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**1 - 5**of**5**### Proof Nets as Processes

, 2012

"... Abstract. We present delta-calculus, a novel interpretation of Linear Logic, in the form of a typed process algebra that enjoys a Curry-Howard correspondence with Proof Nets. Reduction inherits the qualities of the logical objects: termination, deadlock-freedom, determinism, and very importantly, a ..."

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Abstract. We present delta-calculus, a novel interpretation of Linear Logic, in the form of a typed process algebra that enjoys a Curry-Howard correspondence with Proof Nets. Reduction inherits the qualities of the logical objects: termination, deadlock-freedom, determinism, and very importantly, a high degree of parallelism. We obtain the necessary soundness results and provide a propositions-as-types theorem. The ba-sic system is extended in two directions. First, we adapt it to interpret Affine Logic. Second, we propose extensions for general recursion, and introduce a novel form of recursive linear types. As an application we show a highly parallel type-preserving translation from a linear System F and extend it to the recursive variation. Our interpretation can be seen as a more canonical proof-theoretic alternative to several recent works on pi-calculus interpretations of linear sequent proofs (propositions-as-sessions) which exhibit reduced parallelism. 1

### Proof Nets in Process Algebraic Form

"... Abstract. We present δ-calculus, a computational interpretation of Lin-ear Logic, in the form of a typed process algebra whose structures corre-spond to Proof Nets, and where typing derivations correspond to linear sequent proofs. Term reduction shares the properties of cut elimination in the logic, ..."

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Abstract. We present δ-calculus, a computational interpretation of Lin-ear Logic, in the form of a typed process algebra whose structures corre-spond to Proof Nets, and where typing derivations correspond to linear sequent proofs. Term reduction shares the properties of cut elimination in the logic, and immediately we can obtain a number of inherited quali-ties, among which are termination, deadlock-freedom, and determinism. We obtain the expected soundness results and provide a propositions-as-types correspondence theorem. We then propose extensions for general recursion and the removal of the additive fragment while allowing to express similar behaviour, contributing to the theory of proof nets. 1

### A behavioural theory for a pi-calculus with preorders

"... Abstract. We study the behavioural theory of piP, a pi-calculus in the tradition of Fusions and Chi calculi. In contrast with such calculi, re-duction in piP generates a preorder on names rather than an equivalence relation. We present two characterisations of barbed congruence in piP: the first is ..."

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Abstract. We study the behavioural theory of piP, a pi-calculus in the tradition of Fusions and Chi calculi. In contrast with such calculi, re-duction in piP generates a preorder on names rather than an equivalence relation. We present two characterisations of barbed congruence in piP: the first is based on a compositional LTS, and the second is an axiomati-sation. The results in this paper bring out basic properties of piP, mostly related to the interplay between the restriction operator and the preorder on names. Consequently, piP is a calculus in the tradition of Fusion calculi, in which both types and behavioural equivalences can be exploited in order to reason rigorously about concurrent and mobile systems. 1

### This work is licensed under the Creative Commons Attribution License. Soft Session Types

"... We show how systems of session types can enforce interactions to be bounded for all typable pro-cesses. The type system we propose is based on Lafont’s soft linear logic and is strongly inspired by recent works about session types as intuitionistic linear logic formulas. Our main result is the exist ..."

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We show how systems of session types can enforce interactions to be bounded for all typable pro-cesses. The type system we propose is based on Lafont’s soft linear logic and is strongly inspired by recent works about session types as intuitionistic linear logic formulas. Our main result is the existence, for every typable process, of a polynomial bound on the length of any reduction sequence starting from it and on the size of any of its reducts. 1