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BI-hyperdoctrines, higher-order separation logic, and abstraction

Bodil Biering, Lars Birkedal, Noah Torp-Smith
2007 ACM Transactions on Programming Languages and Systems  
Moreover, we introduce the notion of a BI hyperdoctrine and show that it soundly models classical and intuitionistic first-and higher-order predicate BI, and use it to show that we may easily extend separation  ...  logic to higher-order.  ...  ACKNOWLEDGMENTS The authors wish to thank Carsten Butz and the anonymous referees of previous versions of the work in this paper for helpful comments and insights.  ... 
doi:10.1145/1275497.1275499 fatcat:ipusbj3h5rf2vddcwrafhit44e

BI Hyperdoctrines and Higher-Order Separation Logic [chapter]

Bodil Biering, Lars Birkedal, Noah Torp-Smith
2005 Lecture Notes in Computer Science  
Moreover, we introduce the notion of a BI hyperdoctrine and show that it soundly models classical and intuitionistic first-and higher-order predicate BI, and use it to show that we may easily extend separation  ...  logic to higher-order.  ...  Acknowledgements: The authors wish to thank Carsten Butz and the anonymous referees for their helpful comments and suggestions.  ... 
doi:10.1007/978-3-540-31987-0_17 fatcat:z2gmsmoirngnblxwrxmsnv5uye

Local Local Reasoning: A BI-Hyperdoctrine for Full Ground Store [chapter]

Miriam Polzer, Sergey Goncharov
2020 Lecture Notes in Computer Science  
Our main result is a construction of an (intuitionistic) BI-hyperdoctrine, which is arguably the semantic core of higher order logic over local store.  ...  We capitalize on recent progress on categorical semantics of full ground store, in terms of a full ground store monad, to build a corresponding semantics of a higher order logic over the corresponding  ...  art advanced framework for higher order separation logic [8] .  ... 
doi:10.1007/978-3-030-45231-5_28 fatcat:d2deva7z7fdk5gy2rgjuo7ux6e

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

Simon Docherty, David Pym
2018 Electronical Notes in Theoretical Computer Science  
Separation Logic.  ...  Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -in addition to elegant abstraction -they strengthen soundness and completeness  ...  have appeared elsewhere in the literature as BI hyperdoctrines where they were used to prove the existence of models of higher-order variants of Separation Logic [3] .  ... 
doi:10.1016/j.entcs.2018.03.018 fatcat:elprorqfsjcl5ohjfve6zn3tz4

On Models of Higher-Order Separation Logic

Aleš Bizjak, Lars Birkedal
2018 Electronical Notes in Theoretical Computer Science  
higher-order logic.  ...  We show how tools from categorical logic can be used to give a general account of models of higher-order separation logic with a sublogic of so-called persistent predicates satisfying the usual rules of  ...  BI-algebras and BI-hyperdoctrines In this section we begin by calling to mind the notion of a BI-hyperdoctrine, a category-theoretic definition of a model of higher-order separation logic [4] .  ... 
doi:10.1016/j.entcs.2018.03.016 fatcat:xgpv7f4xuvelvmc6cilqeetohy

Local Local Reasoning: A BI-Hyperdoctrine for Full Ground Store [article]

Miriam Polzer, Sergey Goncharov
2020 arXiv   pre-print
Our main result is a construction of an (intuitionistic) BI-hyperdoctrine, which is arguably the semantic core of higher order logic over local store.  ...  We capitalize on recent progress on categorical semantics of full ground store, in terms of a full ground store monad, to build a corresponding semantics of a higher order logic over the corresponding  ...  art advanced framework for higher order separation logic [8] .  ... 
arXiv:2003.05386v1 fatcat:s7ttbzkrpjgkld4pljkn33dh2i

Stone-Type Dualities for Separation Logics [article]

Simon Docherty, David Pym
2019 arXiv   pre-print
This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI).  ...  to both classical and intuitionistic Separation Logic.  ...  [6] to prove the existence of models of higher-order variants of Separation Logic.  ... 
arXiv:1710.03021v2 fatcat:dvrepipdzbctli2aqdar2jg4lu

ModuRes: A Coq Library for Modular Reasoning About Concurrent Higher-Order Imperative Programming Languages [chapter]

