## A Complete Characterization of Complete Intersection-Type Theories (Extended Abstract) (2000)

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Venue: | ACM TOCL |

Citations: | 12 - 5 self |

### BibTeX

@ARTICLE{Dezani-ciancaglini00acomplete,

author = {M. Dezani-ciancaglini and F. Honsell and F. Alessi},

title = {A Complete Characterization of Complete Intersection-Type Theories (Extended Abstract)},

journal = {ACM TOCL},

year = {2000},

volume = {4},

pages = {2003}

}

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### Abstract

M. DEZANI-CIANCAGLINI Universita di Torino, Italy F. HONSELL Universita di Udine, Italy F. ALESSI Universita di Udine, Italy Abstract We characterize those intersection-type theories which yield complete intersection-type assignment systems for l-calculi, with respect to the three canonical set-theoretical semantics for intersection-types: the inference semantics, the simple semantics and the F-semantics. Keywords Lambda Calculus, Intersection Types, Semantic Completeness, Filter Structures. 1 Introduction Intersection-types disciplines originated in [6] to overcome the limitations of Curry 's type assignment system and to provide a characterization of strongly normalizing terms of the l-calculus. But very early on, the issue of completeness became crucial. Intersection-type theories and filter l-models have been introduced, in [5], precisely to achieve the completeness for the type assignment system l" BCD W , with respect to Scott's simple semantics. And this result, ...

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Citation Context ...iderably over the years (e.g. [9, 8, 17, 11, 2, 20, 16], etc.), especially in connection with the study of domain models for l-calculi in the perspective of Abramsky's "domain theory in logical f=-=orm" [1]-=-. Furthermore new semantics have been proposed for intersection-types [14]. The problem of characterizing syntactically the sound and adequate (complete) intersection-type theories, with respect to th... |

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Citation Context ...r than simply as subsets. The arrow constructor is defined as for logical relations: dsA!B d 0 iff for all c; c 0 such that csA c 0 it holds d \Delta csB d 0 \Delta c 0 . The F-quotient set semantics =-=[23]-=-, modifies the quotient set semantics, in the same way as the F-semantics modifies the simple semantics. Namely it requires that all elements of the domain which are equivalent with respect to an arro... |

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Citation Context ...characterization of strongly normalizing terms of the λ-calculus. But very early on, the issue of completeness became crucial. Intersection-type theories and filter λ-models have been introduced, in [=-=Barendregt et al. 1983-=-], precisely to achieve the completeness for the type assignment system λ∩BCD Ω , with respect to Scott’s simple semantics. And this result, together with the conservativity of λ∩BCD Ω , with respect ... |

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Citation Context ...e Scott's conjecture concerning the completeness of the set-theoretic semantics for simple types. The number of type theories of interest in the literature has grown considerably over the years (e.g. =-=[9, 8, 17, 11, 2, 20, 16], etc.), e-=-specially in connection with the study of domain models for l-calculi in the perspective of Abramsky's "domain theory in logical form" [1]. Furthermore new semantics have been proposed for i... |

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Citation Context ...ng the type theories of Figure 2, those adequate for the simple semantics are Σ Ba and Σ BCD . Other adequate type theories in the literature are those of [Honsell and Lenisa 1999; Egidi et al. 1992; =-=Scott 1972-=-; Park 1976; Coppo et al. 1987; Honsell and Ronchi della Rocca 1992]. Non adequate type theories are all those inducing computationally adequate models for the lazy λ-calculus, e.g. Σ AO , or for the ... |

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Citation Context ...ess for intersection-type theories. Filter structures arise naturally in the context of those generalizations of Stone duality that are used in discussing domain theory in logical form (see [1], [9], =-=[26]-=-). This approach provides a conceptually independent semantics to intersection -types, the lattice semantics. Types are viewed as compact elements of domains. The type W denotes the least element, int... |

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Citation Context ...pected with respect to application and which are also canonical representatives of functions. The inference semantics is the counterpart of the inference semantics for polymorphic types introduced in =-=[19]-=-, generalized to suitable applicative structures with an interpretation function, called l-applicative structures. Here the interpretation of arrows is taken with respect to an arbitrary set which inc... |

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Citation Context ...amming General Terms: Theory, Languages Additional Key Words and Phrases: Lambda calculus, Intersection Types, Lambda Models, Completness 1. INTRODUCTION Intersection-types disciplines originated in [=-=Coppo and Dezani-Ciancaglini 1980-=-] to overcome the limitations of Curry’s type assignment system and to provide a Author addresses: M. Dezani-Ciancaglini, Dipartimento di Informatica, Università di Torino, Corso Svizzera 185, 10149 T... |

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Citation Context ... Scott’s conjecture concerning the completeness of the set-theoretic semantics for simple types. The number of type theories of interest in the literature has grown considerably over the years (e.g. [=-=Coppo et al. 1984-=-; Coppo et al. 1987; Honsell and Ronchi della Rocca 1992; Egidi et al. 1992; Abramsky and Ong 1993; Plotkin 1993; Honsell and Lenisa 1999], etc.), especially in connection with the study of domain mod... |

