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The semantics of BI and resource tableaux

D. GALMICHE, D. MÉRY, D. PYM
2005 Mathematical Structures in Computer Science  
As consequences of the relationships between semantics of BI and resource tableaux, we prove two strong new results for propositional BI: its decidability and the finite model property with respect to  ...  The logic of bunched implications, BI, provides a logical analysis of a basic notion of resource rich enough, for example, to form the logical basis for "pointer logic" and "separation logic" semantics  ...  We are grateful to Hongseok Yang, Peter O'Hearn, and the anonymous referees for their help with finding errors in, and with the presentation of, this long and complex paper.  ... 
doi:10.1017/s0960129505004858 fatcat:n2gktzqns5d57pq3phcf6mlvcm

A Substructural Epistemic Resource Logic [chapter]

Didier Galmiche, Pierre Kimmel, David Pym
2016 Lecture Notes in Computer Science  
We give a labelled tableaux calculus and establish soundness and completeness with respect to the resource semantics.  ...  The new modalities can be seen as generalizations of the usual epistemic modalities. The logic combines Boolean BI's resource semantics with epistemic agency.  ...  The soundness and completeness of BI for the semantics given above is established in [20] and via labelled tableaux in [11] , and the completeness of BBI for the partial monoid semantics described above  ... 
doi:10.1007/978-3-662-54069-5_9 fatcat:6theufuk6zednmk6tf7yaemdhi

A Substructural Epistemic Resource Logic: Theory and Modelling Applications [article]

Didier Galmiche, Pierre Kimmel, David Pym
2019 arXiv   pre-print
We also give a labelled tableaux calculus and establish soundness and completeness with respect to the resource semantics.  ...  The logic combines Boolean BI's resource semantics --- we introduce BI and its resource semantics at some length --- with epistemic agency.  ...  Acknowledgements We are grateful to Simon Docherty and to the anonymous referees for their comments on earlier drafts of this paper.  ... 
arXiv:1909.07296v1 fatcat:aksqqhwgn5fbjb3kqyxvw4hyla

OUP accepted manuscript

2019 Journal of Logic and Computation  
We also give a labelled tableaux calculus and establish soundness and completeness with respect to the resource semantics.  ...  The logic combines Boolean BI's resource semantics-we introduce BI and its resource semantics at some length-with epistemic agency.  ...  Acknowledgements We are grateful to Simon Docherty and to the anonymous referees for their comments on earlier drafts of this paper.  ... 
doi:10.1093/logcom/exz024 fatcat:uz2oxrh7evhmrm4rb7oze4osmq

Resource Graphs and Countermodels in Resource Logics

Didier Galmiche, Daniel Méry
2005 Electronical Notes in Theoretical Computer Science  
In this abstract we emphasize the role of a semantic structure called resource graph in order to study the provability in some resource-sensitive logics, like the Bunched Implications Logic (BI) or the  ...  Such a semantic structure is appropriate for capturing the particular interactions between different kinds of connectives (additives and multiplicatives in BI, commutatives and non-commutatives in NL)  ...  and BI logic The development of a mathematical theory of resource is one of the objectives of the programme of study of the logic of bunched implications (BI) [15, 16] .  ... 
doi:10.1016/j.entcs.2004.06.069 fatcat:b3ton3aln5c27kczbelzafclui

Expressivity Properties of Boolean BI Through Relational Models [chapter]

Didier Galmiche, Dominique Larchey-Wendling
2006 Lecture Notes in Computer Science  
properties for BI and BBI, embedding algorithmic solution to BBI constraints solving introduction of the notion of invertible resource Perspectives: implement constraint solving and proof-search for BBI  ...  intuitionistic Kripke semantics for additives relevant Kripke semantics for multiplicatives sound and complete Kripke semantics for BI C for BI and BBI Labels and constraints based methods calculi with  ... 
doi:10.1007/11944836_33 fatcat:bfma3l6cajd7vgcuxyqmuffydi

Page 9555 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
Summary: “In this paper, we study semantic labelled tableaux for the propositional bunched implications logic (BI) that freely combines intuitionistic logic (IL) and multiplicative intuitionistic linear  ...  We prove the soundness and completeness of this calculus with respect to the Kripke resource semantics with emphasis on countermodel construction.  ... 

