Nominal Models of Linear Logic

PhD thesisMore than 30 years after the discovery of linear logic, a simple fully-complete model has still not been established. As of today, models of logics with type variables rely on di-natural transformations, with the intuition that a proof should behave uniformly at variable types. Consequently, the interpretations of the proofs are not concrete. The main goal of this thesis was to shift from a 2-categorical setting to a first-order category. We model each literal by a pool of resources of a certain type, that we encode thanks to sorted names. Based on this, we revisit a range of categorical constructions, leading to nominal relational models of linear logic. As these fail to prove fully-complete, we revisit the fully-complete game-model of linear logic established by Melliès. We give a nominal account of concurrent game semantics, with an emphasis on names as resources. Based on them, we present fully complete models of multiplicative additive tensorial, and then linear logics. This model extends the previous result by adding atomic variables, although names do not play a crucial role in this result. On the other hand, it provides a nominal structure that allows for a nominal relationship between the Böhm trees of the linear lambda-terms and the plays of the strategies. However, this full-completeness result for linear logic rests on a quotient. Therefore, in the final chapter, we revisit the concurrent operators model which was first developed by Abramsky and Melliès. In our new model, the axiomatic structure is encoded through nominal techniques and strengthened in such a way that full completeness still holds for MLL. Our model does not depend on any 2-categorical argument or quotient. Furthermore, we show that once enriched with a hypercoherent structure, we get a static fully complete model of MALL

Event domains, stable functions and proof-nets

We pursue the program of exposing the intrinsic mathematical structure of the &amp;apos;&amp;apos;space of proofs&amp;apos;&amp;apos; of a logical system [S. Abramsky and R. Jagadeesan. Games and Full Completeness for Multiplicative Linear Logic, Journal of Symbolic Logic, (1994), vol. 59, no. 2, 543–574]. We study the case of Multiplicative-Additive Linear Logic (MALL). We use tools from Domain theory to develop a semantic notion of proof net for MALL, and prove a Sequentialization Theorem. This work forms part of a continuation of previous joint work with Radha Jagadeesan [S. Abramsky and R. Jagadeesan. Games and Full Completeness for Multiplicative Linear Logic, Journal of Symbolic Logic, (1994), vol. 59, no. 2, 543–574] and Paul-André Melliès [S. Abramsky and P.-A. Melliès. Concurrent Games and Full Completeness. Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science, 431–44, 1999].</p

Event domains, stable functions and proof-nets

We pursue the program of exposing the intrinsic mathematical structure of the &apos;&apos;space of proofs&apos;&apos; of a logical system [S. Abramsky and R. Jagadeesan. Games and Full Completeness for Multiplicative Linear Logic, Journal of Symbolic Logic, (1994), vol. 59, no. 2, 543–574]. We study the case of Multiplicative-Additive Linear Logic (MALL). We use tools from Domain theory to develop a semantic notion of proof net for MALL, and prove a Sequentialization Theorem. This work forms part of a continuation of previous joint work with Radha Jagadeesan [S. Abramsky and R. Jagadeesan. Games and Full Completeness for Multiplicative Linear Logic, Journal of Symbolic Logic, (1994), vol. 59, no. 2, 543–574] and Paul-André Melliès [S. Abramsky and P.-A. Melliès. Concurrent Games and Full Completeness. Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science, 431–44, 1999].© 2007 Elsevier B.V.Open access under CC BY-NC-ND license.(http://creativecommons.org/licenses/by-nc-nd/3.0/