## Comparing cubes of typed and type assignment systems (1997)

### Cached

### Download Links

- [pubs.doc.ic.ac.uk]
- [dev.pubs.doc.ic.ac.uk]
- [www.loria.fr]
- [www.doc.ic.ac.uk]
- [www.doc.ic.ac.uk]
- [schiele.doc.ic.ac.uk]
- DBLP

### Other Repositories/Bibliography

Venue: | Annals of Pure and Applied Logic |

Citations: | 7 - 3 self |

### BibTeX

@ARTICLE{Bakel97comparingcubes,

author = {Steffen Van Bakel and Luigi Liquori and Simona Ronchi Della Rocca and Paweł Urzyczyn},

title = {Comparing cubes of typed and type assignment systems},

journal = {Annals of Pure and Applied Logic},

year = {1997},

volume = {86}

}

### OpenURL

### Abstract

We study the cube of type assignment systems, as introduced in [13], and confront it with Barendregt’s typed λ-cube [4]. The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address the question whether a judgement, derivable in a type assignment system, is always an erasure of a derivable judgement in a corresponding typed system; we show that this property holds only for the systems without polymorphism. The type assignment systems we consider satisfy the properties ‘subject reduction’ and ‘strong normalization’. Moreover, we define a new type assignment cube that is isomorphic to the typed one.

### Citations

940 | A theory of type polymorphism in programming
- Milner
- 1978
(Show Context)
Citation Context ...studying types is not solely justifiable through the connection between types and logic, as is clearly shown by, for example, the type system developed for ML that models type-constants and recursion =-=[18]-=-, and the intersection type discipline [2]. In our view, the main motivation for TAS comes from the ML-style of approaching types: to have type-free code with type assignment seen as a correctness cri... |

519 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
Citation Context ...u Warszawskiego, ul. Banacha 2, 02-097 Warszawa, Polska, urzy@mimuw.edu.pl Abstract We study the cube of type assignment systems, as introduced in [13], and confront it with Barendregt’s typed λ-cube =-=[4]-=-. The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address the question whether ... |

474 |
The calculus of constructions
- Coquand, Huet
- 1988
(Show Context)
Citation Context ...s in various ways. Examples of typed λ-calculi are the simply typed λcalculus (λ→) of Church, the second order λ-calculus of Girard and Reynolds (λ2) [15, 20], and the calculus of constructions (λPω) =-=[7, 8]-=-. Barendregt gave in [4] a compact and appealing presentation of a class of typed systems (TS), arranging them in a cube. In this cube, every vertex represents ∗ Partly supported by HCM project No. ER... |

364 |
Towards a theory of type structure
- Reynolds
- 1974
(Show Context)
Citation Context ...ped λ-calculi, where terms are decorated with types in various ways. Examples of typed λ-calculi are the simply typed λcalculus (λ→) of Church, the second order λ-calculus of Girard and Reynolds (λ2) =-=[15, 20]-=-, and the calculus of constructions (λPω) [7, 8]. Barendregt gave in [4] a compact and appealing presentation of a class of typed systems (TS), arranging them in a cube. In this cube, every vertex rep... |

245 |
Combinatory Reduction Systems
- Klop
- 1980
(Show Context)
Citation Context ...inition 1.17 satisfies the following property. Property 2.10 (CHURCH-ROSSER FOR TAS) If A →β A ′ and B →β B ′ , then there exists C, such that A ′ →β C and B ′ →β C. Proof: In the terminology of Klop =-=[16]-=-, our β-reduction is a regular combinatory reduction system, and thus the Church-Rosser property follows from Theorem 3.11 in [16]. The following lemma shows that all subterms of typable terms are typ... |

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 ...rough the connection between types and logic, as is clearly shown by, for example, the type system developed for ML that models type-constants and recursion [18], and the intersection type discipline =-=[2]-=-. In our view, the main motivation for TAS comes from the ML-style of approaching types: to have type-free code with type assignment seen as a correctness criterion, or safety means, but always outsid... |

