## Completeness of Intersection and Union Type Assignment Systems for Call-By-Value Lambda-Models (0)

Citations: | 4 - 0 self |

### BibTeX

@MISC{Ishihara_completenessof,

author = {Hajime Ishihara and Toshihiko Kurata},

title = {Completeness of Intersection and Union Type Assignment Systems for Call-By-Value Lambda-Models},

year = {}

}

### OpenURL

### Abstract

We study a version of intersection and union type assignment system, unionelimination rule of which is allowed only when subject of its major premiss is a value of call-by-value -calculus. The system is shown to be sound and complete under some abstract notion of membership relation dened over simple semantics for callby -value -models, and to be invariant under call-by-value -conversion of subjects. We prove it by constructing a lter call-by-value -model. 1 Introduction Coppo et al. [5] introduced an intersection type assignment system as an extension of Curry's simple type assignment system (see [12, 11, 7] for expositions) to deal with the functional characters of solvable -terms. In addition to the simple types constructed from type-variables and ! , the intersection types contain a type-constant ! and types constructed by a constructor ^ (their intended meanings are universe and intersection of Partly supported by a Grant-in-Aid for Scientic Research (C) No.09640253 o...

### Citations

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A filter lambda model and the completeness of type assignment
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(Show Context)
Citation Context ...Japan. 1 two types). The intersection type assignment system has a pleasing property that the set of types assigned to a -term is invariant unders-conversion. Then Hindley [10] and Barendregt et al. [=-=-=-3] independently proved soundness and completeness for the standard semantics in which a type is interpreted by a subset of a -model and a -term having the type is interpreted by an element of the sub... |

63 |
Intersection and Union Types: Syntax and Semantics
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(Show Context)
Citation Context ... to extend the type assignment system further by adding a type constructor _ whose intended meaning is union of two types. The constructor wassrst introduced by MacQueen et al. [13]. Barbanera et al. =-=[2-=-] extensively dealt with type assignment systems with intersection and union types, and they introduced a type theory which is a natural extension of that for intersection types, and a type assignmen... |

35 |
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(Show Context)
Citation Context ...ver simple semantics for callby -value -models, and to be invariant under call-by-values-conversion of subjects. We prove it by constructing aslter call-by-value -model. 1 Introduction Coppo et al. [5=-=-=-] introduced an intersection type assignment system as an extension of Curry's simple type assignment system (see [12, 11, 7] for expositions) to deal with the functional characters of solvable -terms... |

25 |
The simple semantics for Coppo-Dezani-Salle types
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(Show Context)
Citation Context ... Hachioji, Tokyo 192-0397, Japan. 1 two types). The intersection type assignment system has a pleasing property that the set of types assigned to a -term is invariant unders-conversion. Then Hindley [=-=1-=-0] and Barendregt et al. [3] independently proved soundness and completeness for the standard semantics in which a type is interpreted by a subset of a -model and a -term having the type is interprete... |

9 |
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(Show Context)
Citation Context ...tained by allowing union-elimination rule only when the subject of its major premise is an element of Val in TA and of Var in TA , respectively. A call-by-value -model was introduced by Egini et al. [=-=8-=-] consists of a domain D and a subset K of D , the set of values in D , so that a -term in Val is interpreted by an element of K . We will introduce model of our type assignment systems based on a cal... |

7 |
union and intersection types for program extraction
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(Show Context)
Citation Context ... get type invariance unders-conversion of subjects, and prove soundness and completeness for models in which types are interpreted by the Beth type interpretation, or based on intersection types. See =-=[14, 9, 18, 22]-=- for other extensions of the intersection type assignment system. Call-by-value evaluation would be one of the most standard strategy to reduce program expressions, actually being adopted in various p... |

4 | A Minimal Relevant Logic and the Call-by-value Lambda Calculus
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(Show Context)
Citation Context ...ype assignment systems, which is a counterpart of theslter model for classical -calculus due to Barendregt et al. Then the idea was formalized by Motohama [15] (more detailed explanation is given in [=-=1]), an-=-d was presented in the MSJ regional workshop on \Theories of Types and Proofs" held at Tokyo Institute of Technology in 1997. On the other hand, in this paper we shall also introduce two intersec... |

1 |
Extended type structures and lambda models, in: Logic Colloquium '82
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(Show Context)
Citation Context ...3] built aslter -model in which a -term is interpreted by the set of all types that can be assigned to the -term, andslter models turn out to form a very rich class of -models (see, for examples, [4] and [17]). It is natural to extend the type assignment system further by adding a type constructor _ whose intended meaning is union of two types. The constructor wassrst introduced by MacQueen et al... |

1 |
personal communication
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- 1997
(Show Context)
Citation Context ... follow: (x :M )N ! v M [x =N ] (N 2 Val); and the contextual, re exive, symmetric and transitive closure of ! v is denoted by = v . Besides the consideration for union types mentioned above, Dezani [=-=-=-6] pointed out an construction of models of the call-by-value -calculus based on intersection and union type assignment systems, which is a counterpart of theslter model for classical -calculus due to... |

1 |
Intersection types, -models and Bohm-trees, MSJ Memoirs volume 2: theories of types and proofs
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(Show Context)
Citation Context ... prove it by constructing aslter call-by-value -model. 1 Introduction Coppo et al. [5] introduced an intersection type assignment system as an extension of Curry's simple type assignment system (see [=-=12, 11, 7-=-] for expositions) to deal with the functional characters of solvable -terms. In addition to the simple types constructed from type-variables and ! , the intersection types contain a type-constant ! a... |