## Normalisation, Approximation, and Semantics for Combinator Systems (2003)

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Venue: | Theoretical Computer Science |

Citations: | 12 - 11 self |

### BibTeX

@ARTICLE{Bakel03normalisation,approximation,,

author = {Steffen Van Bakel and Maribel Fernández},

title = {Normalisation, Approximation, and Semantics for Combinator Systems},

journal = {Theoretical Computer Science},

year = {2003},

pages = {975--1019}

}

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### Abstract

This paper studies normalization of typeable terms and the relation between approximation semantics and filter models for Combinator Systems. It presents notions of approximants for terms, intersection type assignment, and reduction on type derivations; the last will be proved to be strongly normalizable. With this result, it is shown that, for every typeable term, there exists an approximant with the same type, and a characterization of the normalization behaviour of terms using their assignable types is given. Then the two semantics are defined and compared, and it is shown that the approximants semantics is fully abstract but the filter semantics is not.

### Citations

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Citation Context ... -normal forms of Huet and Lévy [16] (see also [18]). The relation between the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) =-=[6]-=- (see [8, 7, 1, 3]), where it has been proved that they coincide [19, 3]. But, perhaps surprisingly, this has never been studied for more general notions of rewriting, such as Term Rewriting Systems (... |

756 | Rewrite systems
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Citation Context ... [8, 7, 1, 3]), where it has been proved that they coincide [19, 3]. But, perhaps surprisingly, this has never been studied for more general notions of rewriting, such as Term Rewriting Systems (TRS) =-=[12, 17]-=-. Within the framework of orthogonal first-order TRS, a term-like model and an appropriate semantics are defined in [21], interpreting terms by the set of their approximants. For these TRS it is also ... |

569 | Term Rewriting Systems
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Citation Context ... [8, 7, 1, 3]), where it has been proved that they coincide [19, 3]. But, perhaps surprisingly, this has never been studied for more general notions of rewriting, such as Term Rewriting Systems (TRS) =-=[12, 17]-=-. Within the framework of orthogonal first-order TRS, a term-like model and an appropriate semantics are defined in [21], interpreting terms by the set of their approximants. For these TRS it is also ... |

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Citation Context ...submitted to Elsevier Preprint 29 April 2002The relation between the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) [6] (see =-=[8,7,1,3]-=-), where it has been proved that they coincide [19,3]. But, perhaps surprisingly, this has never been studied for more general notions of rewriting, such as Term Rewriting Systems (TRS) [12,17]. Withi... |

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Citation Context ...Rewriting Systems (CuTRS, first-order TRS extended with application). This type system is inspired by the Intersection Type Discipline defined in [8] (see also [7, 1]), an extension of Curry’s system =-=[10, 11]-=- in that, essentially, terms are allowed to have more than one type (using the type constructor ‘ ’). By introducing also the type constant ‘ ’ a type system for LC is obtained Partially supported by ... |

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Citation Context ...tion needed to show that any typeable term in a typeable CS has an approximant of the same type (Theorem 8.2). In [3], this approximation result has been obtained for LC, by a computability technique =-=[20]-=-. A particular problem to solve in this paper is that the approach of [3] cannot be automatically translated to a technique to use in CS, because of the absence of abstraction in CS. In order to prove... |

153 | Semantic domains
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Citation Context ...ing 8.1. , that is of use in the proof of Lemma Definition 7.6 On T , the partial mapping T T T is defined by: The last alternative defines the join on applications in a more general way than that of =-=[15]-=-, which would state that , since it is not always sure if a join of two arbitrary terms exists. However, we will use our more general definition only on terms that are compatible, so the conflict is o... |

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Citation Context ...forms of Huet and Lévy [16] (see also [18]). The relation between the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) [6] (see =-=[8, 7, 1, 3]-=-), where it has been proved that they coincide [19, 3]. But, perhaps surprisingly, this has never been studied for more general notions of rewriting, such as Term Rewriting Systems (TRS) [12, 17]. Wit... |

102 |
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Citation Context ... their normalization properties are wellstudied, there is still no thorough semantic analysis of TRS. As we have already mentioned, there exists some work in this direction, either supported by types =-=[14]-=- or not [21], but, for example, the relation between these models has not been studied. This paper is a first step towards filling that void, by studying two approaches to semantics for CS, the approx... |

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Citation Context ... derivation reduction) If D , then SN D . Proof: If D , then, taking R such that R = , by Theorem 6.4, Comp D . Then, by Lemma 6.2(i), SN D . 7 Approximants Now we will develop, essentially following =-=[22]-=- (see also [6]), a notion of approximant for combinator terms. This will be done by introducing a special symbol into the definition of terms. The general idea is that a term directly approximates a t... |

84 |
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Citation Context ...e set of their approximants, and their interrelation. Approximants are defined as rooted finite sub-trees of the (possibly infinite) normal form, based on the notion of -normal forms of Huet and Lévy =-=[16]-=- (see also [18]). The relation between the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) [6] (see [8, 7, 1, 3]), where it has... |

