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A Higher-Order Calculus for Categories

Mario Jose Cáccamo, Glynn Winskel
2001 BRICS Report Series  
A calculus for a fragment of category theory is presented. The types in the language denote categories and the expressions functors.  ...  The calculus gives a basis for mechanising a theory of categories in a generic theorem prover like Isabelle.  ...  Acknowledgments: This work was inspired by Martin Hyland's Cambridge part III lecture course on category theory and its emphasis on end and coend manipulation.  ... 
doi:10.7146/brics.v8i27.21687 fatcat:hsnotpce6vfvxn5alf2p3immri

Decomposing typed lambda calculus into a couple of categorical programming languages [chapter]

Masahito Hasegawa
1995 Lecture Notes in Computer Science  
These languages are complementary to each other { one of them provides a rst-order programming style whereas the other does higherorder { and are \children" of the simply typed lambda calculus in the sense  ...  This decomposition is a consequence of a semantic analysis on typed lambda calculus due to C. Hermida and B. Jacobs HJ94].  ...  Acknowledgements I thank Claudio Hermida and Bart Jacobs for letting me know their results, and Rod Burstall, Martin Hofmann and Makoto Takeyama for their helpful comments.  ... 
doi:10.1007/3-540-60164-3_28 fatcat:4ylb4nrc3fcmlkshkqdqedoi4q

Review of type-logical semantics

Riccardo Pucella, Stephen Chong
2003 ACM SIGACT News  
The semantic will be given in some higher-order logic as we saw above. We assume that to every basic category corresponds a higher-order logic type.  ...  A (standard) frame for higher-order logic is specified by giving for each basic type τ ∈ BasTyp a domain Dom τ of values of that type.  ... 
doi:10.1145/637437.637442 fatcat:fpmeqj4f6bgg3kc67s4abfaywm

Page 3624 of Mathematical Reviews Vol. , Issue 89G [page]

1989 Mathematical Reviews  
The semantics of ordinary lambda calculus has a nice formula- tion using Cartesian closed categories [J. Lambek and P. J. Scott, Jn- troduction to higher order categorical logic, Cambridge Univ.  ...  Ehrhard have since proposed a syntactical and equational formalism for the higher-order polymor- phic type system [in Category theory and computer science (Edin- burgh, 1987), 40-56, Lecture Notes in Comput  ... 

A Categorical Model of an $$\mathbf {i/o}$$ -typed $$\pi $$ -calculus [chapter]

Ken Sakayori, Takeshi Tsukada
2019 Lecture Notes in Computer Science  
This paper introduces a new categorical structure that is a model of a variant of the i/o-typed π-calculus, in the same way that a cartesian closed category is a model of the λ-calculus.  ...  To the best of our knowledge, no categorical model has been given for the i/o-typed πcalculus, in contrast to session-typed calculi, to which corresponding logic and categorical structure were given.  ...  We would like to thank Naoki Kobayashi, Masahito Hasegawa and James Laird for discussions, and anonymous referees for valuable comments.  ... 
doi:10.1007/978-3-030-17184-1_23 fatcat:dt4csuzmrnf65oe2xz4hnawkcm

Applying Categorical Logic on a Holistic Security Architecture across Distributed Systems

Dimitrios Sisiaridis, Nick Rossiter
2011 2011 15th Panhellenic Conference on Informatics  
We propose the development of a new architecture for organizing in a holistic way the protection measures needed to be taken in a distributed system based on categorical logic. holistic security; process  ...  In typed λ-calculus higher-order functions are generally those with types containing more than one arrow.  ...  The exponentiation feature is adequate for binding system activities in a framework. Cartesian closed categories provide a natural setting for λ-calculus.  ... 
doi:10.1109/pci.2011.35 dblp:conf/pci/SisiaridisR11 fatcat:6ye42owyevhfjjsbgzg2h3viwi

Page 3230 of Mathematical Reviews Vol. , Issue 97E [page]

1997 Mathematical Reviews  
Gabriel Ciobanu (RC-AST-I; Taipei) 97e:68075 68Q55 03B70 68Q10 Lenisa, Marina (l-UDIN-MI; Udine) Final semantics for a higher order concurrent language.  ...  {For the entire collection see MR 97d:68007. } 97e:68073 68Q55 18B30 Fiore, Marcelo P. (4-EDIN-FC; Edinburgh) Order-enrichment for categories of partial maps.  ... 

