## Branching Types (2002)

Citations: | 11 - 5 self |

### BibTeX

@MISC{Wells02branchingtypes,

author = {J. B. Wells and Christian Haack},

title = {Branching Types},

year = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

Although systems with intersection types have many unique capabilities, there has never been a fully satisfactory explicitly typed system with intersection types. We introduce and prove the basic properties of # , a typed #-calculus with branching types and types with quantification over type selection parameters. The new system # an explicitly typed system with the same expressiveness as a system with intersection types. Typing derivations in # use branching types to squash together what would be separate parallel derivations in earlier systems with intersection types.

### Citations

365 |
Towards a Theory of Type Structure
- Reynolds
- 1974
(Show Context)
Citation Context ...ed near the end of the 1970s by Coppo and Dezani [3] and Pottinger [15]. Intersection types provide type polymorphism by listing type instances, differing from the more widely used ∀-quantified type=-=s [8, 16],-=- which provide type polymorphism by giving a type scheme that can be instantiated into various type instances. The original motivation was for analyzing and/or synthesizing λ-models as well as in ana... |

240 |
Interprétation Fonctionnelle et Élimination des Coupures de L’arithmétique D’order Supérieur
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- 1972
(Show Context)
Citation Context ...ed near the end of the 1970s by Coppo and Dezani [3] and Pottinger [15]. Intersection types provide type polymorphism by listing type instances, differing from the more widely used ∀-quantified type=-=s [8, 16],-=- which provide type polymorphism by giving a type scheme that can be instantiated into various type instances. The original motivation was for analyzing and/or synthesizing λ-models as well as in ana... |

109 | Design of the programming language Forsythe
- Reynolds
- 1996
(Show Context)
Citation Context ...e ??? appears, what should be written? This trouble is related to the fact that the ∧ type constructor is not a truth-functional propositional connective. Earlier Approaches. In the language Forsyth=-=e [17], Reynold-=-s annotates the binding of (λx.M) with a list of types, e.g., (λx:σ1| · · · |σn. M). If the abstraction body M is typable with a fixed type τ for each type σi for x, then the abstraction gets... |

93 | What are principal typings and what are they good for - Jim - 1996 |

88 |
An extension of the basic functionality theory for the λ-calculus
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- 1980
(Show Context)
Citation Context ... earlier systems with intersection types. 1 Introduction 1.1 Background and Motivation Intersection Types. Intersection types were independently invented near the end of the 1970s by Coppo and Dezani =-=[3] a-=-nd Pottinger [15]. Intersection types provide type polymorphism by listing type instances, differing from the more widely used ∀-quantified types [8, 16], which provide type polymorphism by giving a... |

86 | The essence of principal typings - Wells - 2002 |

78 | A direct algorithm for type inference in the rank-2 fragment of the second-order λ-calculus
- Kfoury, Wells
- 1994
(Show Context)
Citation Context ... GR/L 36963 and GR/R 41545/01, and Sun Microsystems equipment grant EDUD-7826-990410-US.sand it has been proven for many such type systems that ATI algorithms can not be both complete and terminating =-=[11, 23, 24, 20]-=-. In contrast, ATI algorithms have been proven complete and terminating for the rank-k restriction for every finite k for several systems with intersection types [12, 10]. We use intersection types in... |

61 | Intersection Type Assignment Systems - Bakel - 1995 |

59 | Typability and type checking in system f are equivalent and undecidable
- Wells
- 1998
(Show Context)
Citation Context ... GR/L 36963 and GR/R 41545/01, and Sun Microsystems equipment grant EDUD-7826-990410-US.sand it has been proven for many such type systems that ATI algorithms can not be both complete and terminating =-=[11, 23, 24, 20]-=-. In contrast, ATI algorithms have been proven complete and terminating for the rank-k restriction for every finite k for several systems with intersection types [12, 10]. We use intersection types in... |

54 | A modular, polyvariant and type-based closure analysis - Banerjee - 1997 |

51 | Principality and decidable type inference for finiterank intersection types
- Kfoury, Wells
- 1999
(Show Context)
Citation Context ...d to be the most powerful type system with ∀-quantifiers measured by the set of pure λ-terms it can type. In contrast, this λ-term is typable with intersection types satisfying the rank-3 restrict=-=ion [12]. -=-Better results for automated type inference (ATI) have also been obtained for intersection types. ATI for type systems with ∀-quantifiers that are more powerful than the very-restricted Hindley/Miln... |