61 | Intersection Type Assignment Systems
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Citation Context ...forms of Huet and Lévy [16] (see also [18]). The relation between the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) [6] (see =-=[8, 7, 1, 3]-=-), where it has been proved that they coincide [19, 3]. But, perhaps surprisingly, this has never been studied for more general notions of rewriting, such as Term Rewriting Systems (TRS) [12, 17]. Wit... |

58 |
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Citation Context ...etween the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) [6] (see [8, 7, 1, 3]), where it has been proved that they coincide =-=[19, 3]-=-. But, perhaps surprisingly, this has never been studied for more general notions of rewriting, such as Term Rewriting Systems (TRS) [12, 17]. Within the framework of orthogonal first-order TRS, a ter... |

54 |
Functional characters of solvable terms. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
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(Show Context)
Citation Context ...forms of Huet and Lévy [16] (see also [18]). The relation between the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) [6] (see =-=[8, 7, 1, 3]-=-), where it has been proved that they coincide [19, 3]. But, perhaps surprisingly, this has never been studied for more general notions of rewriting, such as Term Rewriting Systems (TRS) [12, 17]. Wit... |

41 |
Grundlagen der kombinatorischen Logik
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(Show Context)
Citation Context ...s definitions and insight. However, note that the normalization properties of LC do not translate directly to CS, since the mappings between LC and Combinatory Logic (a particular CS defined by Curry =-=[9]-=-) do not preserve normal forms or reductions (see Example 1.9). Although TRS are very popular in language design and their normalization properties are wellstudied, there is still no thorough semantic... |

40 |
Sequentiality in orthogonal term rewriting systems
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(Show Context)
Citation Context ...approximants, and their interrelation. Approximants are defined as rooted finite sub-trees of the (possibly infinite) normal form, based on the notion of -normal forms of Huet and Lévy [16] (see also =-=[18]-=-). The relation between the filter semantics and the approximation semantics has been studied extensively in the setting of the Lambda Calculus (LC) [6] (see [8, 7, 1, 3]), where it has been proved th... |

35 | Principal type schemes for the strict type assignment system
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(Show Context)
Citation Context ...s ( ) ( ) if ( ) if ( ) ( ) ( ) ( ) if ( ) ( ) ( ) ( ) where ( ) ii) If and are type-substitutions, then so is , where . iii) . iv) . For type-substitutions, the following properties hold: Lemma 2.5 (=-=[2]-=-) Let S be a type-substitution. If , then , and if , then . Our operation of expansion is similar to the one defined in [19] for the full intersection system, we just need to make some minor changes t... |

24 | M.: Normalization results for typeable rewrite systems
- Bakel, Fernández
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(Show Context)
Citation Context ...account for polymorphism, or higher order functions, which are standard in functional programming languages. A more general and expressive type system, using intersection types, has been developed in =-=[5]-=- for Curryfied Term Rewriting Systems (CuTRS, first-order TRS extended with application). This type system is inspired by the Intersection Type Discipline defined in [8] (see also [7, 1]), an extensio... |

20 |
Combinatory logic, volume 1
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(Show Context)
Citation Context ...g Systems (CuTRS, first-order TRS extended with application). This type system is inspired by the Intersection Type Discipline for LC, defined in [8] (see also [7, 1]): an extension of Curry’s system =-=[10, 11]-=- in that, essentially, terms are allowed to have more than one type (using the type constructor ‘∩’). By introducing also the type constant ‘ω’ a type system for LC is obtained that is closed under β-... |

16 |
Intersection types for combinatory logic
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(Show Context)
Citation Context ...the length of the reduction sequence increases significantly. Only for particular CS it is also possible to define an interpretation of LC, [[ ]] ; the standard example is that of CL (for details see =-=[11, 6, 13]-=-). One important property of these two translations is that [[ ]] CL for all . There exists no converse of this property; moreover, the mapping does not preserve normal forms or reductions: Example 1.... |

9 | Approximation and Normalization Results for Typeable Term Rewriting Systems
- Bakel, Fernández
- 1996
(Show Context)
Citation Context ...rewriting systems (in particular, the LC and TRS), where it also helps to obtain easier proofs of the characterisation of normalisation properties of typeable terms (for TRS the proof was sketched in =-=[4]-=-). In the future we hope to be able to extend the semantic study presented in this paper to the more general TRS studied in [4].Technical Report 2000/10 Department of Computing, Imperial College 32 A... |

6 |
Full Abstraction and Limiting Completeness in Equational Languages
- Thatte
- 1989
(Show Context)
Citation Context ...for more general notions of rewriting, such as Term Rewriting Systems (TRS) [12, 17]. Within the framework of orthogonal first-order TRS, a term-like model and an appropriate semantics are defined in =-=[21]-=-, interpreting terms by the set of their approximants. For these TRS it is also possible to define a semantics where types are interpreted as multi-sorted algebras [12]. Although these types are enoug... |