From Linear Logic to Cyclic Sharing

Masahito Hasegawa
2019 Electronic Proceedings in Theoretical Computer Science  
We present a translation from Multiplicative Exponential Linear Logic to a simply-typed lambda calculus with cyclic sharing.  ...  This translation is derived from a simple observation on the Int-construction on traced monoidal categories.  ...  Acknowledgements I thank Ken Sakayori and Takeshi Tsukada for stimulating discussions related to this work.  ... 
doi:10.4204/eptcs.292.3 fatcat:cznwpsweuvfnrlguxy4dzdaepy

Page 3545 of Mathematical Reviews Vol. , Issue 2002E [page]

2002 Mathematical Reviews  
This paper introduces a calculus of explicit substitutions for higher- order abstract syntax (HOAS) which allows a high-level descrip- tion of variable bindings in object languages while providing sub-  ...  Now that a higher-order calculus of explicit substitutions has been formulated without de Bruijn indices, the author establishes its confluence in a similar way to establishing the confluence of the calculus  ... 

Page 4530 of Mathematical Reviews Vol. , Issue 96h [page]

1996 Mathematical Reviews  
The author presents a new paradox for type theory by proving inconsistent the system of polymorphic higher-order logic, an ex- tension of minimal higher-order logic, which is provably consistent.  ...  Polymorphic higher-order logic is defined the same way except that the types are taken to be the types of polymorphic second-order lambda calculus.  ... 

Constraint categorial grammars [chapter]

Luís Damas, Nelma Moreira
1995 Lecture Notes in Computer Science  
In this paper we present a higher order feature description calculus based on a typed λ-calculus.  ...  CCLG is a simple formalism, based on categorial grammars, designed to test the practical feasibility of such a calculus.  ...  Acknowledgments The authors would like to thank Sabine Broda and the anonymous reviewers for their valuable comments on an earlier draft of this paper.  ... 
doi:10.1007/3-540-60428-6_29 fatcat:dhcfwrlfsbgbtaejtv5issttly

Constraint Categorial Grammars [article]

Luis Damas, Nelma Moreira
1995 arXiv   pre-print
In this paper we present a higher order feature description calculus based on a typed λ-calculus.  ...  We show how the techniques used in for resolving complex feature constraints can be efficiently extended. is a simple formalism, based on categorial grammars, designed to test the practical feasibility  ...  Acknowledgments The authors would like to thank Sabine Broda and the anonymous reviewers for their valuable comments on an earlier draft of this paper.  ... 
arXiv:cmp-lg/9507001v1 fatcat:da27mlx6nfhxnlnrxxrs4pbzdi

Observable properties of higher order functions that dynamically create local names, or: What's new? [chapter]

Andrew M. Pitts, Ian D. B. Stark
1993 Lecture Notes in Computer Science  
We consider a simple language (equivalent to a fragment of Standard ML) of typed, higher order functions that can dynamically create fresh names.  ...  The research reported in this paper is concerned with the problem of reasoning about properties of higher order functions involving state.  ...  Acknowledgements We are grateful to Eugenio Moggi, Peter O'Hearn, Allen Stoughton and Robert Tennent for making their unpublished work available to us.  ... 
doi:10.1007/3-540-57182-5_8 fatcat:2lp6rpktjbdwdnkswxi3lybiiy

Page 5024 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
Jiri Rosicky (CZ-MAS; Brno) 2003g:18005 18C50 03B35 03B40 68N18 Caccamo, Mario (4-CAMB-C; Cambridge) ; Winskel, Glynn (4-CAMB-C; Cambridge) A higher-order calculus for categories.  ...  The calculus gives a basis for mechanising a theory of categories in a generic theorem prover like Isabelle.”  ... 

Page 6955 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews  
The second language extends this idea by a form of higher order communication, with statements c!s and c?x, for c a channel.  ...  (NL-MATH-ST; Amsterdam) Topological models for higher order control flow.  ... 
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