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Provability in Intuitionistic Linear Logic from a New Interpretation on Petri nets

1998
*
Electronical Notes in Theoretical Computer Science
*

*From*the relationships between the notions of ordered m o n o i d and of quantale we de ne

*a*

*new*

*interpretation*of ILL

*on*

*Petri*

*nets*that is complete and veri es the property of nite models. ...

*In*the latter case, the correspondence between

*Intuitionistic*

*Linear*

*Logic*ILL and

*Petri*

*nets*illustrates the interest of e cient proof search methods for proving speci cations or properties of distributed ... Then we propose,

*from*

*a*study of the relationships between the notions of ordered monoid and of quantale, to de ne

*a*

*new*

*interpretation*of ILL

*on*

*Petri*

*nets*that is complete, by opposition to the

*one*of ...

##
###
The ILLTP Library for Intuitionistic Linear Logic

2019
*
Electronic Proceedings in Theoretical Computer Science
*

Finally, we generate

doi:10.4204/eptcs.292.7
fatcat:hdwnb3hccjevlmfucibkx2vcni
*a*comprehensive set of reachability problems for*Petri**nets*and encode such problems as*linear**logic*sequents, thus enlarging our collection of problems. ... We analyze four different translations of*intuitionistic**logic*into*linear**logic*and compare their proofs using*a**linear**logic*based prover with focusing. ...*nets*as*linear**logic*sequents. ...##
###
A Linear Logic Based Approach to Timed Petri Nets
[chapter]

2008
*
Petri Net, Theory and Applications
*

*In*(Tanabe, 1997) ,

*a*relationship between

*a*timed

*Petri*

*net*and

*a*temporal

*linear*

*logic*was discussed based

*on*quantale models with the soundness theorem for this

*logic*. ...

*In*the present paper,

*a*kind of temporal

*linear*

*logic*, called

*linear*-time

*linear*

*logic*, is used to describe timed

*Petri*

*nets*with timed tokens. ...

*A*

*Linear*

*Logic*Based Approach to Timed

*Petri*

*Nets*,

*Petri*

*Net*, Theory and Applications, Vedran Kordic (Ed.), ISBN: 978-3-902613-12-7, InTech, Available

*from*: http://www.intechopen.com/books/petri_net_theory_and_applications ...

##
###
Page 5799 of Mathematical Reviews Vol. , Issue 96j
[page]

1996
*
Mathematical Reviews
*

As to (1), generalized

*Petri**nets*can include non-deterministic transitions: once such*a*transition is fired it is not known*in*advance which direction*from**a*set of alternative*ones*will be chosen. 96j ...*Petri**nets*, Horn programs,*linear**logic*and vector games. (English summary) Proof theory,*provability**logic*, and computation (Berne, 1994). Ann. Pure Appl.*Logic*75 (1995), no. 1-2, 107-135. ...##
###
Labelled Proof Systems for Intuitionistic Provability
[chapter]

2000
*
Applied Logic Series
*

*In*this paper, we propose

*new*labelled proof systems to analyse the

*intuitionistic*

*provability*

*in*classical and

*linear*

*logics*. ... An alternative approach, based

*on*the notion of proof-

*net*and

*on*the definition of suitable labelled classical proof-

*nets*, allows to directly study the

*intuitionistic*

*provability*by constructing

*intuitionistic*...

*INTUITIONISTIC*

*PROVABILITY*

*IN*

*LINEAR*

*LOGIC*Let us now consider the

*intuitionistic*

*provability*

*from*classical search

*in*the case of

*linear*

*logic*. ...

##
###
Computational Complexity of Deciding Provability in Linear Logic and its Fragments
[article]

2021
*
arXiv
*
pre-print

We prove an original complexity characterization of

arXiv:2110.00562v1
fatcat:a6sidawnbjh5jpf2cvrermfrcy
*a*fragment of the*logic*and present ideas for*a**new*, structural approach to the examination of*provability**in**linear**logic*. ... This*interpretation*makes*linear**logic**a*useful tool for formalisation*in*mathematics and computer science. ... (*Petri**net*).*A*transition*on**a*set is*a*pair( , ) with , ⊆ .*A**Petri**net*= ( , )consists of*a*set of nodes and*a*transition relation . ...##
###
Proofs, concurrent objects and computations in a FILL framework
[chapter]

1996
*
Lecture Notes in Computer Science
*

*In*this paper, we propose and analyze

*a*framework based

*on*Full

*Intuitionistic*

*Linear*

*Logic*(FILL),

*logical*fragment with potentialities for non-determinisms management, as foundation of concurrent object-oriented ...

*In*this context,

*one*aim can be to achieve di erent forms of programming as

*logic*, object-oriented or concurrent

*ones*

*in*

*a*same

*logical*language. ... FILL: Full

*Intuitionistic*

*Linear*

*Logic*Full

*Intuitionistic*

*Linear*

*Logic*(FILL), unlike

*intuitionistic*

*linear*

*logic*(ILL), includes the multiplication disjunction par. ...

