D. Yetter. Quantales and Non-Commutative Linear Logic. (preprint). 24  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Languages ) funded by the Netherlands Organization for the Advancement of Research (N.W.O. 1 the non directed calculus LP and the geometric invariant or balance in the directed calculus L [1] Since LP and L are fragments of the ordinary linear logic LL [2] and the cyclic linear logic CLL [6] respectively, we shall also study similar equivalence relation in both these logics. Let MLL and MCLL denote the multiplicative fragments of LL and CLL. Characterization of the equivalence in these fragments involves a new invariant defined by #p = 0, #(a.b) #a #b, #(a # ) 1 #a, ....

....are the following: # X ; a ; b ; Y ( # X ; a b ; Y # X; a # b; Y ( # X ; a. b ; Y 4 # X; a # a # ; Y (CUT) # X;Y # X; a; b; Y (P) # X; b; a; Y Now we shall consider MCLL the multiplicative fragment of the cyclic (noncommutative) linear logic presented by Yetter in [6]. The connectives are the same as in MLL except that in MCLL there are two linear implications and instead of #. The formulas are defined in the same way as in MLL, but the abbreviations are di#erent. a b # # (a) # b b a # # b (a) # (1) # # # 0 (0) # # # 1 (p # ) # # # p (a.b) # # # ....

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D.N. Yetter. Quantales and noncommutative linear logic. Journal of Symbolic Logic, 55(1):41--64, 1990. 20

....the induction hypothesis and apply the rule (# ) Case 2b: # #= ##, 0# We use Lemma 7.7 and the rules ( # # ( # # ) to reduce this case to the previous one. Remark. Analogous result can be easily established also for the multiplicative fragment of cyclic linear logic defined in [12]. 8 Properties of proof nets Lemma 8.1 Let ## # , A, B, E# be a proof structure. If the graph ## # , ## #A# contains a cycle, then there exists a cycle (# 1 , # 1 , # 2 , # 2 , #n , #n ) such that (i) # i ## # # and # i ## # # for each i # n; ii) # i ## # i for each ....

D. N. Yetter. Quantales and noncommutative linear logic. Journal of Symbolic Logic, 55(1):41--64, 1990. 37

....Then in that logic the two negation operators would coincide, and that would then immediately mean that they coincide in the other logic. It would not be necessary for the tensor and par to be commutative in that logic, but it would mean that that logic is cyclic , in the sense of Yetter [33]. Categorically, there is a tiny bit more happening here and this is discussed fully in [12] The essence is this: if we have a LF : can setting (F between autonomous categories being given by identity) then if one of the two linearly distributive structures is symmetric, the other must be ....

....and transformations, the comonoidal components being implicitly induced by duality. So, a model of NL could be viewed as a category equipped with a symmetric tensor making it into a (symmetric) autonomous category, and a second monoidal structure making it into a cyclic autonomous category [33, 11]. Then the two monoidal structures are related by a monoidal transformation called the entropy map. This is summarized in the following definition, which was presented in [8] and is an example of the approach of the present paper. Definition 3.1 An entropic category is a category C equipped with ....

D. Yetter (1990) "Quantales and (non-commutative) linear logic." Journal of Symbolic Logic 55 41--64. 25

....of linear logic had arisen before in the form of quantales. Indeed Girard s phase semantics in [Gir87] for linear logic uses free quantales. Abramsky and Vickers [AV88] approached quantales from a computer science viewpoint, the hope being that it would lead to a linear process logic . Yetter [Yet] and Rosenthal [Ros] looked at quantales and linear logic more from the perspective of pure mathematics how to represent them and their relationship with other bits of mathematics. This note points out a straightforward way in which a Petri net induces a quantale and so becomes a model for ....

....a logical constant, denoting linear absurdity is fixed on, linear negation (not to be mistaken for intuitionistic negation) is derivable: A = A Gamma , 3 Quantales We have just seen the proof rules of linear intuitionistic logic. What are its models As recognised by several people [AV88, Yet, Ros, Sam], quantales 2 provide an algebraic semantics for linear intuitionistic logic. Quantales are to linear intuitionistic logic as complete Heyting algebras are to intuitionistic logic. A quantale is a commutative monoid on a complete join semilattice. Spelled out: Definition 3.1 A quantale Q is a ....

D. Yetter. Quantales and Non-Commutative Linear Logic. (preprint). 21

....: if we take a general (non commutative) M and an arbitrary , then we get a mess, in which we eventually lose: associativity of the logical conjunction, see F.1. Now, if we require cyclicity, i.e. xy 2 ) yx 2 , we get a very natural system, expounded as cyclic linear logic by Yetter [28]. The fact that cyclicity is natural should be obvious to persons with a basic mathematical culture, think of T r(uv) T r(vu) Concretely the point behind the A naturality B of soundness is that one should be able to produce models in a fluent way : this is one of the key differences with ....

D.N. Yetter. Quantales and non-commutative linear logic. Journal of Symbolic Logic, 55:41--64, 1990.

....: its syntax and semantics 7 one have simultaneously a commutative Times , in which case the relation between both types of conjunctions should be understood. I Linear negation is delicate, since there are several possibilities, e.g. a single negation, like in cyclic linear logic as expounded in [27] or two negations, like the two linear implications, in which case the situation may become extremely intricate. Abrusci, see [2] this volume, proposed an interesting solution with two negations. The problem of finding the non commutative system is delicate, since although many people will ....

D.N. Yetter. Quantales and non-commutative linear logic. Journal of Symbolic Logic, 55:41--64, 1990.

....scope in Do sen (cf. 5, 6] who discusses the general picture, concentrating on proof theoretical properties like embeddability. Independently, the idea was taken up by Morrill et al..ii in [17] who were interested in an extension of the Lambek Calculus with restricted Permutation, and in Yetter [25] where an extension of cyclic linear logic is treated. The point that we want to make here is that there are some problems involved with a straightforward adaptation of the proof calculus for from linear logic to other substructural logics. To discuss these problems, let us assume that we add an ....

Yetter, D.N., "Quantales and (noncommutative) linear logic", Journal of Symbolic Logic 55 (1990) 41--64.

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D. Yetter. Quantales and Non-Commutative Linear Logic. (preprint). 24

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D. Yetter, "Quantales and (non commutative) linear logic", J. Symb. Logic 55 (1990) no. 1, pp. 41--64.

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