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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 ...doi:10.1016/s1571-0661(05)01180-1 fatcat:grziahhbbjb5xgl5gtiz5e6lku
Finally, we generate 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. ...doi:10.4204/eptcs.292.7 fatcat:hdwnb3hccjevlmfucibkx2vcni
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 ...doi:10.5772/5319 fatcat:gw25kquocnemtn3d5kfpenlqb4
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. ...
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. ...doi:10.1007/978-94-011-4040-9_1 fatcat:zsmdap7xrnc7jlpwc5cpduc36u
We prove an original complexity characterization of 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 . ...arXiv:2110.00562v1 fatcat:a6sidawnbjh5jpf2cvrermfrcy
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. ...doi:10.1007/3-540-61487-7_26 fatcat:vsayzwsmeneudocev7dzarujgq
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. ...
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  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 ...doi:10.1016/s0168-0072(96)00024-3 fatcat:ksaovrfevjbfve5griq4bk5uxy
Lecture Notes in Computer Science
This paper describes 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). ...doi:10.1007/3-540-45635-x_29 fatcat:qph3nwkip5darjxkhd3c5gfaxu
BRICS Report Series
This article might serve as 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. ...doi:10.7146/brics.v1i3.21667 fatcat:q5zaidp6nzelhp3rsqr2466ttu
Lecture Notes in Computer Science
This article might serve as 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. ...doi:10.1007/3-540-58043-3_20 fatcat:wdocu4s4rzdcvhmviiafrpe2eq
The strong connections between di erent proof-search methods 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. ...doi:10.1016/s0304-3975(99)00169-3 fatcat:gy4nmmsupfeype23pw3jzbrvwu
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. ...
Specifically, we present 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 ...doi:10.1016/j.entcs.2006.05.043 fatcat:eavrmrbudjbvbhnu2zlxecsd2e
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