##
###
Page 2333 of Mathematical Reviews Vol. , Issue 2002D
[page]

2002
*
Mathematical Reviews
*

*In*this paper, we first construct non-distributive quantales

*from*

*Petri*

*nets*by using

*a*closure operation, and prove completeness of full

*linear*

*logic*without the exponential for them. ...

*One*of the difficulties

*in*proving completeness for full proposi- tional

*intuitionistic*

*linear*

*logic*lies

*in*the distributivity of [ over U, which is not valid

*in*

*linear*

*logic*. ...

##
###
Completeness results for linear logic on Petri nets

1997
*
Annals of Pure and Applied Logic
*

*One*

*logic*considered is the @-free fragment of

*intuitionistic*

*linear*

*logic*without the exponential !. For this fragment

*Petri*

*nets*form

*a*sound and complete model. ... Completeness is shown for several versions of Girard's

*linear*

*logic*with respect to

*Petri*

*nets*as the class of models. ... Introduction

*In*[3] it was shown how

*Petri*

*nets*can naturally be made into models of Girard's

*intuitionistic*

*linear*

*logic*

*in*such

*a*way that many properties

*one*might wish to state of

*nets*become expressible ...

##
###
Logic Programming in a Fragment of Intuitionistic Temporal Linear Logic
[chapter]

2001
*
Lecture Notes in Computer Science
*

This paper describes

doi:10.1007/3-540-45635-x_29
fatcat:qph3nwkip5darjxkhd3c5gfaxu
*a**logic*programming language, called TLLP, based*on**intuitionistic*temporal*linear**logic*. ... This*logic*, an extension of*linear**logic*with some features*from*temporal*logics*, allows the use of the modal operators ' '(next-time) and '2'(always)*in*addition to the operators used*in**intuitionistic*... This program checks the reachability of*a*Timed*Petri**Net**in*Fig. 6*from*the initial marking (*one*token*in*p) to the final marking (*one*token*in*p and two tokens*in*r). ...##
###
Linear Logic on Petri Nets

1994
*
BRICS Report Series
*

This article might serve as

doi:10.7146/brics.v1i3.21667
fatcat:q5zaidp6nzelhp3rsqr2466ttu
*a*tutorial, providing*one**in*-road into Girard's*linear**logic*via*Petri**nets*. With this*in*mind we have added several exercises and their solutions. ... An aim is to use*Petri**nets*to give an understanding of*linear**logic*and give some appraisal of the value of*linear**logic*as*a*specification*logic*for*Petri**nets*. ... derived*from*the*linear**one*, and through the propositions-as-types reading of*logic*it gave*a*n explanation of*intuitionistic**logic**in*terms of*a*more primitive and*new*classical*linear**logic*. ...##
###
Linear logic on Petri nets
[chapter]

1994
*
Lecture Notes in Computer Science
*

This article might serve as

doi:10.1007/3-540-58043-3_20
fatcat:wdocu4s4rzdcvhmviiafrpe2eq
*a*tutorial, providing*one**in*-road into Girard's*linear**logic*via*Petri**nets*. With this*in*mind we have added several exercises and their solutions. ... An aim is to use*Petri**nets*to give an understanding of*linear**logic*and give some appraisal of the value of*linear**logic*as*a*specification*logic*for*Petri**nets*. ... be derived*from*the*linear**one*, and through the propositions-as-types reading of*logic*it gave an explanation of*intuitionistic**logic**in*terms of*a*more primitive and*new*classical*linear**logic*. ...##
###
Proof-search in type-theoretic languages: an introduction

2000
*
Theoretical Computer Science
*

The strong connections between di erent proof-search methods

doi:10.1016/s0304-3975(99)00169-3
fatcat:gy4nmmsupfeype23pw3jzbrvwu
*in**logics*, type theories and*logical*frameworks, together with their impact*on*programming and implementation issues, are central*in*this context ... We do not claim to cover all of the theoretical and implementation issues*in*the study of proof-search*in*type-theoretic languages; rather, we present some key ideas and problems, starting*from*well-motivated ... Acknowledgements We are grateful to Maurice Nivat for encouraging us to produce this article and for acting as*a*referee. Any remaining errors are our responsibility. ...##
###
Page 5161 of Mathematical Reviews Vol. , Issue 96i
[page]

1996
*
Mathematical Reviews
*

iff it is

*provable**in**intuitionistic**logic*. ...*In*the rest of this sec- tion, fuzzy sensors, some aspects of pattern recognition and*Petri**nets*are discussed. ...##
###
Relating State-Based and Process-Based Concurrency through Linear Logic

2006
*
Electronical Notes in Theoretical Computer Science
*

Specifically, we present

doi:10.1016/j.entcs.2006.05.043
fatcat:eavrmrbudjbvbhnu2zlxecsd2e
*a**new*semantics for multiset rewriting founded*on*an alternative view of*linear**logic*. ...*In*the first part of this paper, we review some of these*interpretations*of concurrent languages into*linear**logic*. ... Acknowledgments We are grateful to Frank Pfenning, Dale Miller, Mark-Oliver Stehr, Valeria de Paiva, Gerald Allwein, Steve Zdancevic and Vijay Saraswat for their comments*on*various aspects of this work ...
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