## Intersection Types for λ-Trees

### BibTeX

@MISC{Bakel_intersectiontypes,

author = {Steffen Van Bakel and Franco Barbanera and Mariangiola Dezani-ciancaglini},

title = {Intersection Types for λ-Trees},

year = {}

}

### OpenURL

### Abstract

We introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating λ-terms (Böhm trees, Lévy-Longo trees,...). Then we prove, in an (almost) uniform way, that each type assignment system fully describes the observational equivalences induced by the corresponding tree representation of terms. More precisely, for each family of trees, two terms have the same tree if and only if they get assigned the same types in the corresponding type assignment system.

### Citations

1115 |
The Lambda Calculus: Its Syntax and Semantics
- Barendregt
- 1981
(Show Context)
Citation Context ...ding on the possible notions of stable relevant minimal information; the most commonly used being top trees (or Berarducci trees [6]), weak trees (or Lévy-Longo trees [25]), head trees (or Böhm trees =-=[4]-=-), eta trees and infinite eta trees (infinite eta trees are in one-one correspondence with Nakajima trees [23]). Hence, the various notions of tree represent different notions of meaning of a term (in... |

259 | Domains for denotational semantics - Scott - 1982 |

216 |
A filter lambda model and the completeness of type assignment. J.Symbolic Logic, 48:931--940
- Barendregt, Coppo, et al.
- 1983
(Show Context)
Citation Context ...ature for particular notions of tree. Ronchi della Rocca [24] proved that two terms have the same Böhm tree if and only if they have the same set of types in the standard intersection type discipline =-=[5]-=-. The proof of [24] is based on the notion of principal type of an approximate normal form, which is a type completely describing the approximate normal form. Principal types (as defined in [11] and u... |

210 | Data types as lattices - SCOTT - 1976 |

133 | Full abstraction in the lazy lambda calculus
- Abramsky, Ong
- 1993
(Show Context)
Citation Context ...olvable and it cannot be reduced to a lambda abstraction by means of the reduction relation induced by the β-rule [6]. Such terms are called unsolvables of order 0 in [22] and strongly unsolvables in =-=[1]-=-. Definition 2.1 Given the following axioms and rules: (β) (λx.M)N → M[N/x] (η) λx.Mx → M if x∈ FV(M) (ν) M→N ⇒ ML→ NL (ν)t M→N ⇒ ML→ NL (provided M is not a strong zero term) (ξ) M→N ⇒ λx.M→λx.N we ... |

121 |
Continuous lattices
- Scott
- 1972
(Show Context)
Citation Context ... that two λ-terms have the same tree representation if and only if they are equal in the λ-model. For example, • the infinite eta trees represent the local structure of Scott’s D∞ model as defined in =-=[26]-=- (this result was proved in [29]); • the eta trees represent the local structure of the inverse limit model defined in [12]; • the head trees represent the local structure of Scott’s Pω model as defin... |

95 | The relation between computational and denotational properties for Scott’s D∞-models of the lambda-calculus - Wadsworth - 1976 |

53 | The lazy lambda calculus in a concurrency scenario
- Sangiorgi
- 1994
(Show Context)
Citation Context ...resentations in literature, depending on the possible notions of stable relevant minimal information; the most commonly used being top trees (or Berarducci trees [6]), weak trees (or Lévy-Longo trees =-=[25]-=-), head trees (or Böhm trees [4]), eta trees and infinite eta trees (infinite eta trees are in one-one correspondence with Nakajima trees [23]). Hence, the various notions of tree represent different ... |

52 |
Extended type structure and filter lambda models
- Coppo, Dezani-Ciancaglini, et al.
- 1984
(Show Context)
Citation Context ... terms can be mimicked by suitable type theories: • Each inverse limit λ-model is isomorphic to a filter model, i.e., to a model in which the meaning of terms is a set of derivable intersection types =-=[10]-=-. • Two terms have the same head tree if and only if they have the same set of types in the standard intersection type discipline [5], as proved in [24]. • Two terms have the same weak tree if and onl... |

48 | A syntactic characterization of the equality in some models for the Lambda-Calculus
- Hyland
- 1976
(Show Context)
Citation Context ...if and only if, for all contexts C[ ], the following holds: C[M] has a head normal form if and only if C[N] has a head normal form. The same property holds even considering eta trees and normal forms =-=[18]-=-. By adding a non-deterministic choice operator and an adequate numeral system to the pure calculus, we obtain a language which internally discriminates two λ-terms if and only if they have different ... |

47 |
Principal type schemes and λ-calculus semantics
- Coppo, Dezani-Ciancaglini, et al.
- 1980
(Show Context)
Citation Context ...ipline [5]. The proof of [24] is based on the notion of principal type of an approximate normal form, which is a type completely describing the approximate normal form. Principal types (as defined in =-=[11]-=- and used in [24]) need an infinity of type variables and this agrees with the type syntax of [5]. Another related paper is [16]: it proves that two terms have the same Lévy-Longo tree [22] if and onl... |

