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14
Normalization Results for Typeable Rewrite Systems
, 1997
"... In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and !. Three operations on types  substitution, expansion, and lifting  are used to define type assignment, and are proved to be ..."
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Cited by 30 (26 self)
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In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and !. Three operations on types  substitution, expansion, and lifting  are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the typeconstant ! is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) headnormal form, and that terms whose type does not contain ! are normalizable.
Rank 2 Intersection Type Assignment in Term Rewriting Systems
 Fundamenta Informaticae
, 1996
"... A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the ..."
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Cited by 22 (14 self)
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A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the types of function symbols. Using a modified unification procedure, for each term the principal pair (of basis and type) will be defined in the following sense: from these all admissible pairs can be generated by chains of operations on pairs, consisting of the operations substitution, copying, and weakening. In general, given an arbitrary typeable CuTRS, the subject reduction property does not hold. Using the principal type for the lefthand side of a rewrite rule, a sufficient and decidable condition will be formulated that typeable rewrite rules should satisfy in order to obtain this property.
Interaction Nets and Term Rewriting Systems
, 1998
"... Term rewriting systems provide a framework in which it is possible to specify and program in a traditional syntax (oriented equations). Interaction nets, on the other hand, provide a graphical syntax for the same purpose, but can be regarded as being closer to an implementation since the reductio ..."
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Cited by 12 (8 self)
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Term rewriting systems provide a framework in which it is possible to specify and program in a traditional syntax (oriented equations). Interaction nets, on the other hand, provide a graphical syntax for the same purpose, but can be regarded as being closer to an implementation since the reduction process is local and asynchronous, and all the operations are made explicit, including discarding and copying of data. Our aim is to bridge the gap between the above formalisms by showing how to understand interaction nets in a term rewriting framework. This allows us to transfer results from one paradigm to the other, deriving syntactical properties of interaction nets from the (wellstudied) properties of term rewriting systems; in particular concerning termination and modularity. Keywords: term rewriting, interaction nets, termination, modularity. 1 Introduction Term rewriting systems provide a general framework for specifying and reasoning about computation. They can be regarde...
Head)Normalization of Typeable Rewrite Systems
 Proceedings of RTA '95. 6th International Conference on Rewriting Techniques and Applications
, 1995
"... Abstract. In this paper we study normalization properties of rewrite systems that are typeable using intersection types with and with sorts. We prove two normalization properties of typeable systems. On one hand, for all systems that satisfy a variant of the JouannaudOkada Recursion Scheme, every t ..."
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Cited by 12 (12 self)
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Abstract. In this paper we study normalization properties of rewrite systems that are typeable using intersection types with and with sorts. We prove two normalization properties of typeable systems. On one hand, for all systems that satisfy a variant of the JouannaudOkada Recursion Scheme, every term typeable with a type that is not is head normalizable. On the other hand, nonCurryfied terms that are typeable with a type that does not contain, are normalizable.
Approximation and Normalization Results for Typeable Term Rewriting Systems
 Proceedings of HOA ’95. Second International Workshop on Higher Order Algebra, Logic and Term Rewriting
, 1996
"... We consider an intersection type assignment system for term rewriting systems extended with application, and define a notion of (finite) approximation on terms. We then prove that for typeable rewrite systems satisfying a general scheme for recursive definitions, every typeable term has an approxima ..."
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Cited by 11 (11 self)
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We consider an intersection type assignment system for term rewriting systems extended with application, and define a notion of (finite) approximation on terms. We then prove that for typeable rewrite systems satisfying a general scheme for recursive definitions, every typeable term has an approximant of the same type. This approximation result, and the proof technique developed to obtain it, allow us to deduce in a direct way a headnormalization, a normalization, and a strong normalization theorem, for different classes of typeable terms. 1
Type Assignment and Termination of Interaction Nets
"... Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semisi ..."
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Cited by 9 (4 self)
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Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semisimple interaction nets as a programming language, and present a type assignment system using intersection types. First we show that interactions preserve types (i.e. the system enjoys subject reduction), and we compare this type assignment system with the intersection systems for calculus and term rewriting systems. Then we define a recursion scheme that ensures termination of all interaction sequences. By relaxing the scheme and using the type assignment system, we derive another sufficient condition for termination of interaction nets. Finally, we show that although the type system based on general intersection types is not decidable, its restriction to rank 2 types is, and we give an algo...
On Introducing Higher Order Functions In Abel
, 1998
"... We discuss how the 1'st order specification and programming language ABEL could be extended with higher order functions. Several issues arise, related to subtyping, parameterization, strictness of generators and defined functions, and to the choice between lambda expressions and currying. The p ..."
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Cited by 4 (1 self)
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We discuss how the 1'st order specification and programming language ABEL could be extended with higher order functions. Several issues arise, related to subtyping, parameterization, strictness of generators and defined functions, and to the choice between lambda expressions and currying. The paper can be regarded as an exercise in language design: how to introduce higher order functions under the restrictions enforced by (1'st order) ABEL. A technical result is a soundness proof for covariant subtype replacement, useful when implementing data types under volume constraints imposed by computer hardware.
Rewrite Systems with Abstraction and βrule:Types, Approximants and Normalization
, 1996
"... In this paper we define and study intersection type assignment systems for firstorder rewriting extended withapplication, *abstraction, and fireduction (TRS+fi). One of the main results presented is that, using a suitablenotion of approximation of terms, any typeable term of a TRS+fi that satisf ..."
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In this paper we define and study intersection type assignment systems for firstorder rewriting extended withapplication, *abstraction, and fireduction (TRS+fi). One of the main results presented is that, using a suitablenotion of approximation of terms, any typeable term of a TRS+fi that satisfies a general scheme for recursivedefinitions has an approximant of the same type. From this result we deduce, for different classes of typeable terms, a headnormalization and a normalization theorem.
AND
"... In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and. Three operations on types – substitution, expansion, and lifting – are used to define type assignment, and are proved to be sou ..."
Abstract
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In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and. Three operations on types – substitution, expansion, and lifting – are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the typeconstant is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) headnormal form, and that terms whose type does not contain are normalizable. 1
AND
"... In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and ω. Three operations on types – substitution, expansion, and lifting – are used to define type assignment, and are proved to be s ..."
Abstract
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In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and ω. Three operations on types – substitution, expansion, and lifting – are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the typeconstant ω is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) headnormal form, and that terms whose type does not contain ω are normalizable. 1