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Completeness and partial soundness results for intersection and union typing for λ¯μμ̃

Steffen van Bakel
2010 Annals of Pure and Applied Logic  
These results when combined show that, with respect to full reduction, it is not possible to define a sound and complete intersection and union type assignment system for λµμ.  ...  This paper studies intersection and union type assignment for the calculus λµμ (Curien and Herbelin, 2000 [16]), a proof-term syntax for Gentzen's classical sequent calculus, with the aim of defining a  ...  Summers and especially Dan Dougherty for detailed, critical and constructive discussions on the results of this paper, and Vanessa Loprete for valuable support.  ... 
doi:10.1016/j.apal.2010.04.010 fatcat:ugiqp72vlzcdrogztu6mqaeqhm

Sound and Complete Typing for lambda-mu [article]

Steffen van Bakel
2011 arXiv   pre-print
In this paper we define intersection and union type assignment for Parigot's calculus lambda-mu.  ...  We show that this notion is complete (i.e. closed under subject-expansion), and show also that it is sound (i.e. closed under subject-reduction).  ...  Conclusions We have seen that the calculus λµ is sufficiently limited to allow for the definition of a sound and complete notion of type assignment.  ... 
arXiv:1101.4425v1 fatcat:6ko4jvy4lbfz7n2lpzxxpwmphi

Infinite intersection types

Marcello M. Bonsangue, Joost N. Kok
2003 Information and Computation  
A type theory with infinitary intersection and union types for an extension of the λ-calculus is introduced.  ...  Finally we apply the framework to applicative transition systems, obtaining a sound a complete infinitary intersection type assignment system for the lazy λ-calculus.  ...  In particular we are indebted to one of the referee who found a mistake in the original proof of completeness of the type assignment system and suggested the actual proof presented in this paper.  ... 
doi:10.1016/s0890-5401(03)00143-3 fatcat:qw32q3hk7vekzbkwwdxemanl6y

The root cause of blame: contracts for intersection and union types

Jack Williams, J. Garrett Morris, Philip Wadler
2018 Proceedings of the ACM on Programming Languages (PACMPL)  
We extend the untyped lambda calculus with contracts for monitoring higher-order intersection and union types, for the first time giving a uniform treatment to both.  ...  Gradual typing has emerged as the tonic for programmers with a thirst for a blend of static and dynamic typing.  ...  ACKNOWLEDGMENTS We wish to thank the anonymous OOPSLA reviewers for their detailed comments.  ... 
doi:10.1145/3276504 fatcat:d7ppbeyxgzh5hf34d2wph2t3k4

Subtyping with union types, intersection types and recursive types [chapter]

Flemming M. Damm
1994 Lecture Notes in Computer Science  
In the work reported on here, we generalize the result and develop a general technique for deriving subtyping algorithms for type languages with union, intersection and recursive types.  ...  This paper is a follow-up on previous work by the author on subtyping with (set-theoretic) union, intersection and recursive types.  ...  The complexity results for set constraints are quite discouraging ( NEXPTIME]) AKVW93], but it is not clear what these worst case results mean in practice for our subtyping approach.  ... 
doi:10.1007/3-540-57887-0_121 fatcat:ftzqhqtasvgs3g5rbq6p4dth54

Preciseness of Subtyping on Intersection and Union Types [chapter]

Mariangiola Dezani-Ciancaglini, Silvia Ghilezan
2014 Lecture Notes in Computer Science  
We propose a technique for formalising and proving operational preciseness of the subtyping relation in the setting of a concurrent lambda calculus with intersection and union types.  ...  We then prove soundness and completeness getting that the subtyping relation of this calculus enjoys both denotational and operational preciseness.  ...  Acknowledgments The authors gratefully thank the anonymous referees for their numerous constructive remarks.  ... 
doi:10.1007/978-3-319-08918-8_14 fatcat:7xezjs43w5fzrcwp2dk2relkne

Strong Normalization as Safe Interaction

Colin Riba
2007 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)  
We prove that the union and intersection type discipline without (∨ E) is complete w.r.t. strong normalization.  ...  This allows to show that (∨ E) is safe if and only if an interpretation of types based on biorthogonals is sound for it.  ...  The author thanks Frédéric Blanqui and Claude Kirchner for advices, support and comments.  ... 
doi:10.1109/lics.2007.46 dblp:conf/lics/Riba07 fatcat:cw44ovzqkrezlk5fctckz5na4e

Untangling Typechecking of Intersections and Unions

Jana Dunfield
2011 Electronic Proceedings in Theoretical Computer Science  
The key results are soundness and completeness: a typing derivation (in the system with too many subterm choices) exists for a program if and only if a derivation exists for the let-normalized program.  ...  Intersection and union types denote conjunctions and disjunctions of properties.  ...  Most of the work was done at Carnegie Mellon University with the support of the US National Science Foundation.  ... 
doi:10.4204/eptcs.45.5 fatcat:zmbwswvtcbbgboyan3z3tp4cmi