41 | Lambda-calculi for (strict) parallel functions - Boudol - 1991 |

35 | Principal type schemes for the strict type assignment system
- Bakel
- 1993
(Show Context)
Citation Context ...For the type assignment system ⊢i we need a further property dealing with the types we can deduce for the terms whose infinite eta tree is just one variable. The notion of strict types comes in handy =-=[2]-=-. Definition 4.17 The set of strict types ST⊆Tei is the minimal set such that: i) ω,ζ,ϑ∈ST, ii) α, β1,..., βn ∈ST, n≥1 ⇒ β1∧ . . .∧βn→α∈ST. Property 4.18 For all types α ∈ Tei, there is a set of stric... |

34 | Principal type schemes and -calculus semantics - Coppo, Dezani-Ciancaglini, et al. - 1980 |

32 | Continuous lattices, Toposes, Algebraic Geometry and Logic - Scott - 1971 |

26 | Type Theories, Normal Forms and D1-Lambda-Models. Information and Computation 72 - COPPO, DEZANI-CIANCAGLINI, et al. - 1987 |

17 |
Type theories, normal forms, and D∞-lambda-models
- Coppo, Dezani-Ciancaglini, et al.
- 1987
(Show Context)
Citation Context ...te eta trees represent the local structure of Scott’s D∞ model as defined in [26] (this result was proved in [29]); • the eta trees represent the local structure of the inverse limit model defined in =-=[12]-=-; • the head trees represent the local structure of Scott’s Pω model as defined in [27] (a discussion on this topic can be found in [4], Chapter 19); • the weak trees were introduced by Longo in [22] ... |

16 | Algebras and combinators - Engeler - 1981 |

16 |
Infinite λ-Calculus and Non-Sensible Models
- Berarducci
- 1994
(Show Context)
Citation Context ...nal term. There are many such tree representations in literature, depending on the possible notions of stable relevant minimal information; the most commonly used being top trees (or Berarducci trees =-=[6]-=-), weak trees (or Lévy-Longo trees [25]), head trees (or Böhm trees [4]), eta trees and infinite eta trees (infinite eta trees are in one-one correspondence with Nakajima trees [23]). Hence, the vario... |

14 |
Set-theoretical models of λ-calculus: theories, expansions, isomorphisms
- Longo
- 1983
(Show Context)
Citation Context ... [12]; • the head trees represent the local structure of Scott’s Pω model as defined in [27] (a discussion on this topic can be found in [4], Chapter 19); • the weak trees were introduced by Longo in =-=[22]-=- (following [21]), who proved that they represent the local structure of Engeler’s models as defined in [17]. Orthogonally, the results about observational equivalences confirm this operational intuit... |

13 | The discriminating power of multiplicities in the -calculus - Boudol, Laneve - 1996 |

11 | An algebraic interpretation of the fiK-calculus; and an application of a labelled -calculus - L'evy - 1976 |

10 | Set-theoretical models of -Calculus: theories, expansions, isomorphisms - Longo - 1983 |

10 |
An algebraic interpretation of the λβK-calculus, and an application of a labelled λ-calculus
- Lévy
- 1976
(Show Context)
Citation Context ...d trees represent the local structure of Scott’s Pω model as defined in [27] (a discussion on this topic can be found in [4], Chapter 19); • the weak trees were introduced by Longo in [22] (following =-=[21]-=-), who proved that they represent the local structure of Engeler’s models as defined in [17]. Orthogonally, the results about observational equivalences confirm this operational intuition of dynamical... |

9 | Infinite -calculus and non-sensible models - Berarducci - 1994 |

9 |
Willem Klop, Ronan Sleep, and Fer-Jan de Vries. Transfinite reductions in orthogonal term rewriting systems
- Kennaway
(Show Context)
Citation Context ...makes the different intended notions of stable minimal relevant information explicit. Definition 2.2 i) A top normal form 1 is a term of one of the following three kinds: 1 Called root stable form in =-=[19]-=-.Theoretical Computer Science 272 (Theories of Types and Proofs 1997): 3-40, 2002 5 a) an application term of the form xM1 . . . Mm (m≥0); b) an abstraction term λx.M; c) an application term of the f... |

9 | Infinite normal forms for the -calculus - Nakajima - 1975 |

7 | A filter model for concurrent -calculus - Dezani-Ciancaglini, de’Liguoro, et al. - 1998 |

7 | Böhm’s theorem for Böhm trees
- Dezani-Ciancaglini, Intrigila, et al.
- 1998
(Show Context)
Citation Context ... a non-deterministic choice operator and an adequate numeral system to the pure calculus, we obtain a language which internally discriminates two λ-terms if and only if they have different head trees =-=[14]-=-. Weak trees correspond to the observational equivalence with respect to weak head normal forms in suitably enriched versions of the λ-calculus [25, 9, 16]. We can discriminate terms in the same way t... |