208 |
Feys R. \Combinatory Logic
- Curry
- 1958
(Show Context)
Citation Context ...ules of R. The dependency-free plane of TAS contains some type assignment systems already known in the literature, that are convertible to certain typed systems: the Curry type assignment system (F1) =-=[9]-=- that corresponds to λ→, the polymorphic type assignment system (F2) [17] that corresponds to λ2, and the higher order type assignment system (Fω) [14] that corresponds to λω. The fact that in [13] al... |

144 |
The System F of Variable Types, fifteen years later
- Girard
- 1986
(Show Context)
Citation Context ...ped λ-calculi, where terms are decorated with types in various ways. Examples of typed λ-calculi are the simply typed λcalculus (λ→) of Church, the second order λ-calculus of Girard and Reynolds (λ2) =-=[15, 20]-=-, and the calculus of constructions (λPω) [7, 8]. Barendregt gave in [4] a compact and appealing presentation of a class of typed systems (TS), arranging them in a cube. In this cube, every vertex rep... |

85 |
Polymorphic Type Inference
- Leivant
- 1983
(Show Context)
Citation Context ...t systems already known in the literature, that are convertible to certain typed systems: the Curry type assignment system (F1) [9] that corresponds to λ→, the polymorphic type assignment system (F2) =-=[17]-=- that corresponds to λ2, and the higher order type assignment system (Fω) [14] that corresponds to λω. The fact that in [13] also systems that contain dependencies were considered, was a first attempt... |

60 |
Metamathematical investigations of a calculus of constructions. Rapport de recherche de l’INRIA
- Coquand
- 1989
(Show Context)
Citation Context ...s in various ways. Examples of typed λ-calculi are the simply typed λcalculus (λ→) of Church, the second order λ-calculus of Girard and Reynolds (λ2) [15, 20], and the calculus of constructions (λPω) =-=[7, 8]-=-. Barendregt gave in [4] a compact and appealing presentation of a class of typed systems (TS), arranging them in a cube. In this cube, every vertex represents ∗ Partly supported by HCM project No. ER... |

43 |
Modular proof of strong normalization for the calculus of constructions
- Geuvers, Nederhof
- 1991
(Show Context)
Citation Context ... one commonly used; this should enable the appreciation of the presentation of our cube of type assignment systems in the next subsection. For a complete development of Barendregt’s cube, we refer to =-=[4, 11]-=-. Definition 1.1 Iti) {∗,✷} is the set of sorts. Itii) The sets of typed λ-terms (Λt), typed constructors (Const), and typed kinds (Kindt) are mutually defined by the following grammar, where M, φ, an... |

43 | Extracting F# 's programs from proofs in the Calculus of Constructions - Paulin-Mohring - 1989 |

32 |
Extracting Fω’s programs from proofs in the calculus of constructions
- Paulin-Mohring
- 1989
(Show Context)
Citation Context ...nment systems. For this, we use the function ED that ‘erases dependencies’, i.e., removes the λ-term information in dependent types, as defined in [13], that is based on a similar definition given in =-=[19]-=-. A similar function, erasing term-dependencies in the Theory of Generalized Functionality of [21], can also be found in [5]. Definition 2.19 The function ED : Tu → Tu is defined as follows: Iti) On Λ... |

28 | Introduction to Generalised Type Systems - Barendregt - 1991 |

14 |
On the church-rosser property for expressive type systems and its consequences for their metatheoretic study
- Geuvers, Werner
- 1994
(Show Context)
Citation Context ..., through E, of two typed terms can be β-equivalent, while the originals were not (a thorough investigation on the possible alternative definitions of the (Conv) rule on typed systems can be found in =-=[12]-=-). We will show that it is possible to define another erasing function, named E ′ , that gives rise to a second type assignment cube TAS ′ which is isomorphic to the TS-cube (via E ′ ). Remember that ... |

