B. Jacobs. Semantics of Weakening and Contraction. Preprint, May 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....k implicit. The above de nition, exploits the internal logic of C to formulate k strictness in the natural way. Nevertheless, we remark that the de nition also has a simple categorytheoretic formulation using the (double )strength of the monad, see, e.g. the de nition of bimorphism in [16]. Proposition 4.3 For pointed objects (X 1 ; 1 ) X k ; k ) and (Y; where k 1, any k strict map h : X 1 X k Y is a strict map from (X 1 ; 1 ) X k ; k ) to (Y; 18 This result is a special case of Proposition 4.8, which is proved below. In fact, ....

....) where k 1, any k strict map h : X 1 X k Y is a strict map from (X 1 ; 1 ) X k ; k ) to (Y; 18 This result is a special case of Proposition 4.8, which is proved below. In fact, the above proposition holds, more generally, for all relevant monads in the sense of [16]. The initial algebra I of the endofunctor on C carries a pointed structure = I. The pointed structure on I interacts nicely with the initial algebra property. De ne a successor function s = f g : I I. Proposition 4.4 Suppose that (X; is a pointed object and that f : X X ....

B. Jacobs. Semantics of weakening and contraction. Ann. Pure Appl. Logic, 69:73{ 106, 1994.

....; Delta) a I . ffl Linear negation ( Delta) cuts down to a duality G G op ; in fact (A = A ; A . 5. 2 Exponentials Jacobs has recently investigated the decomposition of the exponentials ; into weakening parts w ; w and contraction parts c ; c [Jac92]. He develops a general theory for this decomposition. We will use a little of this theory to structure our presentation of the exponentials. The reflection and co reflection of Proposition 11 give rise to a monad and a comonad on G respectively, which we denote by w and w . Our reason for ....

B. Jacobs. Semantics of weakening and contraction. Preprint, 1992.

....a complete pretopology semantics for a system of Intuitionistic Linear Logic (commutative or not) where the storage operator is split into a contraction and a weakening component and then recovered again from them. The semantics for weakening and contraction has been explored by Bart Jacobs [13] in a categorical setting. However, a completeness theorem is not given in [13] and the approach taken there does not accommodate the case of non commutative linear logic. Besides, we think it useful to have an intuitive, Kripke type semantics for the bimodal system. Extensions of the ....

....(commutative or not) where the storage operator is split into a contraction and a weakening component and then recovered again from them. The semantics for weakening and contraction has been explored by Bart Jacobs [13] in a categorical setting. However, a completeness theorem is not given in [13] and the approach taken there does not accommodate the case of non commutative linear logic. Besides, we think it useful to have an intuitive, Kripke type semantics for the bimodal system. Extensions of the exponential free linear logic with modalities weaker than Girard s operator have ....

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B. Jacobs, "Semantics of Weakening and Contraction", Annals of Pure and Applied Logic 69 (1994) 73-106.

....silent, while dropping interchange forces the model to distinguish between derivations differing for the order of execution of nested rewrites. A current trend of work is investigating the case of preserving interchange and dealing with some kind of weak cartesianity (along the line of Jacobs [Jac93] for the semantics of linear logic) making housekeeping explicit. Although a detailed discussion about the actual implementability of term rewriting in a way reflecting the degree of concurrency of either models goes beyond the scope of this paper, some comment is in order. Even if the ....

B. Jacobs, Semantics of Weakening and Contraction, draft.

....a complete pretopology semantics for a system of Intuitionistic Linear Logic (commutative or not) where the storage operator is split into a contraction and a weakening component and then recovered again from them. The semantics for weakening and contraction has been explored by Bart Jacobs [13] in a categorical setting. However, a completeness theorem is not given in [13] and the approach taken there does not accommodate the case of non commutative linear logic. Besides, we think it useful to have an intuitive, Kripke type semantics for the bimodal system. Extensions of the ....

....(commutative or not) where the storage operator is split into a contraction and a weakening component and then recovered again from them. The semantics for weakening and contraction has been explored by Bart Jacobs [13] in a categorical setting. However, a completeness theorem is not given in [13] and the approach taken there does not accommodate the case of non commutative linear logic. Besides, we think it useful to have an intuitive, Kripke type semantics for the bimodal system. Extensions of the exponential free linear logic with modalities weaker than Girard s operator have been ....

[Article contains additional citation context not shown here]

B. Jacobs, "Semantics of Weakening and Contraction", Annals of Pure and Applied Logic 69 (1994) 73-106.

....( Delta) cuts down to a duality G Gamma G op ; in fact (A Gamma ) A ) A ) A ) Gamma . 5. 2 Exponentials Jacobs has recently investigated the decomposition of the exponentials ; into weakening parts w ; w and contraction parts c ; c [Jac92]. He develops a general theory for this decomposition. We will use a little of this theory to structure our presentation of the exponentials. 5.2.1 Weakening The reflection and co reflection of Proposition 11 give rise to a monad and a comonad on G respectively, which we denote by w and w . ....

B. Jacobs. Semantics of weakening and contraction. Preprint, 1992.

....can write G = Chu(G Gamma ; I) where the tensor unit I (which is the terminal object in G Gamma ) is used as the dualizing object. 5. 2 Exponentials Jacobs has recently investigated the decomposition of the exponentials ; into weakening parts w ; w and contraction parts c ; c [Jac92]. He develops a general theory for this decomposition. We will use a little of this theory to structure our presentation of the exponentials. 5.2.1 Weakening The reflection and co reflection of Proposition 11 give rise to a monad and a comonad on G respectively, which we denote by w and w . ....

