## Polymorphic Intersection Type Assignment for Rewite Systems with Intersection and beta-rule (Extended Abstract) (2000)

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Venue: | IN TYPES’99. LNCS |

Citations: | 4 - 2 self |

### BibTeX

@INPROCEEDINGS{Bakel00polymorphicintersection,

author = {Steffen van Bakel and Franco Barbanera and Maribel Fernández},

title = {Polymorphic Intersection Type Assignment for Rewite Systems with Intersection and beta-rule (Extended Abstract)},

booktitle = {IN TYPES’99. LNCS},

year = {2000},

pages = {41--60},

publisher = {Springer}

}

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### Abstract

We define two type assignment systems for first-order rewriting extended with application,-abstraction, and-reduction (TRS). The types used in these systems are a combination of (-free) intersection and polymorphic types. The first system is the general one, for which we prove a subject reduction theorem and show that all typeable terms are strongly normalisable. The second is a decidable subsystem of the first, by restricting types to Rank 2. For this system we define, using an extended notion of unification, a notion of principal type, and show that type assignment is decidable.

### Citations

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Citation Context ...an look at such calculi also as extensions of the Curryfied Term Rewriting Systems (CuTRS) considered in [6], by adding -abstraction and a -reduction rule. We assume the reader to be familiar with LC =-=[9]-=- and refer to [23, 14] for rewrite systems. We consider terms built from a set of variables, a set of function symbols, application ( ) and -abstraction, modulo -conversion, as usual. A context is a t... |

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Citation Context ...s of Rank 2, but only an algorithm that finds the most general unifying chain for demanded and provided type. It is defined as a natural extension of Robinson’s well-known unification algorithm unify =-=[27]-=-. Definition 23. Unification of Curry types (extended with non-unifiable variables and type constants), unify C C , is defined by: 2 2sunify unify Id unify Id unify unify unify if does not occur in an... |

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Citation Context ...alculi also as extensions of the Curryfied Term Rewriting Systems (CuTRS) considered in [6], by adding -abstraction and a -reduction rule. We assume the reader to be familiar with LC [9] and refer to =-=[23, 14]-=- for rewrite systems. We consider terms built from a set of variables, a set of function symbols, application ( ) and -abstraction, modulo -conversion, as usual. A context is a term with a hole, and i... |

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Citation Context ...alculi also as extensions of the Curryfied Term Rewriting Systems (CuTRS) considered in [6], by adding -abstraction and a -reduction rule. We assume the reader to be familiar with LC [9] and refer to =-=[23, 14]-=- for rewrite systems. We consider terms built from a set of variables, a set of function symbols, application ( ) and -abstraction, modulo -conversion, as usual. A context is a term with a hole, and i... |

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Citation Context ...m, but lacks principal types, and type inference is undecidable in general (although it is decidable for some subsystems, in particular if we consider types of rank 2 [21]). Intersection type systems =-=[10]-=- are somewhere in the middle with respect to polymorphism (they provide less polymorphism than System F but more than ML) and principal types can be constructed for typeable terms. But type assignment... |

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Citation Context ...ML has the three properties above, i.e., it is a polymorphic type inference system with principal types, but its polymorphism is limited (some programs that arise naturally cannot be typed). System F =-=[18]-=- provides a much more general notion of polymorphism, but lacks principal types, and type inference is undecidable in general (although it is decidable for some subsystems, in particular if we conside... |

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Citation Context ...me the rules to satisfy the general schema, a typeable TRS without critical pairs in R is locally confluent on typeable terms (we omit also this proof for lack of space), and hence, by Newman’s Lemma =-=[25]-=-, we can deduce confluence from strong normalization and local confluence. ,s4 Restriction to Rank 2 In this section, we will present a decidable restriction of the type system, based on types of rank... |

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Citation Context ...ormalization). If the rewrite rules satisfy the general schema, any typeable term is strongly normalisable. The proof, which we omit for lack of space, can be carried on by using Tait-Girard’s method =-=[17]-=- and the techniques devised in [20] in order to cope with some of the difficulties that arise because of the presence of algebraic rewriting. It is possible to show that if we assume the rules to sati... |

