R. L. Constable and S. F. Smith. Partial objects in constructive type theory. In Proceedings of Symposium on Logic in Computer Science, pages 183--193, Ithaca, New York, June 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by them, and must be handled by endowing the typing system with some facilities for discussing computational properties of programs. For example, Constable et al. adopted this approach in their pioneering works to incorporate recursive definitions and partial objects into constructive type theory [10, 11]. However, in this paper, we will pursue another approach such that types themselves can express convergence of programs. Towards the approximation modality. Suppose that we have a recursive program f defined by: f F (f) and want to show that f satisfies a certain specification S. Since the ....

R. L. Constable and S. F. Smith. Partial objects in constructive type theory. In Proceedings of the 2nd IEEE Symposium on Logic in Computer Science, pages 183--193. IEEE Computer Society Press, 1987.

....to obtain the p calculus, i.e. the calculus for reasoning about partial computations (or equivalently, about partial functions) In fact, the p calculus (like the calculus) amounts to a particular c theory . A type theoretic approach to partial functions and computations is attempted in [CS87, CS88] by introducing a new type constructor A, whose intuitive meaning is the set of computations of type A. However, Constable and Smith do not adequately capture the general axioms for (partial) computations as we (and [Ros86] do, since they lack a general notion of model and rely only on domain ....

R.L. Constable and S.F. Smith. Partial objects in constructive type theory. In 2nd LICS Conf. IEEE, 1987.

.... Objects in the Calculus of Constructions Philippe Audebaud LaBRI UFR Math ematiques Informatique 351, Crs de la Lib eration 33405 Talence France e mail : audebaud geocub.greco prog.fr Abstract Our purpose is to provide a typed framework for working with non terminating computations. The basic system is the Calculus of Constructions. It is extended using an original idea proposed by ....

.... Objects in the Calculus of Constructions Philippe Audebaud LaBRI UFR Math ematiques Informatique 351, Crs de la Lib eration 33405 Talence France e mail : audebaud geocub.greco prog.fr Abstract Our purpose is to provide a typed framework for working with non terminating computations. The basic system is the Calculus of Constructions. It is extended using an original idea proposed by R.Constable and S.F.Smith and and ....

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Constable R.L. and Smith S.F. (1987) Partial objects in constructive type theory. In 2nd Conf. on Logic in Comp. Science. IEEE, 1987.

....and Mendler [CM85] introduce the type of partial functions as a new type constructor. A function f is a partial function from A to B, denoted A B, if its domain can be computed as a predicate over A. In order to apply f to an element a in A, we need a proof that a is in the domain of f . In [CS87] Constable and Smith develop a partial type theory. Corresponding to every type T of the underlying total theory there is a type T , which might contain diverging terms. They de ne a termination predicate t in T which asserts that t is a term that terminates and they also present some induction ....

....to every type T of the underlying total theory there is a type T , which might contain diverging terms. They de ne a termination predicate t in T which asserts that t is a term that terminates and they also present some induction principles applicable to partial types. Inspired by the work in [CS87] Audebaud [Aud91] introduces x points to the Calculus of Constructions [CH88] obtaining a conservative extension of it where the desired properties still hold. Acknowledgement. We want to thank Herman Geuvers for carefully reading and commenting on a previous version of this paper. ....

R. L. Constable and S. F. Smith. Partial Objects in Constructive Type Theory. In Logic in Computer Science, Ithaca, New York, pages 183-193, June 1987.

....the system since the book was published are documented in a reference manual distributed with the system. Information on how to obtain the system can be obtained from the author. A number of research papers related to Nuprl have been published. Some extensions to the type theory are described in [1, 2, 3, 6, 10, 19, 20, 24, 26]. Numerous applications of Nuprl have been made; these include [5, 8, 9, 11, 16, 17, 18, 21, 22] In order to illustrate explicit and implicit programming in Nuprl, and to show how the system works, we focus on a simple case study in which we verify Boyer and Moore s majority algorithm. We ....

R. Constable and S. Smith. Partial objects in constructive type theory. Proceedings of the Second Annual Symposium on Logic in Computer Science, pages 183--193, March 1987. (Cornell TR 87-822).

