## Two behavioural lambda models (2003)

Venue: | Types for Proofs and Programs |

Citations: | 5 - 4 self |

### BibTeX

@INPROCEEDINGS{Dezani-ciancaglini03twobehavioural,

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

title = {Two behavioural lambda models},

booktitle = {Types for Proofs and Programs},

year = {2003},

pages = {127--147},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. We build a lambda model which characterizes completely (persistently) normalizing, (persistently) head normalizing, and (persistently) weak head normalizing terms. This is proved by using the finitary logical description of the model obtained by defining a suitable intersection type assignment system.

### Citations

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Citation Context ...λvar.Λ) var ::= x | var ′ We use x,y,z,... ,x1,... for arbitrary term variables and M,N,P,... , M1,... for arbitrary terms. In writing terms we assume the standard conventions on parentheses and dots =-=[5]-=-. FV(M) denotes the set of free variables of a term M. By M[x:= N] we denote the term obtained by substituting the term N for all the free occurrences of the variable x in M, taking into account that ... |

272 |
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Citation Context ...structures and infinite combinatorics” (of the Ministry of Science, Technology, and Development of Serbia).swe define the sets FD and FE of filters respectively on the sets TD and TE. Following Scot=-=t [26], Cop-=-po et al. [8], and Alessi [3], we will show that the sets FD (ordered by subset inclusion) and FE and the corresponding inverse models D∞ and E∞ are isomorphic as ω-algebraic cpos. This isomorphi... |

233 | Domain theory in logical form
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Citation Context ...e general framework of Stone dualities (Johnstone [14]). This framework later received a categorically principled explanation by Abramsky in the broader perspective of “domain theory in logical form=-=” [1]. Th-=-e interest of the above isomorphism lies in the fact that the interpretations of lambda terms in D∞ and E∞ are isomorphic to the filters of types one can derive in the corresponding type assignmen... |

231 |
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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 [28], Tait [29], Girard =-=[13]-=-, Krivine [16], [17], Mitchell [20]). The reducibility method is also used in Leivant [18] and Gallier [11] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, an... |

221 |
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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 =-=[28]-=-, Tait [29], Girard [13], Krivine [16], [17], Mitchell [20]). The reducibility method is also used in Leivant [18] and Gallier [11] for characterizing strongly normalizing terms, normalizing terms, he... |

219 |
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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. =-=[4]-=-, generalizes the one given by Scott for simple types (Scott [24]). The meanings of types are subsets of the domain of discourse, arrow types are defined as logical predicates and intersection is set-... |

145 |
Type systems for programming languages
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Citation Context ...led reducibility method. This method is a generally accepted way for proving the strong normalization property of various type systems (Tait [28], Tait [29], Girard [13], Krivine [16], [17], Mitchell =-=[20]-=-). The reducibility method is also used in Leivant [18] and Gallier [11] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak head normalizing terms by th... |

136 | Full abstraction in the lazy lambda calculus
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- 1993
(Show Context)
Citation Context ... PHN = PWN ∩ HN PN � PHN ∩ N CHN = C ∩ HN CN = C ∩ N C ∩ PHN = ∅ C ∩ PN = ∅. Proof. A persistently weak head normalizing term M is either an unsolvable term of order ∞ (as defined =-=in Abramsky and Ong [2]),-=- i.e. for all n there is N such that M =β λx1 . . . xn.N, or it is a solvable term such that the head variable of its head normal form is free. In fact if M is an unsolvable term of a finite order, ... |

133 |
Continuous lattices
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(Show Context)
Citation Context ...o the persistent versions of these notions, and the sets of closable, closable normalizing and closable head normalizing lambda terms. We build two inverse lambda models D∞ and E∞, according to Sc=-=ott [24]-=-, which completely characterize each of the mentioned sets of terms. More precisely for each one of the above nine sets of terms there is a corresponding element in at least one of these models such t... |

128 |
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 [28], Tait [29], Girard [13], Krivine =-=[16]-=-, [17], Mitchell [20]). The reducibility method is also used in Leivant [18] and Gallier [11] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak head no... |

105 | 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 [29] and Girard [13] for proving the strong normalization property of polymorphic lambda calculus. In Pottinger [23], van Bakel =-=[30]-=-, Krivine [16], [17], Ghilezan [12], Amadio and Curien [4], the reducibility method is applied in order to characterize all and only thesstrongly normalizing lambda terms in lambda calculus with inter... |

