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Set-theoretic foundation of parametric polymorphism and subtyping

Giuseppe Castagna, Zhiwu Xu
2011 SIGPLAN notices  
We define and study parametric polymorphism for a type system with recursive, product, union, intersection, negation, and function types.  ...  In particular, we introduce the notion of "convexity" on which our solution is built up and discuss its connections with parametricity as defined by Reynolds to whose study our work sheds new light.  ...  Nils Gesbert spotted a subtle mistake in our subtyping algorithm. Many persons discussed the contents of this work with us and gave precious suggestions.  ... 
doi:10.1145/2034574.2034788 fatcat:imjvc4cbfrcttjrrc4dot2tpgm

Set-theoretic foundation of parametric polymorphism and subtyping

Giuseppe Castagna, Zhiwu Xu
2011 Proceeding of the 16th ACM SIGPLAN international conference on Functional programming - ICFP '11  
We define and study parametric polymorphism for a type system with recursive, product, union, intersection, negation, and function types.  ...  In particular, we introduce the notion of "convexity" on which our solution is built up and discuss its connections with parametricity as defined by Reynolds to whose study our work sheds new light.  ...  Nils Gesbert spotted a subtle mistake in our subtyping algorithm. Many persons discussed the contents of this work with us and gave precious suggestions.  ... 
doi:10.1145/2034773.2034788 dblp:conf/icfp/CastagnaX11 fatcat:r537a3ocjjgedhke63wufk437u

Page 3496 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews  
by set-theoretic functions.  ...  The last two chapters 10 and 11 deal with some of the mathematics of topics in domain theory, such as powerdomains, and parametric polymorphism.  ... 

Foundations for Virtual Types

Atsushi Igarashi, Benjamin C. Pierce
2002 Information and Computation  
Work on the foundations of object-oriented languages has already established a clear connection between parametric classes and the polymorphic functions found in familiar typed lambda-calculi.  ...  We also investigate how the partial \duality" of virtual types and parametric classes can be understood in terms of translations between universal and (dependent) existential types.  ...  Acknowledgments This work was supported by Indiana University, the University of Pennsylvania, and the National Science Foundation under grant CCR-9701826, Principled Foundations for Programming with Objects  ... 
doi:10.1006/inco.2001.2942 fatcat:seakerekmncv7fmnm76byrexdm

Foundations for Virtual Types [chapter]

Atsushi Igarashi, Benjamin C. Pierce
1999 Lecture Notes in Computer Science  
Work on the foundations of object-oriented languages has already established a clear connection between parametric classes and the polymorphic functions found in familiar typed lambda-calculi.  ...  We also investigate how the partial \duality" of virtual types and parametric classes can be understood in terms of translations between universal and (dependent) existential types.  ...  Acknowledgments This work was supported by Indiana University, the University of Pennsylvania, and the National Science Foundation under grant CCR-9701826, Principled Foundations for Programming with Objects  ... 
doi:10.1007/3-540-48743-3_8 fatcat:lkacl54t4zhsfoticfwaoum5kq

Page 4350 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews  
Summary: “We contribute to the syntactic study of F<, a variant of second-order A-calculus F which appears as a paradigmatic kernel language for polymorphism and subtyping.  ...  At first, an introduction on polymorphic /-calculus is given. Then, a short informal discussion on types as invariants and types as sets follows.  ... 

Types as parameters [chapter]

Giuseppe Longo
1993 Lecture Notes in Computer Science  
This problem is at the core of the distinction between "ad hoe" and proper polymorphism and inspired the large amount of work on "parametricity".  ...  Contents. 1 Types; 2 Parametricity; 3 Genericity; 4 Axiom C and Dinatural Transformations; 5 ~xiom C and the Isomorphisms of Types; 6. Types as Parameters; 7.  ...  The blend of &-terms and subtyping makes this calculus an expressive and original mathematical formalism which shows, we claim, that "ad hoe" polymorphism may have also theoretical relevance.  ... 
doi:10.1007/3-540-56610-4_96 fatcat:4rtpgksokvb23ov6o3zgz7diii

PhD Abstracts

GRAHAM HUTTON
2019 Journal of functional programming  
As a service to the community, twice per year the Journal of Functional Programming publishes the abstracts from PhD dissertations completed during the previous year.  ...  Our set-theoretic types are polymorphic, that is, they contain type variables to allow parametric polymorphism.  ...  Polymorphic Set-Theoretic Types for Functional Languages We extend previous work on set-theoretic types and semantic subtyping by showing how to adapt them to new settings and apply them to type various  ... 
doi:10.1017/s0956796819000078 fatcat:nqlt4e66r5ea5ibvngbewqpvs4

A Theory of Class [chapter]