....view that category theory comes, logically, before the calculus led us to consider a categorical semantics of computations rst, rather than to modify directly the rules of conversion to get a correct calculus. A type theoretic approach to partial functions and computations is attempted in [1] by introducing a type constructor A, whose intuitive meaning is the set of computations of type A. Our categorical semantics is based on a similar idea. Constable and Smith, however, do not adequately capture the general axioms for computations (as we do) since they lack a general notion of ....

R.L. Constable and S.F. Smith. Partial objects in constructive type theory. In 2nd LICS Conf. IEEE, 1987.

....1 . Also a programming language with such a type system is often kept pure because it is at least unwieldy, if not impossible, to combine many realistic programming features with type systems similar to that of NuPrl. This is attested in the works such as allowing unlimited recursion [CS87] introducing recursive types [Men87] and incorporating e ects [HMST95] exceptions [Nak94] 1 We point out that NuPrl is primarily a logic rather than a programming language and it is therefore natural to perform interactive theorem proving during type checking. and input output. This is ....

Robert L. Constable and Scott Fraser Smith. Partial objects in constructive type theory. In Proceedings of Symposium on Logic in Computer Science, pages 183-193. Ithaca, New York, June 1987.

....to obtain the p calculus, i.e. the calculus for reasoning about partial computations (or equivalently, about partial functions) In fact, the p calculus (like the calculus) amounts to a particular c theory . A type theoretic approach to partial functions and computations is attempted in [CS87, CS88] by introducing a new type constructor A, whose intuitive meaning is the set of computations of type A. However, Constable and Smith do not adequately capture the general axioms for (partial) computations as we (and [Ros86] do, since they lack a general notion of model and rely only on domain ....

R.L. Constable and S.F. Smith. Partial objects in constructive type theory. In 2nd LICS Conf. IEEE, 1987.

....[Ros86, Mog86] Our work generalises the categorical account of partiality to other notions of computations, indeed partial cartesian closed categories turn out to be a special case of c models (see De nition 3. 9) A type theoretic approach to partial functions and computations is proposed in [CS87, CS88] by introducing a type constructor A, whose intuitive meaning is the set of computations of type A. Our categorical semantics is based on a similar idea. Constable and Smith, however, do not adequately capture the general axioms for computations (as we do) since their notion of model, based ....

R.L. Constable and S.F. Smith. Partial objects in constructive type theory. In 2nd LICS Conf. IEEE, 1987.

....languages can be found in Plotkin s influential study of PCF [56] and in Mitchell s chapter mentioned above, to name two sources. The metaphor of computations (versus values ) implicit in Plotkin s v calculus [55] was made explicit by Constable and Smith in their partial object type theory [12, 63] and by Moggi in his monadic treatment of computational effects [50] 11 Chapter 2 Recursive Functions 2.1 Introduction In this section we study the extension L rec of the language of Chapter 1 with a means of defining general recursive functions. 2.2 Syntax The syntax of L rec ....

Robert L. Constable and Scott Fraser Smith. Partial objects in constructive type theory. In Second Symposium on Logic in Computer Science, pages 183--193, June 1987.

....of our method, we focus on the part of the proof where our method is used. 4.1 Fixpoints Fixpoint operators enhance otherwise terminating typed calculi with the possibility to define non terminating expressions. The motivations for studying this calculus originate from partial type theory [1, 9] and dependently typed programming languages [2] Theorem 3. S fix j= SN(fi) Proof. By Corollary 2, it is enough to show that the specification is sound. To this end, note that fiOE is confluent and hence = fiOE preserves sorts. Moreover S fix has the Subject Reduction property so we may ....

R.L. Constable and S.F. Smith. Partial objects in constructive type theory. In Proceedings of LICS'87, pages 183--193. IEEE Computer Society Press, 1987.

....more work) for almost every type, including function types (to which the fix rule of K is restricted) It is clear, then, that fix cannot be used to define new members of the basic types. How then can recursive functions be typed The solution is to add a new type constructor for partial types [24, 25, 92]. For any type T , the partial type T is a supertype of T that contains all the elements of T and also all divergent terms. A total type is one that contains only convergent terms. The induction principles on T are different than those on T , so we can safely type fix with the rule: 3 H e ....