Filip Sieczkowski, Aleš Bizjak, Lars Birkedal
2015 Lecture Notes in Computer Science  
We show how the library can be used to construct models of type systems and logics for reasoning about concurrent higher-order imperative programming languages.  ...  It is well-known that it is challenging to build semantic models of type systems or logics for reasoning about concurrent higher-order imperative programming languages.  ...  Acknowledgements The formalization of the general solution of the recursive domain equation is inspired by an earlier, unpublished development by Varming and Birkedal.  ... 
doi:10.1007/978-3-319-22102-1_25 fatcat:fsibeompknf6dfdi6wcqhllmqm

Affine logic for constructive mathematics [article]

Michael Shulman
2021 arXiv   pre-print
This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces.  ...  Affine logic and the antithesis construction thus systematically "constructivize" classical definitions, handling the resulting bookkeeping automatically.  ...  My understanding of constructive mathematics and linear  ... 
arXiv:1805.07518v2 fatcat:t34if2wdzreqvecu7mahhzodj4

A theory of indirection via approximation

Aquinas Hobor, Robert Dockins, Andrew W. Appel
2010 Proceedings of the 37th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '10  
powerful substructural accounting required to model (higher-order) separation logic.  ...  Indirect reference can appear in many guises, such as heap pointers, higher-order functions, object references, and shared-memory mutexes.  ...  We would like to thank Amal Ahmed, Nick Benton, Matthew Parkinson, Andrew Pitts, Peter Sewell, Konrad Slind, and the anonymous referees for their helpful suggestions on earlier drafts of this paper.  ... 
doi:10.1145/1706299.1706322 dblp:conf/popl/HoborDA10 fatcat:cftjenykjbafbejm2ac4rithh4

A theory of indirection via approximation

Aquinas Hobor, Robert Dockins, Andrew W. Appel
2010 SIGPLAN notices  
powerful substructural accounting required to model (higher-order) separation logic.  ...  Indirect reference can appear in many guises, such as heap pointers, higher-order functions, object references, and shared-memory mutexes.  ...  We would like to thank Amal Ahmed, Nick Benton, Matthew Parkinson, Andrew Pitts, Peter Sewell, Konrad Slind, and the anonymous referees for their helpful suggestions on earlier drafts of this paper.  ... 
doi:10.1145/1707801.1706322 fatcat:k66shv3luffvhhsy5mlsmxgtjm

From IF to BI: a tale of dependence and separation [article]

Samson Abramsky, Jouko Vaananen
2011 arXiv   pre-print
We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Vaananen, and their compositional semantics due to Hodges.  ...  We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives.  ...  logic coexisting is the setting for BI logic, the "logic of Bunched Implications" of David Pym and Peter O'Hearn (O'Hearn and Pym, 1999; Pym, 2002) , which forms the basis for Separation logic (Reynolds  ... 
arXiv:1102.1388v1 fatcat:3fmf7xkqu5d23b5lmcp5lnqwc4

From IF to BI

Samson Abramsky, Jouko Väänänen
2008 Synthese  
We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges.  ...  We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives.  ...  logic coexisting is the setting for BI logic, the "logic of Bunched Implications" of David Pym and Peter O'Hearn (O'Hearn and Pym, 1999; Pym, 2002) , which forms the basis for Separation logic (Reynolds  ... 
doi:10.1007/s11229-008-9415-6 fatcat:o6sluhibovgxngsy4ze3kzoevu

Verifying Generics and Delegates [chapter]

Kasper Svendsen, Lars Birkedal, Matthew Parkinson
2010 Lecture Notes in Computer Science  
In this paper, we address this by extending a higher-order separation logic with new assertions for reasoning about delegates and variables.  ...  Recently, object-oriented languages, such as C , have been extended with language features prevalent in most functional languages: parametric polymorphism and higher-order functions.  ...  Acknowledgements We would like to thank our anonymous reviewers for their comments and in particular, for suggesting that lookup could be defined in terms of s →.  ... 
doi:10.1007/978-3-642-14107-2_9 fatcat:i6crxdjcuza4redrfgvgsk2kem

Categorical Harmony and Paradoxes in Proof-Theoretic Semantics [chapter]

Yoshihiro Maruyama
2015 Advances in Proof-Theoretic Semantics  
and related paradoxical logical constants.  ...  Prooftheoretic semantics is a semantic enterprise to articulate an inferentialist account of the meaning of logical constants and inferences within the proof-theoretic tradition of Gentzen, Prawitz, and  ...  Acknowledgments I would like to thank Peter Schroeder-Heister for fruitful discussions in Oxford in spring 2012, which, inter alia, led to the table of degrees of paradoxicality of logical constants.  ... 
doi:10.1007/978-3-319-22686-6_6 fatcat:mzlptyfx7nddjalkuakf4bv4pi
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