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Citation Context ...e Scott's conjecture concerning the completeness of the set-theoretic semantics for simple types. The number of type theories of interest in the literature has grown considerably over the years (e.g. =-=[9, 8, 17, 11, 2, 20, 16], etc.), e-=-specially in connection with the study of domain models for l-calculi in the perspective of Abramsky's "domain theory in logical form" [1]. Furthermore new semantics have been proposed for i... |

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Citation Context ... type theory which validates axiom (W-h). Among the type theories of Figure 2, those adequate for the simple semantics are S Ba and S BCD . Other adequate type theories in the literature are those of =-=[16, 11, 21, 8, 17]-=-. Non adequate type theories are all those inducing computationally adequate models for the lazy l-calculus, e.g. S AO . The same argument used for the inference semantics allows to show that the natu... |

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Citation Context ...omain such that for all good ρ we get that ρ(x) ∈ Φ implies [x] D ρ = [λy.xy] D ρ . Remark that the conditions (1) and (2) of Definition 5.5 are true for all known models of (restricted) λ-calculus ([=-=Hindley and Longo 1980-=-], [Plotkin 1975], [Egidi et al. 1992], [Honsell and Lenisa 1999]). One can easily see that the following holds: Proposition 5.6. (1) All type interpretation domains and all type environments agree wi... |

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Citation Context ...n arbitrary type and W is the universal type, is a derived rule. Yokouchi in [27] shows that if we add two suitable rules (quite similar to Hindley rule) to the intersection-type assignment system of =-=[7]-=- we obtain completeness for the F-semantics. Abramsky and Ong in [2] prove the completeness of the theory S AO , with respect to the F-semantics, over applicative structures with convergence. 14 Proce... |

35 |
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Citation Context ...lus. They differ in the interpretation of the arrow type constructor. In what follows we shall mainly follow the terminology of [14]. The simple and the F-semantics are defined as expected. Following =-=[24]-=-, the quotient set semantics takes into account that we want to consider equivalent two functions iff they give equivalent results when applied to Dezani et al.: A Complete Characterization of Complet... |

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Citation Context ...od ρ we get that ρ(x) ∈ Φ implies [x] D ρ = [λy.xy] D ρ . Remark that the conditions (1) and (2) of Definition 5.5 are true for all known models of (restricted) λ-calculus ([Hindley and Longo 1980], [=-=Plotkin 1975-=-], [Egidi et al. 1992], [Honsell and Lenisa 1999]). One can easily see that the following holds: Proposition 5.6. (1) All type interpretation domains and all type environments agree with AO and with E... |

29 | An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus - Honsell, Rocca - 1992 |

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Citation Context ...a quotiented version of the inference semantics, but this would be treated similarly. Completeness of various intersection-type assignment systems, over various kinds of l-models, have been proved in =-=[5, 13, 9, 8, 3, 2, 18]-=-. As far as the completeness with respect to the F-semantics of intersectiontype assignment systems, over l-models, we can cite [10, 27, 2]. In [10] the intersection-type assignment systems which are ... |

24 |
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Citation Context ...ve ways of interpreting Curry’s types in a model of the untyped λ-calculus. They differ in the interpretation of the arrow type constructor. In what follows we shall mainly follow the terminology of [=-=Hindley 1983-=-a]. The simple and the F-semantics are defined as expected. Following [Scott 1980a], the quotient set semantics takes into account that we want to consider equivalent two functions iff they give equiv... |

21 |
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Citation Context ... concerning the completeness of the set-theoretic semantics for simple types. The number of type theories of interest in the literature has grown considerably over the years (e.g. [Coppo et al. 1984; =-=Coppo et al. 1987-=-; Honsell and Ronchi della Rocca 1992; Egidi et al. 1992; Abramsky and Ong 1993; Plotkin 1993; Honsell and Lenisa 1999], etc.), especially in connection with the study of domain models for λ-calculi i... |

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Citation Context ...e semantics of types is the completeness of type assignment systems. Hindley proved in [14] that Curry's type assignment system is complete for all the mentioned semantics. More specifically [14] and =-=[15]-=- show the completeness for the simple semantics and moreover that: 1. G j= 5 F M : A if and only if G j= 5 s M : A, when A is a Curry type; 2. the simple semantics is a special case of the quotient se... |

13 |
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Citation Context ...ional Logic, Vol. TBD, No. TBD, TBD TBD.A Complete Characterization of Complete Intersection-Type Theories · 3 on the types, rather than on the typing judgments themselves, as had been done earlier [=-=Dezani-Ciancaglini and Margaria 1986-=-]. The task of checking the condition is made therefore considerably more tractable. Lastly we do not restrict attention just to λ-models, but to the more general class of λ-applicative structures. Th... |

