V.R. Pratt. Towards full completeness of the linear logic of chu spaces. In Electronic Notes in Theoretical Computer Science, volume 6, Pittsburgh, 1997. URL: http://www.elsevier.nl/locate/entcs/volume6.html, 18 pages.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....theorem, simple semigroups are in 1 1 correspondence with functions Y Theta X H . Such functions form the basis of a new approach to foundations of concurrency and foundations of computer science in general which is promoted by V. R. Pratt from Stanford under the name of Chu spaces (see, e. g, [5, 6, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]) Thus, a general extension of t norms naturally leads us to Chu spaces. Auxiliary Results and Their Relationship With the Existence and Borderline Character of Classical Truth Values. According to [1] Theorem 1.8, and [11] Theorem 1.4.2, if a compact topological semigroup S is not a group ....

V. R. Pratt, "Towards Full Completeness of the Linear Logic of Chu Spaces", Electronic Notes in Theoretical Computer Science, 1997, Vol. 6 (http://www.elsevier.nl/locate/entcs/volume6.html).

....functor from the autonomous category of cut free proofs of MLL to the autonomous category of binary logical transformations between MLL formulae interpreted as operations Ob(Chu) n Ob(Chu) where Ob(Chu) denotes the objects of Chu, and n is the number of variables in a formula. Elsewhere [Pra97] we demonstrated full completeness for dinatural transformations in Chu of binary MLL formulas, those in which each variable occurs only once with each sign. We also demonstrated the impossibility of increasing the number of occurrences of one variable by exhibiting a spurious dinatural ....

..... B, for coherence spaces [Tan97] For Chu spaces however dinaturality is not strong enough in that it admits certain spurious transformations corresponding to no MLL proof [Pra97]. A further drawback of dinatural transformations is that they do not always compose. In this paper we eliminate all spurious transformations with the help of a stronger condition, binary logicality, which we shall abbreviate to logicality. We extend MLL operations to act not on morphisms but on ....

V.R. Pratt. Towards full completeness of the linear logic of chu spaces. In Electronic Notes in Theoretical Computer Science, volume 6, Pittsburgh, 1997. URL: http://www.elsevier.nl/locate/entcs/volume6.html, 18 pages.

....fragment of linear logic, namely those operations obtainable by composition starting with A and A GammaffiB. The goal has been to identify the dinatural (or if necessary logical) transformations for this fragment, those that are suitably robust under change of variables. Recently we have shown [26] that for pairs of terms with a total of at most two occurrences of each variable, the dinatural transformations between such terms in Chu(Set; 2) are in exact correspondence with the cut free proofs (understood suitably abstractly) of multiplicative linear logic without MIX, a full completeness ....

V.R. Pratt. Towards full completeness of the linear logic of chu spaces. In Electronic Notes in Theoretical Computer Science, volume 6, Pittsburgh, 1997. URL: http://www.elsevier.nl/locate/entcs/volume6.html, 18 pages.

....a functor it is now easy to define A Omega B, namely as (A GammaffiB ) and A . B, as A GammaffiB. In [Pra97] we make a start on the project of proving full completeness of linear logic interpreted over Chu spaces, by showing that for formulas of multiplicative linear logic having at most two occurrences of each variable, cut free proof nets of those formulas are in bijective correspondence with the ....

V.R. Pratt. Towards full completeness of the linear logic of chu spaces. In Electronic Notes in Theoretical Computer Science, volume 6, Pittsburgh, 1997. URL: http://www.elsevier.nl/locate/entcs/volume6.html, 18 pages.

....needed to interpret the proofs between terms that make the system a categorical logic. Morphisms between functors are usually taken to be natural transformations. However linear logic contains functors of mixed variance 1 such as A GammaffiA, for which mere naturality is not enough. Elsewhere [Pra97] we have shown that when the morphisms are taken to be ordinary dinatural transformations, as done by Blute and Scott [BS96a, BS96b] for their Lauchli semantics of linear logic, then Girard s MIX free axiomatization of multiplicative linear logic is sound and fully complete for the fragment ....

....variable may occur both positively and negatively, as in the entailment A GammaffiA A GammaffiA . Here F (a 0 ; a) separates the negative occurrences a 0 from the positive occurrences a. When this diagram commutes for all morphisms f : a b of D is called dinatural. Elsewhere [Pra97] we have shown that the dinatural transformations between functors in Chu built up with tensor and perp soundly interpret MLL in that every cut free MLL proof has a distinct interpretation as a dinatural transformation. We showed furthermore that for Binary MLL, that fragment of MLL having one ....

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V.R. Pratt. Towards full completeness of the linear logic of chu spaces. In Electronic Notes in Theoretical Computer Science, volume 6, Pittsburgh, 1997. URL: http://www.elsevier.nl/locate/entcs/volume6.html, 18 pages.

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