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Citation Context ...cause any term typeable in the full intersection type discipline can be given a strict type and vice-versa. We assume the reader to be familiar with intersection type assignment systems, and refer to =-=[10, 1, 3]-=- for more details. 2.1 Types Definition 3 (Types). Let be a set of type-variables, where is the set of free variables, and the set of bound variables. Let be a set of sorts. s, the set of polymorphic ... |

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Citation Context ...-free fragment of this system corresponds to the system studied in [6]. If sorts are not taken into account, the -free LC-fragment of our type assignment system corresponds to the system presented in =-=[3]-=-, and the intersection-free LC-fragment to System F [18]. For LC, a type assignment system that combines intersection types with polymorphic types has been defined in [19] and its principal type prope... |

60 | Concurrent Clean
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Citation Context ...some work has also been done for TRS, more precisely, for Curryfied TRS (CuTRS) [6] which are first-order TRS with application, that correspond to the TRS that underlie the programming language Clean =-=[26]-=-. The interactions between LC and TRS in the context of type assignment were first studied in [5], where CuTRS extended with -abstraction and -reduction were defined, together with a notion of interse... |

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Citation Context ...s expanded new type variables are generated, and other subtypes might be affected (e.g. the expansion of in might affect also : intuitively, each renamed copy of will have an associated copy of ; see =-=[28]-=- for a detailed explanation). Ground types are not affected by expansions since all renamed copies coincide (and ). Before applying an operation of expansion, we need then to compute the set of types ... |

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52 |
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Citation Context ...plines provide an environment in which rewrite rules and -reduction can be combined without loss of their useful properties. This is supported by a number of results for a broad range of type systems =-=[11, 12, 20, 7, 8, 5]-=-. In this paper we study the combination of LC and TRS as a basis for the design of a programming language. The type systems à la Curry, also called type inference or type assignment systems, are the ... |

51 | Principality and decidable type inference for finite-rank intersection types - Kfoury, Wells - 1999 |

35 | Principal type schemes for the strict type assignment system
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Citation Context ...ansion, which replaces (sub)types by the intersection of a number of renamed copies of that type, and lifting, which replaces basis and type by a smaller basis and a larger type, in the sense of (see =-=[2]-=- for details). These operations are standard in type systems with intersection types, we will extend them to take into account the presence of universal quantifiers. The operation of lifting can be us... |

33 |
Executable higher-order algebraic specification languages
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Citation Context ...plines provide an environment in which rewrite rules and -reduction can be combined without loss of their useful properties. This is supported by a number of results for a broad range of type systems =-=[11, 12, 20, 7, 8, 5]-=-. In this paper we study the combination of LC and TRS as a basis for the design of a programming language. The type systems à la Curry, also called type inference or type assignment systems, are the ... |

25 | H.: Modularity of strong normalization and confluence in the algebraic-λ-cube
- Barbanera, Fernández, et al.
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(Show Context)
Citation Context ...plines provide an environment in which rewrite rules and -reduction can be combined without loss of their useful properties. This is supported by a number of results for a broad range of type systems =-=[11, 12, 20, 7, 8, 5]-=-. In this paper we study the combination of LC and TRS as a basis for the design of a programming language. The type systems à la Curry, also called type inference or type assignment systems, are the ... |

25 | Modular termination of term rewriting systems revisited, in: Recent Trends in Data Type Specification - Fernández, Jouannaud - 1995 |

23 | Normalisation Results for Typeable Rewrite Systems
- Bakel, Fernández
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(Show Context)
Citation Context ... (no type annotations are required). Type assignment disciplines have been widely studied in the context of the LC, and some work has also been done for TRS, more precisely, for Curryfied TRS (CuTRS) =-=[6]-=- which are first-order TRS with application, that correspond to the TRS that underlie the programming language Clean [26]. The interactions between LC and TRS in the context of type assignment were fi... |