....it is sometimes interesting to reason about functions that are known not to terminate for all inputs. In order to be able to assign types to partial functions and nonconvergent objects, we add a new type constructor: ffl Partial type (T ) The partial type T (due to Constable and Smith [24, 25, 92]) is like a lifted version of T : It contains all members of T , plus all nonconvergent terms. Terms are equal in T if they have the same convergence behavior, and if they are equal in T when they converge. That is, t = t 0 2 T if and only if t# , t 0 # and t# ) t = t 0 2 T . With this type ....

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Robert L. Constable and Scott Fraser Smith. Partial objects in constructive type theory. In Second IEEE Symposium on Logic in Computer Science, pages 183--193, Ithaca, New York, June 1987.

....with partial functions. The lack of partial functions seriously limited the scope of those theorem provers, because it made them unable to reason about programs in real programming languages where recursion does not always necessarily terminate. This problem was addressed by Constable and Smith [8], who introduced into their type theory the partial type T , which is like a lifted version of T . The type T contains all members of T as well as all divergent terms. Using the partial type, partial functions from A to B may be given the type A B. That is, when applied to an argument in A, ....

Robert L. Constable and Scott Fraser Smith. Partial objects in constructive type theory. In Second IEEE Symposium on Logic in Computer Science, pages 183--193, Ithaca, New York, June 1987.

....to make an intensional interpretation of his type theory inconsistent. This change enabled the reduction in kinds of judgement. ffl Nuprl s type theory has several extra types including the set type [Con85a] the quotient type [Con85a] recursive types [CM85] and partial function 18 types [CS87] Allen has given a semantics for Nuprl s type theory without the recursive or partial types [All87a, All87b] This semantics takes the form of a second order positive inductive definition that is both classical set theoretically valid and acceptable to most constructivist mathematicians. The ....

Robert L. Constable and Scott F. Smith. Partial objects in constructive type theory. In Proceedings of the Second Annual Symposium on Logic in Computer Science. IEEE, 1987.

....is an empty type; similar inconsistencies may be derived (with a bit more work) for almost every type. It is clear, then, that fix cannot be used to define new members of the basic types. How then can recursive functions be typed The solution is to add a new type constructor for partial types (Constable and Smith, 1987; Smith, 1989; Crary, 1998c) For any type T , the partial type T is a supertype of T that contains all the elements of T and also all divergent terms. A total type is one that contains only convergent terms. The induction principles on T (Smith, 1989; Constable and Crary, 1997) are different ....

Constable, R. L. and Smith, S. F. (1987) Partial objects in constructive type theory.

....with partial functions. The lack of partial functions seriously limited the scope of those theorem provers, because it made them unable to reason about programs in real programming languages where recursion does not always necessarily terminate. This problem was addressed by Constable and Smith [7], who introduced into their type theory the partial type T , which is like a lifted version of T . The type T contains all members of T as well as all divergent terms. Using the partial type, partial functions from A to B may be given the type A B. That is, when applied to an argument in A, such ....

R. L. Constable and S. F. Smith. Partial objects in constructive type theory. In Second IEEE Symposium on Logic in Computer Science, pages 183--193, Ithaca, New York, June 1987.

....more work) for almost every type, including function types (to which the fix rule of K is restricted) It is clear, then, that fix cannot be used to define new members of the basic types. How then can recursive functions be typed The solution is to add a new type constructor for partial types [10, 11, 51, 9, 17]. For any type T , the partial type T is a supertype of T that contains all the elements of T and also all divergent terms. A total type is one that contains only convergent terms. The induction principles on T [51, 9] are different than those on T , so we can safely type fix with the rule: 2 ....

Robert L. Constable and Scott Fraser Smith. Partial objects in constructive type theory. In Second IEEE Symposium on Logic in Computer Science, pages 183--193, Ithaca, New York, June 1987.