52 |
A Realizability Interpretation of the Theory of Species
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- 1975
(Show Context)
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 [28], Tait =-=[29]-=-, Girard [13], Krivine [16], [17], Mitchell [20]). The reducibility method is also used in Leivant [18] and Gallier [11] for characterizing strongly normalizing terms, normalizing terms, head normaliz... |

44 | Set-theoretical and other elementary models of the λ-calculus
- Plotkin
- 1993
(Show Context)
Citation Context ... is the settheoretic intersection. This semantics is at the basis of our application of the reducibility method. The second semantics views types as compact elements of Plotkin’s λstructures (Plotk=-=in [22]-=-). 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... |

44 |
A type assignment for the strongly normalizable λ-terms
- Pottinger
- 1980
(Show Context)
Citation Context ...ormalization property of simply typed lambda calculus. Further it was developed in Tait [29] and Girard [13] for proving the strong normalization property of polymorphic lambda calculus. In Pottinger =-=[23]-=-, van Bakel [30], Krivine [16], [17], Ghilezan [12], Amadio and Curien [4], the reducibility method is applied in order to characterize all and only thesstrongly normalizing lambda terms in lambda cal... |

35 |
Typing and computation properties of lambda expressions
- Leivant
- 1986
(Show Context)
Citation Context ...epted way for proving the strong normalization property of various type systems (Tait [28], Tait [29], Girard [13], Krivine [16], [17], Mitchell [20]). The reducibility method is also used in Leivant =-=[18]-=- and Gallier [11] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak head normalizing terms by their typeability in various intersection type systems. I... |

23 |
Logical relations and the typed λ-calculus
- Statman
- 1985
(Show Context)
Citation Context ...rsistent versions are characterized in Dezani et al. [10]. Furthermore, this method was applied for the proof of the Church-Rosser property (confluence) of the simply typed lambda calculus in Statman =-=[27]-=-, Koletsos [15], and Mitchell [20], [21]. We will adapt the reducibility method, by requiring that the terms typable with the key types listed in Theorem 5 belong to the corresponding sets. In order t... |

21 | Strong normalization and typability with intersection types
- Ghilezan
- 1996
(Show Context)
Citation Context ...us. Further it was developed in Tait [29] and Girard [13] for proving the strong normalization property of polymorphic lambda calculus. In Pottinger [23], van Bakel [30], Krivine [16], [17], Ghilezan =-=[12]-=-, Amadio and Curien [4], the reducibility method is applied in order to characterize all and only thesstrongly normalizing lambda terms in lambda calculus with intersection types. The reducibility met... |

18 | Compositional characterization of λ-terms using intersection types. Theoret
- Dezani-Ciancaglini, Honsell, et al.
(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. =-=[10]-=- the reducibility method is applied to characterizing both the mentioned sets of terms and their persistent versions. In all these papers different properties are characterized by means of different t... |

17 |
Amadio and Pierre-Louis Curien. Domains and lambda-calculi, volume 46 of Cambridge Tracts
- Roberto
- 1998
(Show Context)
Citation Context ...oped in Tait [29] and Girard [13] for proving the strong normalization property of polymorphic lambda calculus. In Pottinger [23], van Bakel [30], Krivine [16], [17], Ghilezan [12], Amadio and Curien =-=[4]-=-, the reducibility method is applied in order to characterize all and only thesstrongly normalizing lambda terms in lambda calculus with intersection types. The reducibility method is also used for ch... |

11 |
Strutture di tipi, teoria dei domini, e modelli del λ-calcolo
- Alessi
- 1991
(Show Context)
Citation Context ... corresponding to the elements of D0, by the function type constructor and the intersection type constructor. Then we define a set F of filters on the set T . According to Coppo et al. [7] and Alessi =-=[2]-=-, the set F of filters and the inverse model D∞ are isomorphic as ω-algebraic lattices. This isomorphism falls in the general framework of Stone dualities (Johnstone [14]). This framework later receiv... |

11 |
Foundation for Programmimg Languages
- Mitchell
- 1996
(Show Context)
Citation Context ...ezani et al. [10]. Furthermore, this method was applied for the proof of the Church-Rosser property (confluence) of the simply typed lambda calculus in Statman [27], Koletsos [15], and Mitchell [20], =-=[21]-=-. We will adapt the reducibility method, by requiring that the terms typable with the key types listed in Theorem 5 belong to the corresponding sets. In order to develop the reducibility method we con... |

