Luca Cardelli. A Polymorphic A-calculus with Type:Type. Technical Report DECS-10, Digital Systems Research Center, May 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....polymorphism and dependent types. A number of approaches have been proposed in this respect, some of them are: ffl Girard s Reynolds s second order typed calculus [Girard] Reynolds 74] ffl early Martin Lof s type theory [Martin Lof 73] ffl recent Martin Lof s type theory [Martin Lof 82] ffl Cardelli s type system [Cardelli] ffl Nuprl [Constable et al. Most authors do not care whether or not their types are sets in the classical sense of the term. This has contributed to a widespread belief that classical set theories are useless as a basis for theoretical computer science; cf. for instance ....

....types. A number of approaches have been proposed in this respect, some of them are: ffl Girard s Reynolds s second order typed calculus [Girard] Reynolds 74] ffl early Martin Lof s type theory [Martin Lof 73] ffl recent Martin Lof s type theory [Martin Lof 82] ffl Cardelli s type system [Cardelli] ffl Nuprl [Constable et al. Most authors do not care whether or not their types are sets in the classical sense of the term. This has contributed to a widespread belief that classical set theories are useless as a basis for theoretical computer science; cf. for instance the discussion whether ....

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Luca Cardelli. "A Polymorphic -calculus with Type:Type". Digital, System Research Center, Palo Alto, California, Report #10.

....the type T type serves as the description of all other types; this is known as the type:type property. The issue of type:type is controversial, particularly in the area of programming languages. Fortunately, some functional language specifications where the type:type property holds have emerged [Car86] From the behavioral perspective, a type only defines the semantics for the behaviors of its instances. To ensure encapsulation, the behavioral model only publicizes the interface of a type while keeping the implementations private. Behaviors are either explicitly defined on a particular type ....

L. Cardelli. A Polymorphic -calculus with Type:Type. Research Report 10, DEC Systems Research Center, 130 Lytton Ave. Palo Alto, California 94301, May 1986.

....collections which containing objects of different types have a single type describing the common behaviors of all objects in the result. The single type approach is advocated by several type theories including Martin Lof type theory [ML82, BCMS89] and those based on the typed lambda calculus [Car86] A model must supply a mechanism for removing objects from the system. The TIGUKAT model allows many references to an object. Therefore, the removal of an object (within a particular scope) consists of severing the link between the reference and the object. This process does not necessarily ....

....the type T type serves as the description of all other types; this is known as the type:type property. The issue of type:type is controversial, particularly in the area of programming languages. Fortunately, some functional language specifications where the type:type property holds have emerged [Car86] These are likely candidates to assist in the development of a programming language for the model and in expanding the semantic descriptions of behaviors. Recall from Section 2.4.2 that behaviors are either explicitly defined by a particular type or are inherited from a supertype. Behaviors that ....

L. Cardelli. A Polymorphic -calculus with Type:Type. Research Report 10, DEC Systems Research Center, May 1986.

....of a type of all types was based on the following ideas. First, according to Russell s doctrine of types 2 Burstall and Lampson proposed such an idea of a type of all types in designing the programming language Pebble [BLam84] with the motivation for modular typeful programming. Also see [Card86] for a further development of this idea. INTRODUCTION 3 [Rus03] the range of significance of a propositional function forms a type, i.e. the class of propositions forms a type; secondly, quantification over propositions and predicates is allowed; finally, propositions and types are ....

L. Cardelli, `A Polymorphic -calculus with Type:Type', manuscript.

.... same compact code by a staged version of the same termrewriting system (in direct style) did not seem possible [TS97] Open Problems: While TDPE is concise, it is not conceptually simple: The TDPE translations use powerful language constructs such as type dependence and multi level j expansions, [Car86, Coq96, DMP95, NN92, GJ95, TS97] the combination of which is not yet a well explored area. Open theoretical problems include: ffl Polymorphism, inductive types, and polyvariant specialization [Dan96b] One particular polyvariance [JGS93] problem is that abstracting primitives forces them to be monovariant. While progress has ....

L. Cardelli. A polymorphic -calculus with Type:Type, 1986.

....of constructions [8] which uses propositions as types but not the Curry Howard isomorphism. Because a necessary condition for the Curry Howard isomorphism is the convergence of all programs since proofs have to terminate, there is no general recursion available in systems using this isomorphism [4]. This seems to be too strong a restriction for a computer algebra environment. 3 Signatures A signature is a sequence of declarations where identifiers are associated to types. Because the type of an entry in a signature can depend on other preceding entries, signatures are syntactic sugar for ....

L. Cardelli. A Polymorphic -calculus with Type:Type. Technical Report 10, DEC Systems Research Center, 1986.

....all type objects is T type and it is accessible in the same manner as any object. Thus types, in addition to serving as descriptions of objects, are objects themselves and the type T type serves as a description for all other types, including itself. This is known as the type:type property [Car86] in programming languages. The state of a type object consists of a structural specification of its instances (a template) references to the encapsulated behaviors it defines, references to its subtypes and supertypes, and a reference to its associated class. Two relationships on types have been ....

L. Cardelli. A polymorphic -calculus with Type:Type. Research Report 10, DEC Systems Research Center, May 1986.

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Luca Cardelli. A Polymorphic A-calculus with Type:Type. Technical Report DECS-10, Digital Systems Research Center, May 1986.

....and to type theories such as Constable s V3. 1 Introduction A number of formulations of intuitionistic type theory have been considered as a basis for studying machine assisted formal proof development, and as a theoretical foundation for the study of programming languages (see, for example, [16, 46, 34, 36, 37, 9, 10, 11, 14, 2, 4, 19], to name but a few. One such system, the Calculus of Constructions (CC) was introduced by Coquand and Huet as a comprehensive basis for the formalization of constructive mathematics. 11, 14] CC may be viewed as the calculus associated, via the propositions as types principle [24] with ....

.... is by now widely recognized [36, 10] that the appropriate type theoretic representation of mathematical structures is as elements of strong sum types 1 introduced by Martin Lof [35, 36, 37] and Howard [24] Strong sums have also been used to model modularity constructs in programming languages [33, 41, 2, 4]. Unfortunately, strong sums are, in a sense, incompatible with impredicativity [12, 23, 41] As a result, it is necessary to extend the calculus with a level of types, and to postulate the closure of this additional level under the formation of strong sums. Mathematical structures are then ....

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Luca Cardelli. A polymorphic -calculus with Type:Type. Technical report, DEC SRC, 1986.

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L. Cardelli. "A Polymorphic -calculus with Type:Type." Private communication (1986).

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