22 | Rank 2 Intersection Type Assignment in Term Rewriting Systems
- Bakel
- 1996
(Show Context)
Citation Context ... System F but more than ML) and principal types can be constructed for typeable terms. But type assignment is again undecidable; decidability is recovered if we restrict ourselves to the rank 2 types =-=[4]-=-. In view of the above results, two questions arise naturally: – Is the rank 2 combination of System F and the Intersection System also decidable? – Does it have principal types for all typeable terms... |

20 | Adding algebraic rewriting to the untyped lambda calculus - Dougherty - 1991 |

17 |
Polymorphic rewriting conserves algebraic confluence
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Citation Context |

17 | Partial Intersection Type Assignment in Applicative Term Rewriting Systems - Bakel - 1993 |

14 |
Filter Models with Polymorphic Types
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Citation Context ...e Intersection System also decidable? – Does it have principal types for all typeable terms? A system for the LC that combines intersection types and System F with principal types has been studied in =-=[19, 24]-=-. In this paper we extend that system to a combination of LC and CuTRS. In other words, we extend the type assignment system of [5] further, adding ‘ ’ as an extra type-constructor (i.e. explicit poly... |

12 |
Type reconstruction in finite-rank fragments of the second-order λcalculus
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Citation Context ...ore general notion of polymorphism, but lacks principal types, and type inference is undecidable in general (although it is decidable for some subsystems, in particular if we consider types of rank 2 =-=[21]-=-). Intersection type systems [10] are somewhere in the middle with respect to polymorphism (they provide less polymorphism than System F but more than ML) and principal types can be constructed for ty... |

7 | Type assignment and termination of interaction nets
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Citation Context ...ella Rocca and Venneri [28] for full intersection types without sorts and universal quantifiers, and adapt it to our system. The extension to deal with types containing sorts has already been done in =-=[15]-=-, here we add quantifiers. For an expansion and a triple , an associated set of types is computed, and the types modified by the expansion are those that ‘end’ with a type in this set. The notion of l... |

7 | Some computational properties of intersection types - Bucciarelli, Lorenzis, et al. - 1996 |

6 |
Intersection type assignment systems with higherorder algebraic rewriting, Theoretical Computer Science 170
- Barbanera, Fernández
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Citation Context |

4 |
Principal Typing in a ∀∩-Discipline
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Citation Context ...e Intersection System also decidable? – Does it have principal types for all typeable terms? A system for the LC that combines intersection types and System F with principal types has been studied in =-=[19, 24]-=-. In this paper we extend that system to a combination of LC and CuTRS. In other words, we extend the type assignment system of [5] further, adding ‘ ’ as an extra type-constructor (i.e. explicit poly... |

4 | de Vries. Syntactic definitions of undefined: on defining the undefined - Ariola, Kennaway, et al. - 1994 |

2 |
A Direct Algorithm for Type Inference
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Citation Context ... the case also in the Rank 2 intersection system of [4], but not in the Rank 2 polymorphic system of [21]. For the latter, a type inference algorithm of the same complexity of that of ML was given in =-=[22]-=-, where the problems that occur due to the lack of principal types are discussed in detail. This paper is organised as follows: In Section 1 we define TRS with application, -abstraction and -reduction... |

2 |
Embedding a Second-Order Type System into an Intersection Type System
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(Show Context)
Citation Context ... will present a decidable restriction of the type system, based on types of rank 2. Although the rank 2 intersection system and the rank 2 polymorphic system for LC type exactly the same set of terms =-=[29]-=-, their combination results in a system with more expressive power: polymorphism can be expressed directly (using the universal quantifier) and, moreover, every typeable term has a principal type, as ... |

2 | Principal Typing in a 8"-Discipline - Margaria, Zacchi - 1995 |

1 |
Rewrite Systems with Abstraction and � -rule: Types, Approximants and Normalization
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Citation Context |

1 | Rewrite Systems with Abstraction and firule: Types, Approximants and Normalization - Bakel, Barbanera, et al. - 1996 |

1 | Principal Typing in a - Margaria, Zacchi - 1995 |