M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... finite sub trees of the (possibly infinite) normal form, based on the notion of Omega normal forms of Huet and Levy [15] see also [17] 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 [18, 3] But, perhaps surprisingly, this has never been studied for more general notions of rewriting, such as Term Rewriting Systems (TRS) 12, 16] Within the framework of orthogonal first order TRS, a term like model and an appropriate semantics ....

....has Partially supported by NATO Collaborative Research Grant CRG 970285 Extended Rewriting and Types . 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 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 that is closed under fi equality, and interpreting terms by ....

M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.

....It is well known that in the study of normalization of reduction systems, the notion of types plays an important role. Many of the now existing type assignment systems for functional languages are based on (extensions of) the Curry system for LC [10, 11] The intersection type discipline (see [8, 7, 1]) is an extension of Curry s system, in that, essentially, terms and term variables are allowed to have more than one type (using the type constructor ) By introducing also the type constant a system is obtained that is closed under fi equality, and interpreting terms by their assignable ....

M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.

....system has drawbacks. It is, in this system, for example not possible to assign a type to the term (x:xx) and although the lambda terms (cd:d) and ( xyz:xz(yz) ab:a) are fi equal, the principal type schemes for these terms are different. The intersection type discipline as presented in [9] (see also [6] and [1] is an extension of Curry s system that does not have these drawbacks. The extension to Curry s system is essentially that terms and term variables are allowed to have more than one type. Intersection types are constructed by adding, next to the type constructor , the ....

....and the type constant . By introducing this extension a system is obtained that is closed under fi equality: if B M :oe and M = fi N , then B N :oe. The first type assignment system with intersection types was presented in [7] the CDV system with intersection types and is introduced in [9] and in [23] The best known intersection type assignment system is the BCD system, as presented in [6] that is an extension of the CDV system: there it is strengthened further by introducing a partial order relation on types as well as adding the type assignment rule ( and a more general ....

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M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.

.... term such that s e : ae; A where ae and A are proper. Then e is normalizable. Proof According to the theorem 4, we have (A) e : ae) and by the lemma 5, A) and (ae) are respectively a proper basis and a proper type. By the normalization theorem of the system D (cf. [7]) e is normalizable.2 Lemma 6 Let e be a term in normal form. There exist a proper type ae and a proper constraint environment A such that s e : ae; A. Proof by induction on the structure of a normal form. ffl If e = x then let ae be a type which has no occurrence of [ We can derive ....

....in intersection type discipline and is the most often used and studied. According to J. L. Krivine in [13] we use here his notations) there are essentially three intersection type systems. Types are formed with the constructors and for system D [3] a universal type for the system D [7], and a partial order on types for the system D [1] The theoretical study of system D led to results about I calculus and to the following characterization: a term is strongly normalizable if and only if it is typable in system D [22, 15] In system D a term is normalizable if and only if it ....

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Mario Coppo, Mariangiola Dezani-Ciancaglini, and Betti Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45-- 58, 1981.

....system has drawbacks. It is, in this system, for example not possible to assign a type to the term (x:xx) and although the lambda terms (cd:d) and ( xyz:xz(yz) ab:a) are fi equal, the principal type schemes for these terms are different. The intersection type discipline as presented in [9] (see also [6] and [1] is an extension of Curry s system that does not have these drawbacks. The extension to Curry s system is essentially that terms and term variables are allowed to have more than one type. Intersection types are constructed by adding, next to the type constructor , the ....

....and the type constant . By introducing this extension a system is obtained that is closed under fi equality: if B M :oe and M = fi N , then B N :oe. The first type assignment system with intersection types was presented in [7] the CDV system with intersection types and is introduced in [9] and in [23] The best known intersection type assignment system is the BCD system, as presented in [6] that is an extension of the CDV system: there it is strengthened further by introducing a partial order relation on types as well as adding the type assignment rule ( and a more general ....

[Article contains additional citation context not shown here]

M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.

.... cost : real Gamma Gamma Gamma Gamma At this point, we have summarized the type structure of Forsythe (then called Idealized Algol ) as it appeared in about 1981 [5] Since then, the language has been generalized, and considerably simplified, by the introduction of intersection types [13, 14, 15]. At the outset, a caution must be sounded that this use of the word intersection can be misleading. If one thinks of types as standing for sets, than the intersection of two types need not stand for the intersection of the two corresponding sets. In earlier papers, we used the term ....