51 | Programming with Intersection Types, Union Types, and Polymorphism
- Pierce
- 1991
(Show Context)
Citation Context ... → τ). However, this approach can not handle dependencies between types of nested variable bindings, e.g., this approach can not give K = (λx.λy.x) the type τK = (σ → (σ → σ)) ∧ (τ →=-= (τ → τ)). Pierce [14] improves on-=- Reynolds’s approach by using a for construct which gives a type variable a finite set of types to range over, e.g., K can be annotatedsas (for α ∈ {σ, τ}.λx:α. λy:α. x) with the type τK. ... |

43 |
type assignment for the strongly normalizable λ-terms,” pp. 561–577
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(Show Context)
Citation Context ...ith intersection types. 1 Introduction 1.1 Background and Motivation Intersection Types. Intersection types were independently invented near the end of the 1970s by Coppo and Dezani [3] and Pottinger =-=[15]. -=-Intersection types provide type polymorphism by listing type instances, differing from the more widely used ∀-quantified types [8, 16], which provide type polymorphism by giving a type scheme that c... |

41 | From polyvariant flow information to intersection and union types
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(Show Context)
Citation Context ...ection types have many unique advantages over ∀-quantified types. They can characterize the behavior of λ-terms more precisely, and can be used to express exactly the results of many program analys=-=es [13, 1, 25, 26]. Type-=- polymorphism with intersection types is also more flexible. For example, Urzyczyn [20] proved the λ-term (λx.z(x(λfu.fu))(x(λvg.gv)))(λy.yyy) to be untypable in the system Fω, considered to be ... |

29 | Strongly typed flow-directed representation transformations
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(Show Context)
Citation Context ...upport useful type-directed program transformations. We use intersection types because they support both more accurate analyses (as mentioned above) and interesting type/flow-directed transformations =-=[5, 19, 7, 6] t-=-hat would be very difficult using ∀-quantified types. When using a TIL, it is important to regularly check that the intermediate program representation is in fact well typed. Provided this is done, ... |

28 | A calculus with polymorphic and polyvariant flow types
- Wells, Dimock, et al.
(Show Context)
Citation Context ...ection types have many unique advantages over ∀-quantified types. They can characterize the behavior of λ-terms more precisely, and can be used to express exactly the results of many program analys=-=es [13, 1, 25, 26]. Type-=- polymorphism with intersection types is also more flexible. For example, Urzyczyn [20] proved the λ-term (λx.z(x(λfu.fu))(x(λvg.gv)))(λy.yyy) to be untypable in the system Fω, considered to be ... |

22 | Typability and type-checking in the second-order λ-calculus are equivalent and undecidable - Wells - 1994 |

22 | A typed intermediate language for flow-directed compilation
- Wells, Dimock, et al.
(Show Context)
Citation Context ...ection types have many unique advantages over ∀-quantified types. They can characterize the behavior of λ-terms more precisely, and can be used to express exactly the results of many program analys=-=es [13, 1, 25, 26]. Type-=- polymorphism with intersection types is also more flexible. For example, Urzyczyn [20] proved the λ-term (λx.z(x(λfu.fu))(x(λvg.gv)))(λy.yyy) to be untypable in the system Fω, considered to be ... |

21 | Relating Typability and Expressibility in Finite-Rank Intersection Type Systems
- Kfoury, Mairson, et al.
- 1999
(Show Context)
Citation Context ...plete and terminating [11, 23, 24, 20]. In contrast, ATI algorithms have been proven complete and terminating for the rank-k restriction for every finite k for several systems with intersection types =-=[12, 10]-=-. We use intersection types in typed intermediate languages (TILs) used in compilers. Using a TIL increases reliability of compilation and can support useful type-directed program transformations. We ... |

21 | Coppo and Mariangiola Dezani-Ciancaglini. Structured Communications with Concurrent Constraints - Mario - 2009 |

18 | Coppo-Dezani Types do not Correspond to Propositional Logic - Hindley - 1984 |

16 |
Type reconstruction in Fω
- Urzyczyn
- 1997
(Show Context)
Citation Context ...λ-terms more precisely, and can be used to express exactly the results of many program analyses [13, 1, 25, 26]. Type polymorphism with intersection types is also more flexible. For example, Urzyczyn=-= [20] proved t-=-he λ-term (λx.z(x(λfu.fu))(x(λvg.gv)))(λy.yyy) to be untypable in the system Fω, considered to be the most powerful type system with ∀-quantifiers measured by the set of pure λ-terms it can t... |

