## Reduction in X does not agree with Intersection and Union Types (2008)

Citations: | 2 - 2 self |

### BibTeX

@TECHREPORT{Bakel08reductionin,

author = {Steffen Van Bakel},

title = {Reduction in X does not agree with Intersection and Union Types},

institution = {},

year = {2008}

}

### OpenURL

### Abstract

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 logic. We show that this notion is the minimal one 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.

### Citations

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Citation Context ... on natural deduction (see also [19]); the advantage of using the sequent approach in this paper is that now we can study and explain various anomalies of union type assignment [8] and quantification =-=[20, 25]-=-. The type system defined here initially will be shown to be the natural one, in that intersection and union play their expected roles for witness expansion (also called completeness). However, we sho... |

1115 | The Lambda Calculus: Its Syntax and Semantics - Barendregt - 1981 |

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216 |
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Citation Context ...t is closed for expansion, we will need to add also union types. The system we define in this paper is a conservative extension of the Intersection Type Assignment System for the λ-calculus (see also =-=[10, 1, 2, 3]-=-), in that lambda terms typeable in that system translate to X -nets, while preserving the type. As was the case for systems with intersection types for the λ-calculus [10, 2], in order to get a notio... |

161 | The duality of computation
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Citation Context ...tors, making the non-standard reductions (essentially context calls, which form part of the reduction in X ) unsafe. As, in part, already been observed in [21] in the context of the calculus λµ ˜µ of =-=[13]-=-, the problem is that the added rules are not logical. The advantage of studying this problem in the context of the highly symmetric sequent calculi will be made clear: intersection and union are trul... |

159 | A judgmental reconstruction of modal logic
- Pfenning, Davies
(Show Context)
Citation Context ...f both intersection and union. This problem also appears in other contexts, such as that of ML with side-effects [20, 29, 25], and that of using intersection and union types in an operational setting =-=[14, 18]-=-. As here, also there the cause of the problem is the lack of accompanying syntax for the added type constructors, making the non-standard reductions (essentially context calls, which form part of the... |

103 | Complete restrictions of the Intersection Type Discipline
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(Show Context)
Citation Context ...t is closed for expansion, we will need to add also union types. The system we define in this paper is a conservative extension of the Intersection Type Assignment System for the λ-calculus (see also =-=[10, 1, 2, 3]-=-), in that lambda terms typeable in that system translate to X -nets, while preserving the type. As was the case for systems with intersection types for the λ-calculus [10, 2], in order to get a notio... |

92 | Simple imperative polymorphism
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(Show Context)
Citation Context ...s) no longer holds, and will reason that this is caused by the non-logical foundation of both intersection and union. This problem also appears in other contexts, such as that of ML with side-effects =-=[20, 29, 25]-=-, and that of using intersection and union types in an operational setting [14, 18]. As here, also there the cause of the problem is the lack of accompanying syntax for the added type constructors, ma... |

65 | Preservation of strong normalisation in named lambda calculi with explicit substitution and garbage collection
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Citation Context ...lculus X In this section we will give the definition of the X -calculus which has been proven to be a fine-grained implementation model for various well-known calculi [5], like the λ-calculus [9], λx =-=[11]-=-, λµ [26] and λµ ˜µ [21]. As discussed in the introduction, the calculus X is inspired by the sequent calculus; the system we will consider in this section has only implication, no structural rules an... |

61 | Intersection Type Assignment Systems
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- 1995
(Show Context)
Citation Context ...t is closed for expansion, we will need to add also union types. The system we define in this paper is a conservative extension of the Intersection Type Assignment System for the λ-calculus (see also =-=[10, 1, 2, 3]-=-), in that lambda terms typeable in that system translate to X -nets, while preserving the type. As was the case for systems with intersection types for the λ-calculus [10, 2], in order to get a notio... |

58 |
Intersection and union types: Syntax and semantics
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Citation Context ...ulus [9] which is based on natural deduction (see also [19]); the advantage of using the sequent approach in this paper is that now we can study and explain various anomalies of union type assignment =-=[8]-=- and quantification [20, 25]. The type system defined here initially will be shown to be the natural one, in that intersection and union play their expected roles for witness expansion (also called co... |

