I. Mackie, L. Roman, and S. Abramsky. An internal language for autonomous categories. Applied Categorical Structures, 1:311-343, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... in n] let m be z z 2 in C[n] C[let m be z z 2 in n] provided that in the last conversion C does not bind or contain z 1 and z 2 . We recall that a sound and complete semantics for RLL is provided by autonomous (or monoidal closed) categories. Moreover RLL is also the internal language [MRA93,KO99,MMdPR01] for autonomous categories. 2.1 Reduction rules for RLL To perform computations in RLL we need to agree on which reduction rules to consider, given that we want a con uent and strongly normalizing calculus. If we simply orient the equations into reduction rules we can easily get a nonnormalizing ....

I. Mackie, L. Roman, and S. Abramsky. An internal language for autonomous categories. Applied Categorical Structures, 1:311-343, 1993.

....: k M k : A k a 1 : A 1 ; a k : A k N : B 1 ; 2 ; k promote M i for a i in N : B Where applicable ; are disjoint and ; denotes any permutation of its constituents. This calculus has explicit rules (and terms) for copying and discarding assumptions, following [19]. It also has explicit substitutions in the promotion rule, to cope with the lack of substitutivity in previous systems. For the equations, see Bierman s thesis [7] This calculus is correct, but verbose and cumbersome 2 . 3.3 Linear non Linear logic LNL Benton s type theory LNL [5, 6] ....

....been derived from the categorical semantics. The proof of the theorem mirrors the construction of the equational theory. Theorem 7. The type theory RLL provides the internal language for SMC st . Part of this result, i.e. for every autonomous category S we have S C(L(S) was rst stated in [19]. The work in [19] does not consider V L(C(V) as part of the de nition of internal language. Moreover, another major di erence between the theorem above and the work in [19] is that we take as the equational theory of the language of a category all the equations that are semantically valid in ....

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I. Mackie, L. Roman, and S. Abramsky. An internal language for autonomous categories. Applied Categorical Structures, 1:311-343, 1993.

.... l) x : A; y : B; s : C z : A B; hz A B =x A y B is : C ( l) s : A y : B; t : C z : A ( B; hz A(B ; s=y B it : C Couching them in the familiar but perhaps more cumbersome let notation common to many functional programming languages (e.g. see the let notation in [13]) the rst two let constructs would look like: unit let hx = is letx be in s tensor let hz A B =x A y B is let z A B be x A y B in s In the tensor let above, any free occurrence of x and y in s (it can be shown that there is exactly one each if s is well typed) is ....

....in uenced by Linear Logic. Broadly speaking, two main approaches can be distinguished. Benton, Bierman, de Paiva and Hyland [3] have used the standard terms and let constructs to represent proofs of the Intuitionistic Linear Logic. At about the same time, Mackie, Roman and Abramsky have shown in [13] that a similar type theory, in the style of Natural Deduction, can serve as an internal language for autonomous categories. Our work in this paper has evolved from an attempt to extend their language to represent autonomous categories, but we have arrived at a type theory quite di erent from ....

I. Mackie, L. Roman, and S. Abramsky. An internal language for autonomous categories. Journal of Applied Categorical Structures, 1(3):311-343, 1993.

.... Gamma hz A Omega B =x A Omega y B is : C ( l) Gamma s : A y : B; Delta t : C z : A ( B; Gamma; Delta hz A(B ; s=y B it : C Couching them in the familiar but perhaps more cumbersome let notation common to many functional programming languages (e.g. see the let notation in [17]) the first two let constructs would look like: unit let hx = is j letx be in s tensor let hz A Omega B =x A Omega y B is j let z A Omega B be x A Omega y B in s In the tensor let above, any free occurrence of x and y in s (it can be shown that there is exactly one each if ....

....approach Building on the work of Abramsky in [1] Benton, Bierman, de Paiva and Hyland [4] have used the standard terms and let constructs to represent proofs of the Intuitionistic Linear Logic. At about the same time, Mackie, Roman and Abramsky (we shall call them mra) have shown in [17] that a similar typed calculus can serve as an internal language for autonomous categories. The mra type theory is in the style of Gentzen s Natural Deduction. Our work in this paper has evolved from an attempt to extend their language to represent autonomous categories, but we have arrived at a ....