7 |
A filter model for concurrent λ -calculus
- Dezani-Ciancaglini, de’Liguoro, et al.
- 1998
(Show Context)
Citation Context ... standard intersection type discipline [5], as proved in [24]. • Two terms have the same weak tree if and only if they have the same set of types in the type discipline with union and intersection of =-=[13]-=-, as proved in [16]. • Two terms have the same top tree if and only if they have the same set of types in a type assignment system with applicative types [7]. In the present paper we will design one t... |

7 |
Infinite normal forms for the λ-calculus
- Nakajima
- 1975
(Show Context)
Citation Context ...or Berarducci trees [6]), weak trees (or Lévy-Longo trees [25]), head trees (or Böhm trees [4]), eta trees and infinite eta trees (infinite eta trees are in one-one correspondence with Nakajima trees =-=[23]-=-). Hence, the various notions of tree represent different notions of meaning of a term (in particular, they specify different notions of undefined value [20]). This apparently vague intuition is subst... |

6 |
Characterization theorems for a filter lambda model
- Rocca
- 1982
(Show Context)
Citation Context ... terms is a set of derivable intersection types [10]. • Two terms have the same head tree if and only if they have the same set of types in the standard intersection type discipline [5], as proved in =-=[24]-=-. • Two terms have the same weak tree if and only if they have the same set of types in the type discipline with union and intersection of [13], as proved in [16]. • Two terms have the same top tree i... |

3 | Jerzy Tiuryn, and Pawe/l Urzyczyn. Discrimination by parallel observers: the algorithm - Dezani-Ciancaglini - 1999 |

3 |
Fer-Jan de Vries. Meaningless terms in rewriting
- Kennaway, Oostrom
- 1999
(Show Context)
Citation Context ...one-one correspondence with Nakajima trees [23]). Hence, the various notions of tree represent different notions of meaning of a term (in particular, they specify different notions of undefined value =-=[20]-=-). This apparently vague intuition is substantiated by results starting with [29], which show that there exist precise correspondences between the tree representations of terms and the local structure... |

2 |
Mariangiola Dezani-Ciancaglini, and Fer-Jan de Vries. Types for trees
- Barbanera
- 1998
(Show Context)
Citation Context ... Section 5, instead, contains our main result: our type assignment systems can be used to analyze the observational behaviour represented by trees. A preliminary version of this paper has appeared in =-=[3]-=-, where almost all proofs were omitted. Abbreviations Below, we will use the following abbreviations for lambda terms. Y ≡ λ f . f((λx. f(xx))(λx. f(xx))) R ≡ λzxy.x(zzy) n { }} { I ≡ λx.x ∆n ≡ λx. x ... |

2 |
and Mariangiola Dezani-Ciancaglini. Infinite λ-calculus and types
- Berarducci
- 1999
(Show Context)
Citation Context ...discipline with union and intersection of [13], as proved in [16]. • Two terms have the same top tree if and only if they have the same set of types in a type assignment system with applicative types =-=[7]-=-. In the present paper we will design one type assignment system for each of the five families of trees mentioned above (more precisely, a type assignment system (almost) parametric with respect to th... |

2 |
Fer-Jan de Vries. Böhm’s theorem for Berarducci trees
- Dezani-Ciancaglini, Severi
(Show Context)
Citation Context ...equivalence with respect to weak head normal forms in suitably enriched versions of the λ-calculus [25, 9, 16]. We can discriminate terms in the same way that top trees do, using two powerful δ-rules =-=[15]-=-. It is clear that most of the relevant properties of terms pertain, more or less strongly, to the field of dynamics, i.e., to their computational behaviour. This, however, does not mean that we have ... |

2 | Type theories, normal forms, andD1-lambda-models - Coppo, Dezani-Ciancaglini, et al. - 1987 |

2 |
Jerzy Tiuryn, and Paweł Urzyczyn. Discrimination by parallel observers: the algorithm
- Dezani-Ciancaglini
- 1999
(Show Context)
Citation Context ...terms if and only if they have different head trees [14]. Weak trees correspond to the observational equivalence with respect to weak head normal forms in suitably enriched versions of the λ-calculus =-=[25, 9, 16]-=-. We can discriminate terms in the same way that top trees do, using two powerful δ-rules [15]. It is clear that most of the relevant properties of terms pertain, more or less strongly, to the field o... |

2 | The discriminating power of multiplicities in the λ-calculus
- Boudol, Laneve
- 1996
(Show Context)
Citation Context ...terms if and only if they have different head trees [14]. Weak trees correspond to the observational equivalence with respect to weak head normal forms in suitably enriched versions of the λ-calculus =-=[25, 9, 16]-=-. We can discriminate terms in the same way that top trees do, using two powerful δ-rules [15]. It is clear that most of the relevant properties of terms pertain, more or less strongly, to the field o... |

1 | Infinite normal forms for the -calculus. In -calculus and computer science theory (Proceedings Symposium - Nakajima - 1975 |

1 | An algebraic interpretation of theK-calculus, and an application of a labeled -calculus - Lévy - 1976 |