58 | Classical Logic and Computation
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Citation Context ...1, . . . , An is to be understood as A1∧ . . . ∧An and B1, . . . , Bm is to be understood as B1∨ . . . ∨Bm, and has implicit weakening and contraction. Starting from different approaches in that area =-=[13, 28]-=-, in [24] the calculus X was introduced, and shown to be equivalent to the λµ ˜µ-calculus in terms of expressivity. Using this correspondence, a strong normalization result is shown for λµ ˜µ. In fact... |

54 |
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Citation Context ... extensionality. The type assignment system presented here is based on the BCD-system defined by H. Barendregt, M. Coppo and M. Dezani-Ciancaglini in [10], in turn based on the system as presented in =-=[12]-=-. The BCD-system treats the two type constructors ‘→’ and ‘∩’ the same, allowing, in particular, intersection to occur at the right of arrow types. It also introduced a partial order relation ‘ ≤ ’ on... |

50 | Programming with Intersection Types, Union Types, and Polymorphism - Pierce - 1991 |

46 | Call-by-value is dual to call-by-name - Wadler |

36 | Tridirectional Typechecking
- Dunfield, Pfenning
- 2004
(Show Context)
Citation Context ...f both intersection and union. This problem also appears in other contexts, such as that of ML with side-effects [20, 29, 25], and that of using intersection and union types in an operational setting =-=[14, 18]-=-. As here, also there the cause of the problem is the lack of accompanying syntax for the added type constructors, making the non-standard reductions (essentially context calls, which form part of the... |

30 |
Investigations into logical deduction. In The Collected Papers of Gerhard Gentzen
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Citation Context ...is paper will present a notion of intersection and union type assignment for the (untyped) calculus X , as first defined in [24] and later extensively studied in [5]. is based on the sequent calculus =-=[19]-=-, in contrast to the λ-calculus [9] which is based on natural deduction (see also [19]); the advantage of using the sequent approach in this paper is that now we can study and explain various anomalie... |

30 |
Call-by-value, call-by-name, and strong normalization for the classical sequent calculus
- Lengrand
- 2003
(Show Context)
Citation Context ...duction as well, making it unsuitable to define a semantics. Introduction This paper will present a notion of intersection and union type assignment for the (untyped) calculus X , as first defined in =-=[24]-=- and later extensively studied in [5]. is based on the sequent calculus [19], in contrast to the λ-calculus [9] which is based on natural deduction (see also [19]); the advantage of using the sequent ... |

29 | The language X : circuits, computations and Classical Logic
- Bakel, Lengrand, et al.
- 2005
(Show Context)
Citation Context ...to define a semantics. Introduction This paper will present a notion of intersection and union type assignment for the (untyped) calculus X , as first defined in [24] and later extensively studied in =-=[5]-=-. is based on the sequent calculus [19], in contrast to the λ-calculus [9] which is based on natural deduction (see also [19]); the advantage of using the sequent approach in this paper is that now we... |

22 | C’est maintenant qu’on calcule: au cœur de la dualité. Mémoire de habilitation, Université Paris 11, Décembre 2005 - Herbelin |

18 |
Coppo-Dezani Types do not Correspond to Propositional Logic
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- 1984
(Show Context)
Citation Context ...alone that causes problems, but that the problem is much more profound: although the idea behind both intersection and union might be the (logical) and and or, the fact that they are both not logical =-=[22]-=- destroys the soundness, both for a system based on intersection, as for a system based on union. This also explains why, for ML with side-effects, quantification is no longer sound [20, 25]: also the... |

16 | Computation with Classical Sequents - Bakel, Lescanne |

14 | Cut-Elimination in the Strict Intersection Type Assignment System is Strongly Normalising. Notre Dame journal of Formal Logic
- Bakel
(Show Context)
Citation Context |

12 |
Characterizing strong normalization in the CurienHerbelin symmetric lambda calculus: extending the Coppo-Dezani heritage
- Dougherty, Ghilezan, et al.
(Show Context)
Citation Context ...ains why, for ML with side-effects, quantification is no longer sound [20, 25]: also the (∀I) and (∀E) rules of ML are not logical. Intersection and union have been studied in the context of λµ ˜µ in =-=[15, 21, 16, 17]-=-, and all these systems suffer from the same kind of problem with respect to reduction. In this paper we will improve on those results by showing all proofs in detail, and presenting two more 1sexpres... |