Intuitionistic Layered Graph Logic [chapter]

Simon Docherty, David Pym
2016 Lecture Notes in Computer Science  
We give a soundness and completeness theorem for a labelled tableaux system with respect to a Kripke semantics on graphs.  ...  Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature.  ...  In particular, in BI we have both the standard conjunction ∧ and the multiplicative conjunction * . This connective can be understood through an interpretation known as resource semantics.  ... 
doi:10.1007/978-3-319-40229-1_32 fatcat:ackw5d3w2ras7m3vb7odrv5hiy

Intuitionistic Layered Graph Logic

Simon Docherty, David Pym
2017 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence  
We give a soundness and completeness theorem for a labelled tableaux system with respect to a Kripke semantics on graphs.  ...  Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature.  ...  In particular, in BI we have both the standard conjunction ∧ and the multiplicative conjunction * . This connective can be understood through an interpretation known as resource semantics.  ... 
doi:10.24963/ijcai.2017/673 dblp:conf/ijcai/DochertyP17 fatcat:pjvteytgcbaxrmmkpz4ovcyzzi

Modular Tableaux Calculi for Separation Theories [chapter]

Simon Docherty, David Pym
2018 Lecture Notes in Computer Science  
Parametric soundness and completeness of the framework is proved by a novel representation of tableaux systems as coherent theories, suggesting a strategy for implementation and a tentative first step  ...  Generalizing systems utilized in the proof theory of bunched logics, we propose a framework of tableaux calculi that are generically extendable by rules that correspond to separation theories axiomatized  ...  This analysis is known as resource semantics. A sound interpretation of BI is given by extending the standard poset semantics for propositional intuitionistic logic.  ... 
doi:10.1007/978-3-319-89366-2_24 fatcat:mxobe4hck5cuphdtn4idj3wlmm

Intuitionistic Layered Graph Logic: Semantics and Proof Theory [article]

Simon Docherty, David Pym
2018 arXiv   pre-print
We indicate the utility of predicate ILGL with a resource-labelled bigraph model.  ...  We give soundness and completeness theorems for a labelled tableaux system with respect to a Kripke semantics on graphs.  ...  This means we can prove properties of the layered graph semantics algebraically and yields the equivalence of the Hilbert and labelled tableaux systems.  ... 
arXiv:1702.05795v3 fatcat:euas2qm3avhlzgfhozrt2z27ie

Exploring the relation between Intuitionistic BI and Boolean BI: an unexpected embedding

DOMINIQUE LARCHEY-WENDLING, DIDIER GALMICHE
2009 Mathematical Structures in Computer Science  
Finally we deduce, as main and unexpected result, a sound and faithful embedding of BI into BBI. 2 logic (Ishtiaq and O'Hearn 2001) denoted SL here, and the product of resource and process composition  ...  and their solutions grasp the semantic properties of these logics.  ...  Acknowledgments The authors wish to thank the two anonymous referees for useful comments and suggestions about the overall organization of the paper, their detailed review of the proofs and the typos we  ... 
doi:10.1017/s0960129509007567 fatcat:47s7n4wgjfhaxoq3tgojg4fa5m

A Stone-type Duality Theorem for Separation Logic Via its Underlying Bunched Logics

Simon Docherty, David Pym
2018 Electronical Notes in Theoretical Computer Science  
This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualises the 'resource semantics' interpretation underpinning Separation Logic amongst them.  ...  We give a systematic treatment of Stone-type duality theorems for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar Boolean BI, and concluding with  ...  The semantics of BI Pointer Logic, a theory of (first-order) BBI, is an instance of BBI's resource semantics in which the monoid of resources is constructed from the program's heap.  ... 
doi:10.1016/j.entcs.2018.03.018 fatcat:elprorqfsjcl5ohjfve6zn3tz4

A logic of separating modalities

Jean-René Courtault, Didier Galmiche, David Pym
2016 Theoretical Computer Science  
We give a proof system based on a labelled tableaux calculus with countermodel extraction, establishing its soundness and completeness with respect to the semantics.  ...  LSM's modalities, which generalize those of S4, combine, within a quite general relational semantics, BI's resource semantics with modal accessibility.  ...  Concurrent separation logic is built upon the resource semantics of bunched logic and handles concurrent processes in the style of Hoare logic.  ... 
doi:10.1016/j.tcs.2016.04.040 fatcat:c3ly2twv25hu5e2ofizqtutdca

Possible worlds and resources: the semantics of BI

David J. Pym, Peter W. O'Hearn, Hongseok Yang
2004 Theoretical Computer Science  
This semantics may be motivated from a basic model of the notion of resource. We explain BI's proof-theoretic, categorical and semantic origins.  ...  We give an extensive discussion of BI as a semantically based logic of resources, giving concrete models based on Petri nets, ambients, computer memory, logic programming, and money.  ...  These models give concrete examples of the resource interpretation of BI-algebras and BI's forcing semantics.  ... 
doi:10.1016/j.tcs.2003.11.020 fatcat:bjvwjz2nmvdjvhvz6abs5zrsmq
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