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We show that our semantics yields a categorical model of Linear Logic and prove <jats:italic>full completeness<\/jats:italic> for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of <jats:italic>history-free<\/jats:italic> strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass, et al.<\/jats:p>","DOI":"10.2307\/2275407","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:51:53Z","timestamp":1146941513000},"page":"543-574","source":"Crossref","is-referenced-by-count":209,"title":["Games and full completeness for multiplicative linear logic"],"prefix":"10.1017","volume":"59","author":[{"given":"Samson","family":"Abramsky","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Radha","family":"Jagadeesan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200019976_ref041","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/092\/1003210"},{"key":"S0022481200019976_ref040","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(77)90044-5"},{"key":"S0022481200019976_ref038","first-page":"43","volume-title":"Proceedings of the sixth annual symposium on logic in computer science","author":"Lafont","year":"1991"},{"key":"S0022481200019976_ref036","first-page":"95","volume-title":"Proceedings of the seventeenth ACM symposium on principles of programming languages","author":"Lafont","year":"1990"},{"key":"S0022481200019976_ref035","article-title":"Remarques sur la theorie des jeux a deux personnes","volume":"1","author":"Joyal","year":"1977","journal-title":"Gazette des sciences mathematiques du Quebec"},{"key":"S0022481200019976_ref032","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0040266"},{"key":"S0022481200019976_ref031","volume-title":"Communicating sequential processes","author":"Hoare","year":"1985"},{"key":"S0022481200019976_ref027","first-page":"221","volume-title":"Logic Colloquium 88","author":"Girard","year":"1989"},{"key":"S0022481200019976_ref023","first-page":"223","volume-title":"Proceedings of the seventh symposium on logic in computer science","author":"Gonthier","year":"1992"},{"key":"S0022481200019976_ref022","volume-title":"Categories, allegories","volume":"39","author":"Freyd","year":"1990"},{"key":"S0022481200019976_ref021","unstructured":"Fleury A. and Retor\u00e9 C. , The MIX rule, unpublished note, 1990."},{"key":"S0022481200019976_ref017","unstructured":"Curien P. L. , Concrete data structures, sequential algorithms and linear logic, e-mail communication on types mailing list, (1992)."},{"key":"S0022481200019976_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(90)90151-7"},{"key":"S0022481200019976_ref008","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129500001274"},{"key":"S0022481200019976_ref004","first-page":"211","volume-title":"Proceedings of the seventh symposium on logic in computer science","author":"Abramsky","year":"1992"},{"key":"S0022481200019976_ref018","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/092\/1003194"},{"key":"S0022481200019976_ref028","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/092\/1003197"},{"key":"S0022481200019976_ref006","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129500000189"},{"key":"S0022481200019976_ref024","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(86)90044-7"},{"key":"S0022481200019976_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0064579"},{"key":"S0022481200019976_ref015","first-page":"344","volume-title":"Third workshop on the mathematical foundations of programming language semantics","author":"Coquand","year":"1987"},{"key":"S0022481200019976_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(92)90073-9"},{"key":"S0022481200019976_ref001","volume-title":"Proofs as processes","author":"Abramsky","year":"1991"},{"key":"S0022481200019976_ref037","unstructured":"Lamarche F. , Sequential algorithms, games and linear logic, unpublished lecture, 1992."},{"key":"S0022481200019976_ref029","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129500001328"},{"key":"S0022481200019976_ref009","first-page":"35","volume-title":"Theory and practice of sequential algorithms: the kernel of the applicative language CDS, Algebraic semantics","author":"Berry","year":"1985"},{"key":"S0022481200019976_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(93)90181-R"},{"key":"S0022481200019976_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-56287-7_113"},{"key":"S0022481200019976_ref026","first-page":"76","volume-title":"International Conference on Computer Logic, COLOG 88","volume":"417","author":"Girard","year":"1988"},{"key":"S0022481200019976_ref011","doi-asserted-by":"publisher","DOI":"10.4064\/fm-77-2-151-166"},{"key":"S0022481200019976_ref005","doi-asserted-by":"crossref","unstructured":"Abramsky S. , A strong completeness theorem for multiplicative linear logic: Preliminary announcement, E-mail communication on types mailing list, 1992.","DOI":"10.1007\/3-540-56287-7_113"},{"key":"S0022481200019976_ref034","unstructured":"Jacobs B. , Semantics of weakening and contraction, 1992, preprint."},{"key":"S0022481200019976_ref012","unstructured":"Blass A. , personal communication (1992)."},{"key":"S0022481200019976_ref014","volume-title":"Linear logic, coherence, and dinaturality","author":"Blute","year":"1992"},{"key":"S0022481200019976_ref039","first-page":"157","volume-title":"Logic colloquium, Bristol 1973","author":"Milner","year":"1975"},{"key":"S0022481200019976_ref020","doi-asserted-by":"publisher","DOI":"10.1007\/BF01622878"},{"key":"S0022481200019976_ref025","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(87)90045-4"},{"key":"S0022481200019976_ref033","unstructured":"Hyland J. 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