[Article contains additional citation context not shown here]

I. Mackie, L. Rom'an, and S. Abramsky. An internal language for autonomous categories. Journal of Applied Categorical Structures, 1(3):311--343, 1993.

....as internal languages for autonomous (or symmetric monoidal closed) and autonomous categories, in the same sense that the standard simply typed calculus with surjective pairing is the internal language for cartesian closed categories. A precursor of our work is the calculus of Mackie et al. in [13], though our research has led us to consider type theories that are rather different. In our approach, the rules for the typing judgements are presented in the style of Gentzen s Sequent Calculus. We use non standard let constructs, 1 thong wei.koh wdr.com 2 Homepage: ....

.... Gamma hz A Omega B =x A Omega y B is : C ( l) Gamma s : A y : B; Delta t : C z : A ( B; Gamma; Delta hz A(B ; s=y B it : C Couching them in the familiar but perhaps more cumbersome let notation common to many functional programming languages (e.g. see the let notation in [13]) the first two let constructs would look like: unit let hx = is let x be in s tensor let hz A Omega B =x A Omega y B is let z A Omega B be x A Omega y B in s In the tensor let above, any free occurrence of x and y in s (it can be shown that there is exactly one each if s is ....

[Article contains additional citation context not shown here]

I. Mackie, L. Rom'an, and S. Abramsky. An internal language for autonomous categories. Journal of Applied Categorical Structures, 1(3):311--343, 1993.

.... Gamma hz A Omega B =x A Omega y B is : C ( l) Gamma s : A y : B; Delta t : C z : A ( B; Gamma; Delta hz A(B ; s=y B it : C Couching them in the familiar but perhaps more cumbersome let notation common to many functional programming languages (e.g. see the let notation in [12]) the first two let constructs would look like: unit let hx = is letx be in s tensor let hz A Omega B =x A Omega y B is let z A Omega B be x A Omega y B in s 1 We adopt the Variable Convention in Barendregt s book: bound variables are renamed afreshed whenever necessary ....

....influenced by Linear Logic. Broadly speaking, two main approaches can be distinguished. Benton, Bierman, de Paiva and Hyland [3] have used the standard terms and let constructs to represent proofs of the Intuitionistic Linear Logic. At about the same time, Mackie, Roman and Abramsky have shown in [12] that a similar type theory, in the style of Natural Deduction, can serve as an internal language for autonomous categories. Our work in this paper has evolved from an attempt to extend their language to represent autonomous categories, but we have arrived at a type theory quite different from ....

I. Mackie, L. Rom'an, and S. Abramsky. An internal language for autonomous categories. Journal of Applied Categorical Structures, 1(3):311--343, 1993.

.... Lane, 1982 ] In the reverse direction of applications of category theory to proof theory, general categorical machinery, particularly enriched category theory, has provided useful guidelines for what constitutes a model of proofs in Girard s Linear Logic (see [ Seely, 1989 ] Barr, 1991 ] Mackie et al. 1993 ] Bierman, 1994 ] for example 86 Andrew M. Pitts in [ Benton et al. 1993 ] such considerations facilitated the discovery of a well behaved Natural Deduction formulation of intuitionistic linear logic. Categorical combinators The essentially algebraic nature of the category theory ....

I. Mackie, L. Rom'an, and S. Abramsky. An internal language for autonomous categories. Applied Categorical Structures, 1:311--343, 1993. Categorical Logic 89

....sums and a linear fixpoint operator; so the presence of a linear fixpoint operator in a linear category is consistent with the presence of finite sums. Thus, the inconsistency of recursion with this standard construct vanish when we go to a linear context, which is in accordance with [Plo93] In [MRA93] a different approach to recursion in a linear context is taken. The (I; Omega ; fragment of the linear calculus is extended with natural numbers, corresponding to a weak natural numbers object in the categorical model. The discussion above implies that this approach is consistent with ours. ....

I. Mackie, L. Rom'an, and S. Abramsky. An internal language for autonomous categories. Journal of Applied Categorical Structures, 1, 1993.

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