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Proof Nets and Explicit Substitutions
[chapter]

2000
*
Lecture Notes in Computer Science
*

We refine the simulation technique introduced in [10] to show strong normalization of λ-calculi with

doi:10.1007/3-540-46432-8_5
fatcat:474vipvuojghfis7jtgwfibxli
*explicit**substitutions*via termination of cut elimination in*proof**nets*[13] . ... We first propose a notion of equivalence relation for*proof**nets*that extends the one in [9],*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. ...##
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Extending the Explicit Substitution Paradigm
[chapter]

2005
*
Lecture Notes in Computer Science
*

We present a simple term language with

doi:10.1007/978-3-540-32033-3_30
fatcat:n4xb3zldzveddnrsuoyn76amf4
*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

2013
*
International Conference on Rewriting Techniques and Applications
*

The insight indeed comes from the theory of

doi:10.4230/lipics.rta.2013.39
dblp:conf/rta/Accattoli13
fatcat:niytow7rsbhuxabwthipfbwlti
*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*. ...##
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Proof nets and the call-by-value lambda-calculus

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

doi:10.4204/eptcs.113.5
fatcat:g3mz6ate4bgp3keejyomgl7dge
*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. ...##
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Labelled Lambda-calculi with Explicit Copy and Erase

2010
*
Electronic Proceedings in Theoretical Computer Science
*

also keep track of exponential information present in call-by-value

doi:10.4204/eptcs.22.5
fatcat:i4dj6ftqufhnbhlrffimzawsba
*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

2012
*
Journal of Logic and Computation
*

into Linear Logic

doi:10.1093/logcom/exs021
fatcat:lsmztkky7vc77dssnyybzwr3hu
*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

2015
*
Theoretical Computer Science
*

This paper gives a detailed account of the relationship between (a variant of) the call-by-value lambda calculus

doi:10.1016/j.tcs.2015.08.006
fatcat:ahlkhtswrzfhfftaku6gi3qboi
*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. ...##
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Compressing Polarized Boxes

2013
*
2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
*

The parallel nature of

doi:10.1109/lics.2013.49
dblp:conf/lics/Accattoli13
fatcat:vg26sbwapne7fchwqpkd56rh4y
*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 [23] 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. ...##
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Strong Normalization of Proof Nets Modulo Structural Congruences
[chapter]

1999
*
Lecture Notes in Computer Science
*

commutative binary operator that can float freely in

doi:10.1007/3-540-48685-2_6
fatcat:kc763aeowzf6zkeblb6ojul75a
*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 [5] to simulate a calculus with*explicit**substitution*. ...##
###
Proof Nets and the Linear Substitution Calculus
[article]

2018
*
arXiv
*
pre-print

The two systems however have different granularities, in particular

arXiv:1808.03395v1
fatcat:g4x23aiusvfcxi5dgttxdqoc7q
*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 [18] . ...##
###
Proof Nets and the Linear Substitution Calculus
[chapter]

2018
*
Lecture Notes in Computer Science
*

The two systems however have different granularities, in particular

doi:10.1007/978-3-030-02508-3_3
fatcat:37gfbyblnfgkxp2o4c2s7eopry
*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 [18] . ...##
###
Proof Nets for First-Order Additive Linear Logic

2019
*
International Conference on Rewriting Techniques and Applications
*

We present two versions of our

doi:10.4230/lipics.fscd.2019.22
dblp:conf/rta/HeijltjesHS19
fatcat:oayjro45zfdr7fqr54bgeu3jam
*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. ...##
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Encoding an Object Calculus into Interaction Nets

2005
*
Electronical Notes in Theoretical Computer Science
*

We propose an encoding of an object calculus into interaction

doi:10.1016/j.entcs.2005.03.024
fatcat:szxs6v3farbmxafxypcmfcgcue
*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*. ...##
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Resource operators for λ-calculus

2007
*
Information and Computation
*

We present a simple term calculus with an

doi:10.1016/j.ic.2006.08.008
fatcat:3rmvqhcrpjbd3gs24yqzmvgvtm
*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] . ...##
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Unification nets: canonical proof net quantifiers
[article]

2018
*
arXiv
*
pre-print

These results extend beyond MLL1 via a broader methodological insight: for canonical quantifiers, the standard parallel/sequential dichotomy of

arXiv:1802.03224v1
fatcat:3mz5xcqzuvbzljrllx5lzajgwq
*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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