10 |
Lambda-terms as total or partial functions on normal forms
- Böhm, Dezani-Ciancaglini
- 1975
(Show Context)
Citation Context ...free variable. For each of the above properties, we shall consider also the corresponding persistent version (see Definition 3). Persistently normalizing terms have been introduced in Böhm and Dezani =-=[6]-=-. Definition 3. (Persistent normalization properties) i) A term M is persistently normalizing, M ∈ PN, if M → N ∈ N for all terms → N in N. 35s36 ii) A term M is persistently head normalizing, M ∈ PHN... |

10 |
Church-Rosser theorem for typed functionals
- Koletsos
- 1985
(Show Context)
Citation Context ...ns are characterized in Dezani et al. [10]. Furthermore, this method was applied for the proof of the Church-Rosser property (confluence) of the simply typed lambda calculus in Statman [27], Koletsos =-=[15]-=-, and Mitchell [20], [21]. We will adapt the reducibility method, by requiring that the terms typable with the key types listed in Theorem 5 belong to the corresponding sets. In order to develop the r... |

10 |
Lecture note on the domain interpretation of type theory
- Martin-Löf
- 1983
(Show Context)
Citation Context ...s as information systems in Scott [26], 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-L=-=öf [19]. -=-As stated first in Coppo et al. [8] 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) the e... |

8 | Typing untyped λ-terms, or reducibility strikes again
- Gallier
- 1998
(Show Context)
Citation Context ...ving the strong normalization property of various type systems (Tait [28], Tait [29], Girard [13], Krivine [16], [17], Mitchell [20]). The reducibility method is also used in Leivant [18] and Gallier =-=[11]-=- 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 |
Open problem
- Scott
- 1975
(Show Context)
Citation Context ...iterature and that the present paper deals with both of them. The set-theoretical semantics, originally introduced in Barendregt et al. [5], generalizes the one given by Scott for simple types (Scott =-=[25]-=-). The meanings of types are subsets of the domain of discourse, arrow types are defined as logical predicates and intersection is the settheoretic intersection. This semantics is at the basis of our ... |

6 | A lambda model characterizing computational behaviours of terms
- Dezani-Ciancaglini, Ghilezan
(Show Context)
Citation Context ...resent paper (dealing only with the first six sets of terms) was presented at the International Workshop on Rewriting in Proof and Computation (RPC’01, Tohoku University 25-27/10/2001, Sendai, Japan=-=) [9] -=-and at the Types Workshop (TYPES 2002 24-28/04/2002, Nijmegen, The Netherlands). 2 The Models We use standard notations for lambda terms and beta reductions. Definition 1 (The set Λ of lambda terms).... |

6 |
types and models. Ellis Horwood
- Lambda-calculus
- 1993
(Show Context)
Citation Context ...pply the so called reducibility method. This method is a generally accepted way for proving the strong normalization property of various type systems (Tait [28], Tait [29], Girard [13], Krivine [16], =-=[17]-=-, Mitchell [20]). The reducibility method is also used in Leivant [18] and Gallier [11] for characterizing strongly normalizing terms, normalizing terms, head normalizing terms, and weak head normaliz... |

5 |
Mariangiola Dezani-Ciancaglini, Furio Honsell, and Giuseppe Longo. Extended type structures and filter lambda-models
- Coppo
- 1984
(Show Context)
Citation Context ...nite combinatorics” (of the Ministry of Science, Technology, and Development of Serbia).swe define the sets FD and FE of filters respectively on the sets TD and TE. Following Scott [26], Coppo et al=-=. [8], and-=- Alessi [3], we will show that the sets FD (ordered by subset inclusion) and FE and the corresponding inverse models D∞ and E∞ are isomorphic as ω-algebraic cpos. This isomorphism falls in the ge... |

1 |
Domain theory in logical form, Ann
- Abramsky
- 1991
(Show Context)
Citation Context ...e general framework of Stone dualities (Johnstone [14]). This framework later received a categorically principled explanation by Abramsky in the broader perspective of “domain theory in logical form” =-=[1]-=-. In Kegelman [15] one can find a great tour through various Stone dualities in the category of continuous domains. The interest of the above isomorphism lies in the fact that the interpretations of l... |

1 |
Curien P.L.; Domains and Lambda-Calculi
- Amadio
- 1998
(Show Context)
Citation Context ... strong normalization property of various typessystems (Tait [26], [27], Girard [13], Mitchell [19], [20], Leivant [17], Pottinger [22], Krivine [16], van Bakel [28], Ghilezan [12], Amadio and Curien =-=[3]-=-, Gallier [11], Dezani et al. [10]). In all these papers different properties are characterized by means of different type assignment systems: so the novelty of the present approach is that we charact... |