## A Lambda Model Characterizing Computational Behaviours of Terms (2001)

Venue: | PROCEEDINGS OF THE AND LIKAVEC INTERNATIONAL WORKSHOP REWRITING IN PROOF AND COMPUTATION |

Citations: | 6 - 4 self |

### BibTeX

@INPROCEEDINGS{Dezani-Ciancaglini01alambda,

author = {Mariangiola Dezani-Ciancaglini and Silvia Ghilezan},

title = {A Lambda Model Characterizing Computational Behaviours of Terms},

booktitle = {PROCEEDINGS OF THE AND LIKAVEC INTERNATIONAL WORKSHOP REWRITING IN PROOF AND COMPUTATION},

year = {2001},

pages = {100--119},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms.

### Citations

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Citation Context ...ee Johnstone [12]). Other very important examples are the descriptions of�-algebraic complete lattices as intersection type theories in Coppo et al. [7], Scott domains as information systems in Scott =-=[23]-=-, and SFP domains as pre-locales in Abramsky [1]. It is worthwhile to mention also Martin-Löf’s domain interpretation of intuitionistic type theory in Martin-Löf [16]. As stated first in Coppo et al. ... |

229 |
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Citation Context ...f this property we apply the so called reducibility method. This method is a generally accepted way for proving the strong normalization property of various type systems (Tait [25], Tait [26], Girard =-=[11]-=-, Krivine [14], Mitchell [17]). The reducibility method is also used in Leivant [15], Gallier [9] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak hea... |

228 | Domain theory in logical form
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Citation Context ...e general framework of Stone dualities (Johnstone [12]). This framework later received a categorically principled explanation by Abramsky in the broader perspective of “domain theory in logical form” =-=[1]-=-. The interest of the above isomorphism lies in the fact that the interpretations of lambda terms in�are isomorphic to the filters of types one can derive in this type 1sassignment system (Alessi [3])... |

216 |
A filter lambda model and the completeness of type assignment. J.Symbolic Logic, 48:931--940
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Citation Context ...are essentially two semantics for intersection types in the literature and that the present paper deals with both of them. The set-theoretical semantics, originally introduced in by Barendregt et al. =-=[5]-=-, generalizes the one given by Scott for simple types (Scott [22]). The meanings of types are subsets of the domain of discourse, arrow types are defined as logical predicates and intersection is set-... |

213 |
Intensional Interpretations of Functionals of Finite Type
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Citation Context ...rder to prove one part of this property we apply the so called reducibility method. This method is a generally accepted way for proving the strong normalization property of various type systems (Tait =-=[25]-=-, Tait [26], Girard [11], Krivine [14], Mitchell [17]). The reducibility method is also used in Leivant [15], Gallier [9] for characterizing strongly normalizing terms, normalizing terms, head normali... |

141 |
Type systems for programming languages
- Mitchell
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(Show Context)
Citation Context ...so called reducibility method. This method is a generally accepted way for proving the strong normalization property of various type systems (Tait [25], Tait [26], Girard [11], Krivine [14], Mitchell =-=[17]-=-). The reducibility method is also used in Leivant [15], Gallier [9] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak head normalizing terms by their ... |

133 | Full abstraction in the lazy lambda calculus
- Abramsky, Ong
- 1993
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Citation Context ...ss the inclusions between persistent sets. Example 2.4 shows that all inclusions are proper. A persistently weak normalizing termÅis either an unsolvable term of order (as defined in Abramsky and Ong =-=[2]-=-), i.e. for allÒthere isÆsuch thatÅ�¬�Ü���ÜÒ�Æ, or it is a solvable term such that the head variable of its head normal form is free. In fact ifÅis an unsolvable term of a finite order, is unsolvable ... |

123 |
Lambda-calcul types et modèles
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Citation Context ...y we apply the so called reducibility method. This method is a generally accepted way for proving the strong normalization property of various type systems (Tait [25], Tait [26], Girard [11], Krivine =-=[14]-=-, Mitchell [17]). The reducibility method is also used in Leivant [15], Gallier [9] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak head normalizing ... |

103 | Complete restrictions of the Intersection Type Discipline
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(Show Context)
Citation Context ...perty of simply typed lambda calculus. Further it was developed in Tait [26] and Girard [11] for proving the strong normalization property of polymorphic lambda calculus. In Pottinger [20], van Bakel =-=[27]-=-, Krivine [14], Ghilezan [10], Amadio and Curien [4], the reducibility method is applied in order to characterize all and only the strongly normalizing lambda terms in lambda calculus with intersectio... |

75 |
Stone spaces
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Citation Context ...rding to Coppo et al. [7] and Alessi [3], the set�of filters and the inverse model�are isomorphic as�-algebraic lattices. This isomorphism falls in the general framework of Stone dualities (Johnstone =-=[12]-=-). This framework later received a categorically principled explanation by Abramsky in the broader perspective of “domain theory in logical form” [1]. The interest of the above isomorphism lies in the... |