....that a procedure has the typing one expects, rather than just some typing that makes the overall program type correct. But a more stringent answer is that it has been proven that there is no algorithm that can typecheck an arbitrary implicitly typed program in the intersection type discipline [13, 14, 15]. Thus the Forsythe implementation requires some explicit type information to be provided. The exact nature of this requirement is described in Appendix C. Next there are constructions for denoting objects and selecting their fields: ffl Object Construction p : j p) ffl ....

Coppo, M., Dezani-Ciancaglini, M., and Venneri, B. Functional Characters of Solvable Terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, vol. 27 (1981), pp. 45--58.

....types ( 25] and bounded polymorphism ( 7] The system presented here is an implicitly typed version. More information about syntactic, semantic, and pragmatic details can be found in [21] Information about intersection types or bounded polymorphism can be found in [4, 22, 25, 26] or [6, 7, 8, 9, 10, 12, 20]. The set of F types is defined by: oe : j ns j oe 1 oe 2 j 8 oe 1 : oe 2 j oe 1 oe 2 Here is the binary intersection operator, ns denotes the null intersection, and 8 oe 1 : oe 2 is for bounded polymorphism. We denote 8 ns: oe 2 as 8 : oe 2 . A type context is a sequence Theta = 1 oe 1 ....

M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.

....has some drawbacks. In Curry s system it is, for example, not possible to assign a type to the term (x:xx) moreover, although the lambda terms (cd:d) and ( xyz:xz(yz) ab:a) are fi equal, the principal type schemes for these terms are different. The Intersection Type Discipline as presented in [5] (a more enhanced system was presented in [4] is an extension of Curry s system that does not have these drawbacks. The extension being made consists mainly of allowing for term variables (and terms) to have more than one type. Intersection types are constructed by adding, next to the type ....

....under fi equality: if B M :oe and M = fi N , then B N :oe. Because of this power, in the intersection system (and even in the system that does not contain ) type assignment is undecidable. The type assignment system presented in [4] the BCD system) is based on the system as presented in [5]; it defines the set of intersection types T in a more general way, and is strengthened further by introducing a partial order relation on types as well as adding the type assignment rule ( and a more general form of the rules concerning intersection. The rule ( as well as the more general ....

Coppo M., M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.

....once in M [x : N ] and these occurrences were typed in the derivation for B M [x : N ] oe with different types. A first introduction of a type assignment system with intersection types can be found in [Coppo Dezani Ciancaglini 80] a system with intersection types and is introduced in [Coppo et al. 81] and in [Sall e 78] In [Barendregt et al. 83] the system as presented in [Coppo et al. 81] was strengthened further by introducing a partial order relation on types as well as adding the type assignment rule ( and a more general form of the rules concerning intersection. The rule ( is ....

....with different types. A first introduction of a type assignment system with intersection types can be found in [Coppo Dezani Ciancaglini 80] a system with intersection types and is introduced in [Coppo et al. 81] and in [Sall e 78] In [Barendregt et al. 83] the system as presented in [Coppo et al. 81] was strengthened further by introducing a partial order relation on types as well as adding the type assignment rule ( and a more general form of the rules concerning intersection. The rule ( is introduced mainly to prove completeness of type assignment. This is achieved by showing that ....

[Article contains additional citation context not shown here]

Coppo M., M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.

....described in this report builds on previous studies of two well known calculi: the intersection type discipline and the polymorphic calculus. 2. 1 Intersection Types Intersection types in the pure calculus have been extensively studied by researchers at the university of Turin and elsewhere [14, 13, 15, 2, 39, 26, 10]. More recently, Reynolds has showed how intersection types can be used as the basis for the type system of a practical programming language, called Forsythe [38] The core Forsythe type system can be viewed as consisting of the following components: 1. a collection of primitive types and, for ....

M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.

....application of the fixed point operator to the method vector of an object is performed at messagesending time rather than within the new function. More speculatively, it may be possible to extend our approach to include multiple inheritance by working in a variant of F with intersection types [24]. A secondorder fragment of this calculus is studied in [35] Acknowledgements We are grateful for conversations with Dave Berry, Kim Bruce, Luca Cardelli, Giuseppe Castagna, William Cook, Giorgio Ghelli, Bob Harper, Robin Milner, John Mitchell, Kevin Mitchell, Didier R emy, Nick Rothwell, and ....

M. Coppo, M. Dezani-Ciancaglini, and B. Venneri. Functional characters of solvable terms. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 27:45--58, 1981.

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