Anthony J. H. Simons
1997 OOIS'96  
The theory is general, in that it encompasses many different approaches to type abstraction, such as type constructors, generic parameters, classes, inheritance and polymorphism.  ...  The theory is elegant, in that it is based on a simple generalisation of F-bounds. The theory has timely implications for emerging OMG standards and future language designs.  ...  ), constrained generic and anchored types (both kinds of F-bounded polymorphism); whereas C++ has subtyping and templates (a kind of parametric polymorphism).  ... 
doi:10.1007/978-1-4471-0973-0_4 dblp:conf/oois/Simons96 fatcat:4qbgtblthjbo7awknq6z6eepoa

Consistent Subtyping for All [chapter]

Ningning Xie, Xuan Bi, Bruno C. d. S. Oliveira
2018 Lecture Notes in Computer Science  
The first goal of this paper is to propose a generalization of consistent subtyping that is adequate for polymorphic subtyping, and subsumes the original definition by Siek and Taha.  ...  Polymorphic typesà la System F also induce a subtyping relation that relates polymorphic types to their instantiations. However Siek and Taha's definition is not adequate for polymorphic subtyping.  ...  We thank Ronald Garcia and the anonymous reviewers for their helpful comments. This work has been sponsored by the Hong Kong Research Grant Council projects number 17210617 and 17258816.  ... 
doi:10.1007/978-3-319-89884-1_1 fatcat:zlxtuitbrvga5eak4n2hsu64w4

The Theory of Classification, Part 1: Perspectives on Type Compatibility

Anthony J.H. Simons
2002 Journal of Object Technology  
However, the theoretical concept of polymorphism is widely misunderstood and the term mistakenly applied, by objectoriented programmers, variously to describe dynamic binding or subtyping.  ...  (such as arithmetic) which the Ordinals don't allow (although strictly the set-theoretic model only enumerates the membership of a type and does not describe how elements behave).  ... 
doi:10.5381/jot.2002.1.1.c5 fatcat:usttkkow2fde3lmzybpd2flaaa

Towards an Accurate Mathematical Model of Generic Nominally-Typed OOP [article]

Moez A. AbdelGawad
2016 arXiv   pre-print
The construction of GNOOP as a domain-theoretic model of generic nominally-typed OOP is currently underway.  ...  The presented concepts and suggestions are particularly geared towards enabling GNOOP to offer a precise and simple view of so-far-hard-to-analyze features of generic OOP such as variance annotations (  ...  Features such as Java wildcards [40], in particular, while designed to ameliorate the conceptual mismatch between parametric polymorphism and OO subtyping polymorphism, have proven to be difficult to accurately  ... 
arXiv:1610.05114v3 fatcat:y2sevsf4c5af5ax7mwzkayltfu

The essence of Reynolds

Stephen Brookes, Peter W. O'Hearn, Uday Reddy
2014 Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages - POPL '14  
John Reynolds (1935Reynolds ( -2013 was a pioneer of programming languages research. In this paper we pay tribute to the man, his ideas, and his influence.  ...  Technically, the paper is notable for formalizing the Abstraction Theorem and for introducing Relational Parametricity. • Polymorphism is not Set-Theoretic.  ...  Typed λ-calculus with subtypes, which has been used significantly in the foundations of object-oriented languages and elsewhere.  ... 
doi:10.1145/2535838.2537851 dblp:conf/popl/BrookesOR14 fatcat:gf5zdcx4j5dyhebxwbgwpvhere

Distributive Disjoint Polymorphism for Compositional Programming [chapter]

Xuan Bi, Ningning Xie, Bruno C. d. S. Oliveira, Tom Schrijvers
2019 Lecture Notes in Computer Science  
These improvements are due to the combination of three features: disjoint intersection types with a merge operator ; parametric (disjoint) polymorphism; and BCD-style distributive subtyping.  ...  Besides coherence, we show several other important meta-theoretical results, such as type-safety, sound and complete algorithmic subtyping, and decidability of the type system.  ...  This work has been sponsored by the Hong Kong Research Grant Council projects number 17210617 and 17258816, and by the Research Foundation -Flanders.  ... 
doi:10.1007/978-3-030-17184-1_14 fatcat:qugktkxnlbbg5keuq4imyfbio4

The Essence of Reynolds

Stephen Brookes, Peter W. O'Hearn, Uday Reddy
2014 Formal Aspects of Computing  
John Reynolds (1935Reynolds ( -2013 was a pioneer of programming languages research. In this paper we pay tribute to the man, his ideas, and his influence.  ...  Technically, the paper is notable for formalizing the Abstraction Theorem and for introducing Relational Parametricity. • Polymorphism is not Set-Theoretic.  ...  Typed λ-calculus with subtypes, which has been used significantly in the foundations of object-oriented languages and elsewhere.  ... 
doi:10.1007/s00165-014-0295-x fatcat:cf3w7y647ratnebxbisrrw6oee
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