....to be derived from more basic properties of computations. 1.1 Predacessors to this work This work grew from a desire to extend the Nuprl type theory developed at Cornell University by Constable and others [CAB 86] to encompass nontotal computations. Initial results were published in [CS87] Nuprl is a descendant of the Intuitionistic Type Theory of Martin Lof[Mar73, Mar82, Mar84] and in particular of [Mar82, Mar84] which we refer to as ETT ( Extensional type theory ) while his 1973 theory is ITT ( Intensional Type Theory ) The Nuprl theory is built upon ETT in that it 2 has ....

....reasoning about fixed points exist. Crole and Pitts [CP90] fill this gap by 44 adding a fixed point object to the monad, from which fixed points may be typed. A full programming logic has yet to be built using this methodology. Acknowledgements The original work this paper is built on [CS87] was carried out jointly with my thesis advisor Robert Constable, who also suggested the problem. Stuart Allen and Doug Howe also gave many useful comments and criticisms. ....

R. L. Constable and S. F. Smith. Partial objects in constructive type theory. In Proceedings of the Second Annual Symposium on Logic in Computer Science. IEEE, 1987.

....computations which might not terminate added. This is accomplished by adding a type constructor: given a type A, the type A ( A bar ) is the type of computations over A. A naive interpretation of A would be A [ f g, where is a never terminating computation. In a rich enough partial type theory [6, 17] it is possible to directly extract fixed point computations from proofs. The object of this paper is to show how to carry out extraction of fixedpoint programs in a type theoretic setting. We first review basic concepts of type theory, then outline the extensions that allow fixed point programs ....

....This principle also extends the extraction paradigm to allow fixed point programs to be extracted from proofs. One complication of this principle is the necessity of showing types are admissible, because the principle is not valid for all types. The original presentations of partial type theory [6, 15] had too restrictive an admissibility condition: dependent products could not appear at all, meaning existential quantifiers could not be used. For instance, the example given at the end of this paper would not be expressible in the earlier versions of partial type theory. Recently the collection ....

R. L. Constable and S. F. Smith. Partial objects in constructive type theory. In Proceedings of the Second Annual Symposium on Logic in Computer Science. IEEE, 1987.

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R. L. Constable and S. F. Smith. Partial objects in constructive type theory. In Proceedings of the Second Annual Symposium on Logic in Computer Science. IEEE, 1987.

....This standard theory accepts Church s Thesis, and in Roger s book[22] it is explicitly used to develop the theory. We want to challenge the assumption that this is the only acceptable view. We have discovered through our attempts to provide a formal foundational theory for computer science[4, 5, 23, 24] that there is an interesting, perhaps compelling, alternative to the standard theory. The goal of this paper is to explain this alternative. One of the requirements for a theory of the kind we imagine is that it be adequate to explain all of the basic notions of computation and, where ....

Robert L. Constable and Scott Fraser Smith, Partial Objects in Constructive Type Theory. In Symposium on Logic in Computer Science, 1987, pages 183--193.

....most general laws of substitution; and, two, the need to treat functions intensionally (as algorithms) in order to state the most useful rules of program reasoning and to express complexity. There have been several attempts at this reconciliation, some of which are theoretically quite interesting [11, 42, 3, 29, 30]. Nevertheless, none of these are very practical, nor have any been implemented. We set as our goal producing a practical and adequate account of partial functions, computational induction and computational complexity in the class of constructive type theories such as those of Martin Lof or ....

....induction and complexity. The resulting extension to the type theory is lightweight but still quite expressive. 1.2 Salient Points The first technical step is to extend the type theory with an extensional notion of partial functions. We take the approach introduced by Constable and Smith [11] of adding partial types. An object t is said to be in the partial type T exactly when its termination implies it is in T . The partial functions from A to B are denoted by A B. However, we must extend the theory of Constable and Smith to deal with equality, which they did not consider. The ....

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Robert L. Constable and Scott Fraser Smith. Partial objects in constructive type theory. In Second IEEE Symposium of Logic in Computer Science, pages 183--193, Ithaca, New York, June 1987.

No context found.

R. L. Constable and S. F. Smith. Partial objects in constructive type theory. In Proceedings of Symposium on Logic in Computer Science, pages 183--193, Ithaca, New York, June 1987.

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R.Constable and S.Fraser Smith, Partial Objects in Constructive Type Theory, in: Proceedings of the Second LICS Symposium, IEEE, 1987.

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