12 | The completeness theorem for typing -terms - Hindley - 1983 |

9 |
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Citation Context ...a quotiented version of the inference semantics, but this would be treated similarly. Completeness of various intersection-type assignment systems, over various kinds of l-models, have been proved in =-=[5, 13, 9, 8, 3, 2, 18]-=-. As far as the completeness with respect to the F-semantics of intersectiontype assignment systems, over l-models, we can cite [10, 27, 2]. In [10] the intersection-type assignment systems which are ... |

9 |
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Citation Context ...ics for simple types. The number of type theories of interest in the literature has grown considerably over the years (e.g. [Coppo et al. 1984; Coppo et al. 1987; Honsell and Ronchi della Rocca 1992; =-=Egidi et al. 1992-=-; Abramsky and Ong 1993; Plotkin 1993; Honsell and Lenisa 1999], etc.), especially in connection with the study of domain models for λ-calculi in the perspective of Abramsky’s “domain theory in logica... |

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8 |
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Citation Context ...iontypes. In this paper we solve completely the characterization problem as far as the three canonical set-theoretical semantics for intersection-types: the inference semantics, the simple semantics [=-=Scott 1975-=-] and the F-semantics [Scott 1980b]. These are the semantics which arise by interpreting types as subsets of applicative structures, and by taking as interpretation of the intersection constructor, ∩,... |

7 | Semantical analysis of perpetual strategies in -calculus. Theoret - Honsell, Lenisa - 1999 |

7 |
The Y -combinator in Scott’s λ-calculus models (revised version). Theory of Computation Report
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Citation Context ...theories of Figure 2, those adequate for the simple semantics are Σ Ba and Σ BCD . Other adequate type theories in the literature are those of [Honsell and Lenisa 1999; Egidi et al. 1992; Scott 1972; =-=Park 1976-=-; Coppo et al. 1987; Honsell and Ronchi della Rocca 1992]. Non adequate type theories are all those inducing computationally adequate models for the lazy λ-calculus, e.g. Σ AO , or for the call-by-val... |

6 | Semantical analysis of perpetual strategies in λ-calculus
- Honsell, Lenisa
- 1999
(Show Context)
Citation Context ...erest in the literature has grown considerably over the years (e.g. [Coppo et al. 1984; Coppo et al. 1987; Honsell and Ronchi della Rocca 1992; Egidi et al. 1992; Abramsky and Ong 1993; Plotkin 1993; =-=Honsell and Lenisa 1999-=-], etc.), especially in connection with the study of domain models for λ-calculi in the perspective of Abramsky’s “domain theory in logical form” [Abramsky 1991]. Furthermore new semantics have been p... |

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3 |
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(Show Context)
Citation Context ... characterization of strongly normalizing terms of the l-calculus. But very early on, the issue of completeness became crucial. Intersection-type theories and filter l-models have been introduced, in =-=[5], precisel-=-y to achieve the completeness for the type assignment system l" BCD W , with respect to Scott's simple semantics. And this result, together with the conservativity of l" BCD W , with respect... |

2 |
Dezani-Ciancaglini and M. Zacchi [1987]. Type theories, normal forms
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Citation Context ...e Scott's conjecture concerning the completeness of the set-theoretic semantics for simple types. The number of type theories of interest in the literature has grown considerably over the years (e.g. =-=[9, 8, 17, 11, 2, 20, 16], etc.), e-=-specially in connection with the study of domain models for l-calculi in the perspective of Abramsky's "domain theory in logical form" [1]. Furthermore new semantics have been proposed for i... |

2 |
A Characterization of F-Complete Type Assignments, Theoret
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Citation Context ...ndly the characterizations are all given just in terms of simple closure conditions on the preorder relation , , on the types, rather than on the typing judgments themselves, as had been done earlier =-=[10]-=-. The task of checking the condition is made therefore considerably more tractable. Lastly we do not restrict attention just to l-models, but to the more general class of l-applicative structures. Thu... |

2 |
Della Rocca [1992]. Operational, denotational and logical descriptions: a case study, Fund
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Citation Context |

2 |
1999]. Semantical analysis of perpetual strategies in λ-calculus, Theoret
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Citation Context |

2 |
Della Rocca [1992]. An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus
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Citation Context ...j= 5 F M : A implies G ` 5 M : A iff either S 5 is an F-type theory or S 5 is a natural type theory which validates axiom (W-h). The theories S EH R , S AO and S BCD , as well as the type theories of =-=[16, 21, 8, 17]-=- are adequate for the F-semantics. Moreover for the last five and S BCD the simple semantics coincides with the F-semantics. The theory S Ba is an example of a theory which is not adequate with respec... |

2 | A characterisation of -complete type assignments - Dezani-Ciancaglini, Margaria - 1986 |

2 | Typed λ-calculus and applications - Barendregt - 2000 |