52 |
Extended type structure and filter lambda models
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- 1984
(Show Context)
Citation Context ...ted from atomic types corresponding to the elements of�, by the function type constructor and the intersection type constructor. Then we define a set�of filters on the setÌ. According to Coppo et al. =-=[7]-=- and Alessi [3], the set�of filters and the inverse model�are isomorphic as�-algebraic lattices. This isomorphism falls in the general framework of Stone dualities (Johnstone [12]). This framework lat... |

49 |
A realizability interpretation of the theory of species
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- 1975
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Citation Context ...ve one part of this property we apply the so called reducibility method. This method is a generally accepted way for proving the strong normalization property of various type systems (Tait [25], Tait =-=[26]-=-, Girard [11], Krivine [14], Mitchell [17]). The reducibility method is also used in Leivant [15], Gallier [9] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms,... |

40 | Set-theoretical and other elementary models of the λ-calculus
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Citation Context ...heoretic intersection. This semantics is at the basis of our application of the reducibility method in Section 5. The second semantics views types as compact elements of Plotkin’s�structures (Plotkin =-=[19]-=-). According to this interpretation, the universal type denotes the least element, intersections denote joins of compact elements, and arrow types allow to internalize the space of continuous endomorp... |

36 |
A type assignment for the strongly normalizable terms
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Citation Context ...od method is a generally accepted way for proving the strong normalization property of various type systems (Tait [23], Tait [24], Girard [11], Mitchell [16] [17],Leivant [14], Gallier [9], Pottinger =-=[19]-=-, Krivine [13], van Bakel [25], Ghilezan [10], Amadio and Curien [3], Leivant [14], Gallier [9], Dezani et al. [8]). In all these papers different properties are characterized by means of different ty... |

35 |
Typing and computation properties of lambda expressions
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Citation Context ...ly accepted way for proving the strong normalization property of various type systems (Tait [25], Tait [26], Girard [11], Krivine [14], Mitchell [17]). The reducibility method is also used in Leivant =-=[15]-=-, Gallier [9] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak head normalizing terms by their typeability in various intersection type systems. In De... |

32 |
Continuous lattices, Toposes, Algebraic Geometry and Logic
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Citation Context ...the set of normalizing, head normalizing, weak head normalizing lambda terms, and those corresponding to the persistent versions of such notions. We build an inverse lambda model�, according to Scott =-=[21]-=-, which completely characterizes each of the six sets of terms mentioned. More precisely for each one of the above six sets of terms there is a corresponding element in the model such that a term belo... |

21 | Strong normalization and typability with intersection types
- Ghilezan
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Citation Context ...calculus. Further it was developed in Tait [26] and Girard [11] for proving the strong normalization property of polymorphic lambda calculus. In Pottinger [20], van Bakel [27], Krivine [14], Ghilezan =-=[10]-=-, Amadio and Curien [4], the reducibility method is applied in order to characterize all and only the strongly normalizing lambda terms in lambda calculus with intersection types. The reducibility met... |

17 |
Amadio and Pierre-Louis Curien. Domains and Lambda Calculi, volume 46 of Cambridge Tracts
- Roberto
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Citation Context ... developed in Tait [26] and Girard [11] for proving the strong normalization property of polymorphic lambda calculus. In Pottinger [20], van Bakel [27], Krivine [14], Ghilezan [10], Amadio and Curien =-=[4]-=-, the reducibility method is applied in order to characterize all and only the strongly normalizing lambda terms in lambda calculus with intersection types. The reducibility method is also used for ch... |

11 |
Lecture note on the domain interpretation of type theory
- Martin-Löf
- 1983
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Citation Context ...s as information systems in Scott [23], and SFP domains as pre-locales in Abramsky [1]. It is worthwhile to mention also Martin-Löf’s domain interpretation of intuitionistic type theory in Martin-Löf =-=[16]-=-. As stated first in Coppo et al. [7] and proved in Alessi [3], we can describe an inverse limit model�by taking: �� ¯the types freely generated by closing (a set of atomic �� types corresponding to) ... |

11 |
Foundation for Programmimg Languages
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Citation Context ...ani et al. [8]. Furthermore, this method was applied for the proof of the Church-Rosser property (confluence) of the simply typed lambda calculus in Statman [24], Koltesos [13], and Mitchell [17] and =-=[18]-=-. The general idea of the reducibility method is to interpret types by suitable sets (saturated and stable sets in Tait [25] and Krivine [14] and admissible relations in Mitchell [17] and [18]) of lam... |

10 |
Strutture di tipi, teoria dei domini e modelli del lambda calcolo
- Alessi
- 1991
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Citation Context ... types corresponding to the elements of�, by the function type constructor and the intersection type constructor. Then we define a set�of filters on the setÌ. According to Coppo et al. [7] and Alessi =-=[3]-=-, the set�of filters and the inverse model�are isomorphic as�-algebraic lattices. This isomorphism falls in the general framework of Stone dualities (Johnstone [12]). This framework later received a c... |