8 |
Strong Normalization of the Dual Classical Sequent Calculus
- Dougherty, Ghilezan, et al.
- 2005
(Show Context)
Citation Context ...ains why, for ML with side-effects, quantification is no longer sound [20, 25]: also the (∀I) and (∀E) rules of ML are not logical. Intersection and union have been studied in the context of λµ ˜µ in =-=[15, 21, 16, 17]-=-, and all these systems suffer from the same kind of problem with respect to reduction. In this paper we will improve on those results by showing all proofs in detail, and presenting two more 1sexpres... |

8 | Approaches to Polymorphism in Classical Sequent Calculus
- Summers, Bakel
- 2006
(Show Context)
Citation Context ...V, we limit intersection types to values, i.e. nets that introduce plugs; this is reminiscent of the limitation in ML of quantification of types to terms that are values [20, 25], and is used also in =-=[27]-=-. We define P ·· Γ ⊢ N ∆ as in Definition 5.4, where we change the applicability of rule (∪L), and add a rule to treat left-activated cuts: Definition 9.1 i) The context assignment rules for ⊢ N are: ... |

6 |
Intersection and Union Types for X
- Bakel
- 2004
(Show Context)
Citation Context ..., that are either closed for Call-By-Name or Call-By-Value reduction. Being restrictions, for these systems now witness-expansion collapses. We draw our conclusions in Section 11. This paper corrects =-=[4]-=-. 1 The calculus X In this section we will give the definition of the X -calculus which has been proven to be a fine-grained implementation model for various well-known calculi [5], like the λ-calculu... |

6 | Intersection and Union Types in the λµ ˜µ-calculus
- Dougherty, Ghilezan, et al.
- 2004
(Show Context)
Citation Context ...ains why, for ML with side-effects, quantification is no longer sound [20, 25]: also the (∀I) and (∀E) rules of ML are not logical. Intersection and union have been studied in the context of λµ ˜µ in =-=[15, 21, 16, 17]-=-, and all these systems suffer from the same kind of problem with respect to reduction. In this paper we will improve on those results by showing all proofs in detail, and presenting two more 1sexpres... |

6 |
ML with callcc is unsound. Post to TYPES mailing list
- Harper, Lillibridge
- 1991
(Show Context)
Citation Context ... on natural deduction (see also [19]); the advantage of using the sequent approach in this paper is that now we can study and explain various anomalies of union type assignment [8] and quantification =-=[20, 25]-=-. The type system defined here initially will be shown to be the natural one, in that intersection and union play their expected roles for witness expansion (also called completeness). However, we sho... |

5 |
Capture avoidance and garbage collection for X
- Bakel, Raghunandan
(Show Context)
Citation Context ...† �y(P�β † �zR) (�yP�β·α)�α † �x(Q�γ [x] �zR) → (Q�γ † �yP)�β † �zR for CBN. This way, we obtain two notions of reduction that are clearly confluent: all rules are left-linear and non-overlapping. In =-=[6, 7]-=- some basic properties are shown, which essentially show that the calculus is wellbehaved, as well as the relation between X and a number of other calculi. These results motivate the formulation of ad... |

5 |
Introduction to Metamathematics. Études et Recherches en Informatique
- Kleene
- 1952
(Show Context)
Citation Context ...cut)-rule. A number of variants exist for Gentzen’s calculus for classical logic LK; the one we will consider in this paper, and that lies at the basis of the calculus X , is the calculus known as G3 =-=[23]-=-. G3 can be defined by: (Ax) : Γ, A ⊢ A, ∆ Γ ⊢ A, ∆ Γ, B ⊢ ∆ (→L) : Γ, A→B ⊢ ∆ Γ ⊢ A, ∆ Γ, A ⊢ ∆ (cut) : Γ ⊢ ∆ Γ, A ⊢ B, ∆ (→R) : Γ ⊢ A→B, ∆ It allows sequents of the form A1, . . . , An ⊢ B1, . . . ,... |

1 | Computation with classical sequents. MSCS - Bakel, Lescanne - 2008 |