Geodesic
The Dialectica interpretation of first-order classical affine logic
Theory and applications of categories, Chu spaces: theory and applications, Tome 17 (2006), pp. 49-79.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

We give a Dialectica-style interpretation of first-order classical affine logic. By moving to a contraction-free logic, the translation (a.k.a. D-translation) of a first-order formula into a higher-type $\exists\forall$-formula can be made symmetric with respect to duality, including exponentials. It turned out that the propositional part of our D-translation uses the same construction as de Paiva's dialectica category GC and we show how our D-translation extends GC to the first-order setting in terms of an indexed category. Furthermore the combination of Girard's ?!-translation and our D-translation results in the essentially equivalent $\exists\forall$-formulas as the double-negation translation and Godel's original D-translation.
Classification : 03B47
Keywords: linear logic, dialectica interpretation, categorical logic 
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     author = {Masaru Shirahata},
     title = {The {Dialectica} interpretation of first-order classical affine logic},
     journal = {Theory and applications of categories},
     pages = {49--79},
     publisher = {mathdoc},
     volume = {17},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2006_17_a3/}
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Masaru Shirahata. The Dialectica interpretation of first-order classical affine logic. Theory and applications of categories, Chu spaces: theory and applications, Tome 17 (2006), pp. 49-79. http://geodesic.mathdoc.fr/item/TAC_2006_17_a3/