B. Jacobs. Semantics of weakening and contraction. Preprint, 1992.

....[14] there are also reasons to be interested in the Eilenberg Moore category. Under certain conditions the Eilenberg Moore category is bicartesian, symmetric monoidal closed, with a comonad and thus models intuitionistic linear logic (an observation due to Gordon Plotkin and Bart Jacobs, see [9] for details) So the Eilenberg Moore category has the potential to model quite sophisticated type systems. In order to model recursive types we would like to apply the analysis of Section 5 to the Eilenberg Moore category. The work of Kock [13] leads us to believe that, whenever C is cartesian ....

B.P.F. Jacobs. Semantics of weakening and contraction. Submitted manuscript, forthcoming, 1992.

....model for the following reason. In this category, as we have previously observed, products and coproducts do not coincide. Thus, the isomorphism: A Omega B = A Phi B) is not useful for modelling the additive fragment. We obtain instead a weakening cotriple in the sense of Jacobs [J93]. Jacobs denotes a weakening cotriple by w . Such a cotriple satisfies all of the axioms of modelling , except that the coalgebras will not have the comonoid structure necessary to model contraction. Thus, we have syntax of the following form: Gamma; A Gamma; w B A We Gamma; B ....

Jacobs, B., "The Semantics of Weakening and Contraction", preprint, (1993)

....in C. Equation (10) expresses statically the fundamental requirement that evaluating new produces something new. Equations (11) and (12) correspond respectively to properties (ii) and (i) in Corollary 2.4. They are automatically satisfied if the monad is respectively commutative and affine see [5]. Given the above structure in the category C, for each nu calculus type oe one gets an object [ oe] of C by defining: o] def = 1 1 [ def = N [ oe oe 0 ] def = oe] T ( oe 0 ] And for each valid typing assertion s; Gamma M : oe one can define, by induction on ....

B. Jacobs. Semantics of Weakening and Contraction. Preprint, May 1992.

.... Phi N 6= M Omega N) Omega (M jN) for automata M;N 2 DA. The units for these tensors are 1 = h; fg 1 Theta fg 0 i for Omega ; Phi and I = h; fg 1 Theta fg 1 i for j: And since 1 2 DA is the terminal object, one has that Omega and Phi are affine tensors (with projections, see [7]) On the category DB of deterministic behaviours we can define similar symmetric monoidal structures ( Omega ; 1) j; I) and ( Phi; 1) describing forms of parallel composition for behaviours. We define: A; B; s) Omega (C; D; t) A C; B Theta D; s Omega t: A C) B Theta D) ....

B. Jacobs. Semantics of weakening and contraction. Ann. Pure & Appl. Logic, 69(1):73--106, 1994.

....T = ii) And we say that T has a ( nitely) ane (predicate) lifting if ( T preserves non empty nite supremema. This amounts to (P 1 [ P 2 ) T = P T 1 [ P T 2 for each pair of predicates P 1 ; P 2 X on a set X . Strict and ane functions between complete lattices are considered in [Jac94] as one of the running examples giving categories having tensors with diagonals or with projections, and with exponential operators introducing only weakening or only contraction. The issue, like here, is the distinction between at least most once. See also [Jac93] for examples of models of ....

....tensor product X Y of two metric space. It has the Cartesian product X Y as underlying set, with distance function d( x; y) x 0 ; y 0 ) d(x; x 0 ) d(y; y 0 ) The projection function : X Y X is then non expansive, so that we have a tensor with projections X Y X , see [Jac94] The interval [0; 1] with it usual order forms a complete Heyting algebra, with max as least upper bound and min as greatest lower bound. The implication r s for r; s 2 [0; 1] is given as r s = 1 if r s, and r s = s otherwise. Then minft; rg s , t r s. And thus for a subset S ....

B. Jacobs. Semantics of weakening and contraction. Ann. Pure & Appl. Logic, 69(1):73-106, 1994.

....= ii) And we say that T has a ( nitely) a ne (predicate) lifting if ( T preserves non empty nite supremema. This amounts to (P 1 [ P 2 ) T = P T 1 [ P T 2 for each pair of predicates P 1 ; P 2 X on a set X . Strict and a ne functions between complete lattices are considered in [Jac94] as one of the running examples giving categories having tensors with diagonals or with projections, and with exponential operators introducing only weakening or only contraction. The issue, like here, is the distinction between at least most once. See also [Jac93] for examples of models of ....

....tensor product X Y of two metric space. It has the Cartesian product X Y as underlying set, with distance function d( x; y) x 0 ; y 0 ) d(x; x 0 ) d(y; y 0 ) The projection function : X Y X is then non expansive, so that we have a tensor with projections X Y X , see [Jac94] The interval [0; 1] with it usual order forms a complete Heyting algebra, with max as least upper bound and min as greatest lower bound. The implication r s for r; s 2 [0; 1] is given as r s = 1 if r s, and r s = s otherwise. Then minft; rg s , t r s. And thus for a subset S ....

B. Jacobs. Semantics of weakening and contraction. Ann. Pure & Appl. Logic, 69(1):73-106, 1994.

....of Computer Science Springer LNCS 813, 1994, p. 173 183. 2 Present address: CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands. Email: bjacobs cwi.nl. The paper was written at: Mathematical Institute, University of Utrecht. The comonads that we study all arise in the same way (as in [7]) namely as coming from the standard adjunction between a category of algebras and the underlying category. From a purely categorical point of view we can give the following alternative motivation for our work. Consider a category C with a monad T on it. Then we can form the category of algebras ....

B. Jacobs. Semantics of weakening and contraction. Ann. Pure Appl. Logic, 69(1):73--106, 1994.

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B. Jacobs. Semantics of Weakening and Contraction. Preprint, May 1992.

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B. Jacobs. Semantics of weakening and contraction. Manuscript, May 1992.

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Jacobs, B. (1994). Semantics of weakening and contraction. Annals of Pure and Applied Logic, 69: 73--106.

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