Numeric domains meet algebraic data types

Santiago Bautista, Thomas Jensen, Benoît Montagu
2020 Proceedings of the 9th ACM SIGPLAN International Workshop on Numerical and Symbolic Abstract Domains  
We report on the design and formalization of a novel abstract domain, called numeric path relations (NPRs), that combines numeric relational domains with algebraic data types.  ...  The construction employs projection paths on algebraic values, and in particular projections on variant cases, whose sound treatment is subtle due to mutual exclusiveness.  ...  For any NPRs 1 and 2 , if 1 ⊑ NPR 2 , then NPR ( 1 ) ⊆ NPR ( 2 ). Theorem 4.3 (Soundness of union and intersection).  ... 
doi:10.1145/3427762.3430178 fatcat:yo2yc7xve5acvmmy5ti65xmiwe


Mariangiola Dezani-Ciancaglini, Mitsu Okada, Masako Takahashi
2002 Theoretical Computer Science  
Completeness of Type Assignment Systems with Intersection, Union, and Type Quantiÿers by Hirofumi Yokouchi develops type assignment systems with intersection and union types, and type quantiÿers and shows  ...  Completeness of Intersection and Union Type Assignment Systems for call-by-value -models by Hajime Ishihara and Toshihiko Kurata studies a version of intersection and union type assignment system, which  ... 
doi:10.1016/s0304-3975(00)00345-5 fatcat:bc3qw2ifpncfvdxt3qhyvcoue4

Denotational and Operational Preciseness of Subtyping: A Roadmap [chapter]

Mariangiola Dezani-Ciancaglini, Silvia Ghilezan, Svetlana Jakšić, Jovanka Pantović, Nobuko Yoshida
2016 Lecture Notes in Computer Science  
The soundness and the completeness, together referred to as the preciseness of subtyping, can be considered from two different points of view: denotational and operational.  ...  The latter preciseness has been recently developed with respect to type safety, i.e. the safe replacement of a term of a smaller type when a term of a bigger type is expected.  ...  with intersection and union types (dubbed λ ⊕ ) introduced in [14] .  ... 
doi:10.1007/978-3-319-30734-3_12 fatcat:a7d35babmbc6jgvxp5e2qk5liy

Reduction in X does not agree with Intersection and Union Types (Extended abstract) [article]

Steffen van Bakel
2011 arXiv   pre-print
This paper defines intersection and union type assignment for the calculus X, a substitution free language that enjoys the Curry-Howard correspondence with respect to Gentzen's sequent calculus for classical  ...  We show that this notion is closed for subject-expansion, and show that it needs to be restricted to satisfy subject-reduction as well, making it unsuitable to define a semantics.  ...  Acknowledgement I would like to thank Philippe Audebaud, Mariangiola Dezani and Alexander Summers for fruitfull discussions, and especially thank Vanessa Loprete for valuable support.  ... 
arXiv:1109.4570v1 fatcat:o7x3eehvpbbnblizryh5pav7i4

Page 3936 of Mathematical Reviews Vol. , Issue 93g [page]

1993 Mathematical Reviews  
Moreover, this system was proved to be sound and complete.  ...  In the intersection type discipline the usual ‘—’-based type language for A-calculus was extended by adding a constant w as a universal type and a new connective ‘A’ for the intersection of two types.  ... 

Elaborating intersection and union types

2014 Journal of functional programming  
We describe a foundation for compiling unrestricted intersection and union types: an elaboration type system that generates ordinary lambda-calculus terms.  ...  We also describe a prototype implementation and applications of unrestricted intersections and unions: records, operator overloading, and simulating dynamic typing.  ...  Finally, I had useful discussions about this work with Yan Chen, Matthew A. Hammer, Scott Kilpatrick, Neelakantan R. Krishnaswami, and Viktor Vafeiadis.  ... 
doi:10.1017/s0956796813000270 fatcat:ctnnaravtfgxlf3pcog5pkbzae

Subtyping Union Types [chapter]

Jérôme Vouillon
2004 Lecture Notes in Computer Science  
Subtyping of union types can be fairly complex due to interactions with function and pair types.  ...  Using inference rules, we specify a subtyping relation which is both sound and complete with respect to the family of calculi.  ...  Frisch, Castagna and Benzaken [14] use an approach similar to ours to design a subtyping relation for a typed calculus with union and intersection types.  ... 
doi:10.1007/978-3-540-30124-0_32 fatcat:bt3rw5o2vbd7vowo6snxnegdae
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