10 |
The Undecidability of Typability in the Lambda-Pi-Calculus
- Dowek
- 1993
(Show Context)
Citation Context ...ncy-free part of the cubes TAS and TAS ′ have been extensively studied in the literature. The only type assignment system with dependent types already defined in the literature is the system of Dowek =-=[10]-=-, which is based on the typed system λP. Strictly speaking, this is not a type assignment system in the usual sense. In [10], there is no formal system to derive judgements; instead, a valid judgement... |

9 | della Rocca. Characterization of Typings in Polymorphic Type Discipline - Giannini, Ronchi - 1988 |

6 |
Towards a Mathematical Analysis of Type Dependence in Coquand–Huet Calculus of Constructions and the Other Systems in Barendregt’s Cube
- Berardi
- 1988
(Show Context)
Citation Context ...died for Barendregt’s cube, and has been clearly established for the plane of the cube without dependencies. However, in the opposite plane, this relation is less clear, as demonstrated by Berardi in =-=[6]-=-. As mentioned above, in this paper, we show an example of a inhabited type in TAS, that cannot be obtained through erasure of an inhabited type in TS. This negative result of course implies that the ... |

6 |
della Rocca. Type inference: some results, some problems
- Giannini, Honsell, et al.
- 1993
(Show Context)
Citation Context ...onchi}@di.unito.it 3 Instytut Informatyki Uniwersytetu Warszawskiego, ul. Banacha 2, 02-097 Warszawa, Polska, urzy@mimuw.edu.pl Abstract We study the cube of type assignment systems, as introduced in =-=[13]-=-, and confront it with Barendregt’s typed λ-cube [4]. The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from te... |

5 | Comparing Cubes
- Bakel, Liquori, et al.
- 1994
(Show Context)
Citation Context ...ignment systems is presented. In that section, we will show that these type assignment systems are isomorphic to the systems in Barendregt’s cube. A preliminary version of this paper was presented in =-=[1]-=-. Notational conventions: In this paper, a term will be either an (un)typed λ-term, a constructor, a kind, or a sort. The symbols M, N, P , Q, ... range over (un)typed λ-terms; φ, ψ, ξ, µ, ... range o... |

2 |
Progress Report on Generalised Functionality
- Seldin
- 1979
(Show Context)
Citation Context ...rm information in dependent types, as defined in [13], that is based on a similar definition given in [19]. A similar function, erasing term-dependencies in the Theory of Generalized Functionality of =-=[21]-=-, can also be found in [5]. Definition 2.19 The function ED : Tu → Tu is defined as follows: Iti) On Λ. Itii) On Cons. Itiii) On Kind. ED (M) = M. ED (α) = α, ED (Πx:φ.ψ) = Πx:ED (φ).ED (ψ), ED (Πα:K.... |

2 |
g-Stratification is Equivalent to f -Stratification
- Ben-Yelles
- 1981
(Show Context)
Citation Context ... types, as defined in [13], that is based on a similar definition given in [19]. A similar function, erasing term-dependencies in the Theory of Generalized Functionality of [21], can also be found in =-=[5]-=-. Definition 2.19 The function ED : Tu → Tu is defined as follows: Iti) On Λ. Itii) On Cons. Itiii) On Kind. ED (M) = M. ED (α) = α, ED (Πx:φ.ψ) = Πx:ED (φ).ED (ψ), ED (Πα:K.ψ) = Πα:ED (K).ED (ψ), ED ... |

1 |
g-Stratification is Equivalent to f-Stratification
- Ben-Yelles
- 1981
(Show Context)
Citation Context ... types, as defined in [13], that is based on a similar definition given in [19]. A similar function, erasing term-dependencies in the Theory of Generalized Functionality of [21], can also be found in =-=[5]-=-. Definition 2.19 The function ED : Tu → Tu is defined as follows: Iti) On Λ. Itii) On Cons. Itiii) On Kind. ED (M) = M. ED (α) = α, ED (Πx:φ.ψ) = Πx:ED (φ).ED (ψ), ED (Πα:K.ψ) = Πα:ED (K).ED (ψ), ED ... |