13 |
Intersection Types as Logical Formulae
- Venneri
- 1994
(Show Context)
Citation Context ...henever subterms are transformed. Venneri succeeded in completely removing the (∧-intro) rule from a type system with intersection types, but this was for combinatory logic rather than the λ-calcul=-=us [21, 4], -=-and the approach seems unlikely to be transferable to the λ-calculus. 1.2 Contributions of this Paper Our Approach: Branching Types. In this paper, we define and prove the basic properties of λ B , ... |

12 |
The “Relevance” of Intersection and Union Types
- DEZANI-CIANCAGLINI, GHILEZAN, et al.
- 1997
(Show Context)
Citation Context ...henever subterms are transformed. Venneri succeeded in completely removing the (∧-intro) rule from a type system with intersection types, but this was for combinatory logic rather than the λ-calcul=-=us [21, 4], -=-and the approach seems unlikely to be transferable to the λ-calculus. 1.2 Contributions of this Paper Our Approach: Branching Types. In this paper, we define and prove the basic properties of λ B , ... |

11 | Faithful translations between polyvariant flows and polymorphic types
- Amtoft, Turbak
- 2000
(Show Context)
Citation Context |

10 |
A linearization of the lambda-calculus
- Kfoury
(Show Context)
Citation Context ... this could be obtained indirectly via their proofs of equivalence with traditional systems with intersection types. HL is reported in [18] to have a typed version of an untyped calculus like that in =-=[9]-=-, but in fact there is no significant connection between [2] and [9] and there is no explicitly typed calculus associated with HL. For both IL and HL, there are proofs of equivalence with earlier syst... |

9 | Program representation size in an intermediate language with intersection and union types
- Dimock, Westmacott, et al.
- 2001
(Show Context)
Citation Context ...upport useful type-directed program transformations. We use intersection types because they support both more accurate analyses (as mentioned above) and interesting type/flow-directed transformations =-=[5, 19, 7, 6] t-=-hat would be very difficult using ∀-quantified types. When using a TIL, it is important to regularly check that the intermediate program representation is in fact well typed. Provided this is done, ... |

9 | Typability is undecidable for F+eta
- Wells
- 1996
(Show Context)
Citation Context ... GR/L 36963 and GR/R 41545/01, and Sun Microsystems equipment grant EDUD-7826-990410-US.sand it has been proven for many such type systems that ATI algorithms can not be both complete and terminating =-=[11, 23, 24, 20]-=-. In contrast, ATI algorithms have been proven complete and terminating for the rank-k restriction for every finite k for several systems with intersection types [12, 10]. We use intersection types in... |

7 | Functioning without closure: type-safe customized function representations for Standard ML
- Dimock, Westmacott, et al.
- 2001
(Show Context)
Citation Context ...upport useful type-directed program transformations. We use intersection types because they support both more accurate analyses (as mentioned above) and interesting type/flow-directed transformations =-=[5, 19, 7, 6] t-=-hat would be very difficult using ∀-quantified types. When using a TIL, it is important to regularly check that the intermediate program representation is in fact well typed. Provided this is done, ... |

5 | Compiling with polymorphic and polyvariant flow types
- Turbak, Dimock, et al.
- 1997
(Show Context)
Citation Context |

4 | Proof-functional connectives - Lopez-Escobar - 1985 |

3 |
Intersection logic
- S, Roversi
- 2001
(Show Context)
Citation Context ... Finally, it reduces to ΛP1.λx σ .x, removing the trivial P ′ 2 and A′ by a (∗Λ)-step and a (∗A)-step. Recent Related Work. Ronchi Della Rocca and Roversi have a system called Intersection=-= Logic (IL) [18] -=-which is similar to λ B , but has nothing correspondingsto our explicitly typed terms. IL has a meta-level operation corresponding to our equivalence for function types. IL has nothing corresponding ... |

3 | Strong conjunction and intersection types - Alessi, Barbanera - 1991 |

2 | Proof-functional connectives and realizability - Barbanera, Martini - 1994 |

1 |
parallel deductions and intersection types
- Hyperformulae
(Show Context)
Citation Context ...sented, but the proof terms are pure λ-terms and thus the proof terms do not represent entire derivations. Capitani, Loreti, and Venneri have designed a similar system called HL (Hyperformulae Logic)=-= [2]-=-. HL is quite similar to IL, although it seems overall to have a less complicated presentation. HL has nothing corresponding to our equivalences on types. The set of properties proved for HL in [2] is... |

1 | The completeness of provable realizability - Mints - 1989 |