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### Proof Nets and Explicit Substitutions [chapter]

Roberto Di Cosmo, Delia Kesner, Emmanuel Polonovski
2000 Lecture Notes in Computer Science
We refine the simulation technique introduced in  to show strong normalization of λ-calculi with explicit substitutions via termination of cut elimination in proof nets  .  ...  We first propose a notion of equivalence relation for proof nets that extends the one in , and we show that cut elimination modulo this equivalence relation is terminating.  ...  Acknowledgments We would like to thank Bruno Guillaume and Pierre-Louis Curien for their interesting remarks.  ...

### Extending the Explicit Substitution Paradigm [chapter]

Delia Kesner, Stéphane Lengrand
2005 Lecture Notes in Computer Science
We present a simple term language with explicit operators for erasure, duplication and substitution enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic's Proof  ...  This formalism is the first term calculus with explicit substitutions having full composition and preserving strong normalisation.  ...  Explicit operators and Proof Nets Much work on explicit substitutions has been done in the last 10 years, for example [1, 5, 8, 32] .  ...

### Linear Logic and Strong Normalization

Beniamino Accattoli, Marc Herbstritt
2013 International Conference on Rewriting Techniques and Applications
The insight indeed comes from the theory of explicit substitutions, and from looking at the exponentials as a substitution device.  ...  In this paper we give a new presentation of MELL proof nets, without any commutative cut-elimination rule.  ...  Acknowledgements To Delia Kesner, who taught me the art of strong normalization proofs, and with whom I had uncountable discussions on commutative rules, normalization, and proof nets.  ...

### Proof nets and the call-by-value lambda-calculus

Beniamino Accattoli
2013 Electronic Proceedings in Theoretical Computer Science
This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets.  ...  In this way, we obtain an algebraic reformulation of proof nets. Moreover, we provide a simple correctness criterion for our proof nets, which employ boxes in an unusual way.  ...  To Stefano Guerrini, for introducing me to proof nets, correctness and the representation of λ -terms, and to Delia Kesner, for helping with the financial support of this work.  ...

### Labelled Lambda-calculi with Explicit Copy and Erase

Maribel Fernández, Nikolaos Siafakas
2010 Electronic Proceedings in Theoretical Computer Science
also keep track of exponential information present in call-by-value and call-by-name translations of the lambda-calculus into linear logic proof-nets.  ...  We present two rewriting systems that define labelled explicit substitution lambda-calculi.  ...  In Section 2 we review the syntax of the calculus of explicit substitutions (λ c -terms) and introduce basic terminology regarding linear logic proof-nets.  ...

### Labelled calculi of resources

M. Fernandez, N. Siafakas
2012 Journal of Logic and Computation
into Linear Logic proof nets, respectively.  ...  Since we need to track copy and erasing of substitutions, we will use a calculus where not only substitutions are explicit, but also copy and erase operations are part of the syntax.  ...  In Section 2, we review the syntax of the calculus of explicit substitutions (λ c -terms) and introduce basic terminology regarding proof nets.  ...

### Proof nets and the call-by-value λ -calculus

Beniamino Accattoli
2015 Theoretical Computer Science
This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus and linear logic proof nets.  ...  In this way, we obtain an algebraic reformulation of proof nets.  ...  Explicit Substitutions, or Sharing: proof nets implement a β-step by (at least) two cut-elimination steps.  ...

### Compressing Polarized Boxes

Beniamino Accattoli
2013 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
The parallel nature of proof nets, instead, requires the introduction of explicit boxes, which are global and synchronous constraints on the structure of graphs.  ...  The content of the box can then be recovered locally and unambiguously. Moreover, implicit boxes are more parallel than explicit boxes, as they realize a larger quotient.  ...  Its 0-depth net is in Fig. 4 .c. In  Laurent proves that an explicit net is the translation of a proof, i.e. it is an explicit proof-net, if and only if it is correct.  ...

### Strong Normalization of Proof Nets Modulo Structural Congruences [chapter]

Roberto Di Cosmo, Stefano Guerrini
1999 Lecture Notes in Computer Science
commutative binary operator that can float freely in and out of proof net boxes.  ...  or without explicit substitutions, which needs a notion of reduction more flexible than those present in the literature.  ...  and Kesner  to simulate a calculus with explicit substitution.  ...

### Proof Nets and the Linear Substitution Calculus [article]

Beniamino Accattoli
2018 arXiv   pre-print
The two systems however have different granularities, in particular proof nets have an explicit notion of sharing---the exponentials---and a micro-step operational semantics, while the λ-calculus has no  ...  Finally, such a detailed analysis of the relationship between terms and proof nets, suggests a new, abstract notion of proof net, based on rewriting considerations and not necessarily of a graphical nature  ...  A turning point was the link between explicit substitutions and linear logic proof nets by Di Cosmo and Kesner in  .  ...

### Proof Nets and the Linear Substitution Calculus [chapter]

Beniamino Accattoli
2018 Lecture Notes in Computer Science
The two systems however have different granularities, in particular proof nets have an explicit notion of sharing-the exponentials-and a micro-step operational semantics, while the λcalculus has no sharing  ...  Finally, such a detailed analysis of the relationship between terms and proof nets, suggests a new, abstract notion of proof net, based on rewriting considerations and not necessarily of a graphical nature  ...  A turning point was the link between explicit substitutions and linear logic proof nets by Di Cosmo and Kesner in  .  ...

### Proof Nets for First-Order Additive Linear Logic

Willem B. Heijltjes, Dominic J. D. Hughes, Lutz Straßburger, Michael Wagner
2019 International Conference on Rewriting Techniques and Applications
We present two versions of our proof nets. One, witness nets, retains explicit witnessing information to existential quantification.  ...  We present canonical proof nets for first-order additive linear logic, the fragment of linear logic with sum, product, and first-order universal and existential quantification.  ...  Dominic Hughes thanks his hosts, Wes Holliday and Dana Scott, at the UC Berkeley Logic Group.  ...

### Encoding an Object Calculus into Interaction Nets

Fabien Fleutot
2005 Electronical Notes in Theoretical Computer Science
We propose an encoding of an object calculus into interaction nets in two stages. First, we make the calculus fully explicit, i.e. with explicit substitutions, duplications and erasures.  ...  Then, we use this explicit calculus to produce an interaction net encoding of objects.  ...  Property 1 Explicit substitution in our calculus terminates. Proof.  ...

### Resource operators for λ-calculus

Delia Kesner, Stéphane Lengrand
2007 Information and Computation
We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic's proof-nets  ...  simulation of -reduction and full composition.  ...  Explicit control of resources and proof-nets Much work on explicit substitutions has been done in the last 15 years, for example [3, 7, 10, 37] .  ...

### Unification nets: canonical proof net quantifiers [article]

Dominic J. D. Hughes
2018 arXiv   pre-print
These results extend beyond MLL1 via a broader methodological insight: for canonical quantifiers, the standard parallel/sequential dichotomy of proof nets fails; an implicit/explicit witness dichotomy  ...  Work in progress extends unification nets to additives and uses them to extend combinatorial proofs [Proofs without syntax, Annals of Mathematics, 2006] to classical first-order logic.  ...  Along the north-south axis we have an implicit/explicit witness dichotomy: sequent calculus and Girard nets have redundant explicit existential witnesses (north); unification calculus and unification nets  ...
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