girard, Lafont, and Taylor [8] and for further details on linear logic see the helpful ....

R. A. G. Seely, Linear logic, -autonomous categories, and cofree coalgebras. In Categories in Computer Science and Logic, June 1989. AMS Contemporary Mathematics 92.

....deeper connections underneath. Have the different topographies been shaped by a single underlying pressure The intuitionistic fragment of Girard s formalism may be modeled in a symmetric monoidal closed category with a comonad and some addditional structure. An early model was sketched by Seely [See89], and a later model detailed by Benton, Bierman, de Paiva, and Hyland [BBdPH92] dubbed the Gang of Four . The two models are similar, and their relation has been detailed by Bierman [Bie94] Meanwhile, Moggi s formalism may be modelled in a cartesian closed category with a monad T and some ....

....third part then follows once one has shown jt ffi j l = t for every lambda term t by induction on t. 2 The two translations yield equivalent semantics. This is hardly surprising, as it was the motivation behind Girard s formulation, and has been a touchstone of the categorical models of Seely [See89] and Benton, Bierman, de Paiva, and Hyland [BBdPH92] We spell out the equivalence here, for comparison with the later development. The equivalence has two parts: the two translations of types are isomorphic, and the two translations of terms are related by this isomorphism. This level of detail ....

R. A. G. Seely, Linear logic, -autonomous categories, and cofree coalgebras. In Categories in Computer Science and Logic, June 1989. AMS Contemporary Mathematics 92.

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R.A.G. Seely. Linear logic, *-autonomous categories, and cofree coalgebras. In: Categories in Computer Science and Logic, June 1989.

No context found.

R. A. G. Seely, Linear logic, -autonomous categories, and cofree coalgebras. In Categories in Computer Science and Logic, June 1989. AMS Contemporary Mathematics 92.

No context found.

R. A. G. Seely. Linear Logic, *-Autonomous Categories, And Cofree Coalgebras. Contemp. Math. 92, pp. 371-382.

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R. A. G. Seely. Linear Logic, *-Autonomous Categories, And Cofree Coalgebras. Contemp. Math. 92, pp. 371-382.

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