10 |
Church-Rosser theorem for typed functionals
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Citation Context ...ons are characterized in Dezani et al. [8]. Furthermore, this method was applied for the proof of the Church-Rosser property (confluence) of the simply typed lambda calculus in Statman [24], Koltesos =-=[13]-=-, and Mitchell [17] and [18]. The general idea of the reducibility method is to interpret types by suitable sets (saturated and stable sets in Tait [25] and Krivine [14] and admissible relations in Mi... |

10 |
Lambda-terms as total or partial functions on normal forms
- Böhm, Dezani-Ciancaglini
- 1975
(Show Context)
Citation Context ...ee variable. For each of the above properties, we shall consider also the corresponding persistent version (see Definition 2.3). Persistently normalizing terms have been introduced in Böhm and Dezani=-= [5]. Definition 2.3 (Persistent normalization properties-=-) i) A � term is persistently ������� normalizing, , � � ����� if for all � terms � in . ii) A � term is persistently head �����©��� normalizing, ,... |

10 |
Compositional characterization of -terms using intersection types. Internal report
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Citation Context ...follow, but the case of persistently normalizing terms. For that reason we develop a new characterization of these terms which is less general but much simpler than the one presented in Dezani et al. =-=[8]-=-. The proofs of the only if parts require the set-theoretic semantics of intersection types using saturated sets. The reducibility method method is a generally accepted way for proving the strong norm... |

8 |
Open problem
- Scott
- 1975
(Show Context)
Citation Context ...rature and that the present paper deals with both of them. The set-theoretical semantics, originally introduced in by Barendregt et al. [5], generalizes the one given by Scott for simple types (Scott =-=[22]-=-). The meanings of types are subsets of the domain of discourse, arrow types are defined as logical predicates and intersection is set-theoretic intersection. This semantics is at the basis of our app... |

5 |
Typing untyped -terms, or reducibility strikes again
- Gallier
- 1998
(Show Context)
Citation Context ...ducibility method method is a generally accepted way for proving the strong normalization property of various type systems (Tait [23], Tait [24], Girard [11], Mitchell [16] [17],Leivant [14], Gallier =-=[9]-=-, Pottinger [19], Krivine [13], van Bakel [25], Ghilezan [10], Amadio and Curien [3], Leivant [14], Gallier [9], Dezani et al. [8]). In all these papers different properties are characterized by means... |

1 |
Mariangiola Dezani-Ciancaglini.�-terms as total or partial functions on normal forms. In�-calculus and computer science theory
- Böhm
- 1975
(Show Context)
Citation Context ...ee variable. For each of the above properties, we shall consider also the corresponding persistent version (see Definition 2.3). Persistently normalizing terms have been introduced in Böhm and Dezani =-=[6]-=-. Definition 2.3 (Persistent normalization properties) i) A termÅis persistently normalizing,ÅÈÆ, ifÅ�ÆÆfor all terms�Æ inÆ. ii) A termÅis persistently head normalizing,ÅÈÀÆ, ifÅ�ÆÀÆfor all terms�Æ. i... |

1 |
Compositional characterization of�-terms using intersection types
- Dezani-Ciancaglini, Honsell, et al.
- 2000
(Show Context)
Citation Context ...] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak head normalizing terms by their typeability in various intersection type systems. In Dezani et al. =-=[8]-=- the reducibility method is applied to characterizing both the mentioned terms and their persistent versions. In all these papers different properties are characterized by means of different type assi... |

1 |
Typing untyped�-terms, or reducibility strikes again
- Gallier
- 1998
(Show Context)
Citation Context ...y for proving the strong normalization property of various type systems (Tait [25], Tait [26], Girard [11], Krivine [14], Mitchell [17]). The reducibility method is also used in Leivant [15], Gallier =-=[9]-=- for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak head normalizing terms by their typeability in various intersection type systems. In Dezani et al. [... |

1 |
A type assignment for the strongly normalizable�-terms
- Pottinger
- 1980
(Show Context)
Citation Context ...ormalization property of simply typed lambda calculus. Further it was developed in Tait [26] and Girard [11] for proving the strong normalization property of polymorphic lambda calculus. In Pottinger =-=[20]-=-, van Bakel [27], Krivine [14], Ghilezan [10], Amadio and Curien [4], the reducibility method is applied in order to characterize all and only the strongly normalizing lambda terms in lambda calculus ... |

1 |
Logical relations and the typed�-calculus
- Statman
- 1985
(Show Context)
Citation Context ...ersistent versions are characterized in Dezani et al. [8]. Furthermore, this method was applied for the proof of the Church-Rosser property (confluence) of the simply typed lambda calculus in Statman =-=[24]-=-, Koltesos [13], and Mitchell [17] and [18]. The general idea of the reducibility method is to interpret types by suitable sets (saturated and stable sets in Tait [25] and Krivine [14] and admissible ... |