B. Nordstrom, K. Petersson, and J. Smith. Programming in MartinL of's Type Theory: An Introduction. Oxford University Press, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Q checks the positive branch P under the assumption that the names x and y are the same; we formalize this by substituting y for x in the type environment and the process P . Given that names are the only kind of value, this technique is simpler than the standard one from dependent type theory [NPS90, Bar92] of defining typing judgments with respect to an equivalence relation on values. Honda, Vasconcelos, and Yoshida [HVY00] also use the technique of applying substitutions to environments while type checking. In their study of a distributed # calculus, Hennessy and Riely [HR98] propose an ....

B. Nordstrom, K. Petersson, and J. Smith. Programming in MartinL of's Type Theory: An Introduction. Oxford University Press, 1990.

....[36] which consist of a simple realization of the required behaviour. However, the theoretical basis of algebraic specification is largely in terms of constructions on algebraic models, so it is at the same time much more model oriented than approaches such as those based on type theory (see e.g. [58]) where the emphasis is almost entirely on syntax and formal systems of rules, and semantic models are absent or regarded as of secondary importance. The past 25 years has seen a great deal of research on the theory and practice of algebraic specification. Overviews of this material include ....

Bengt Nordstrom, Kent Petersson, and Jan Smith. Programming in MartinL of's Type Theory: An Introduction. Oxford Univ. Press, 1990.

....# terms as proofs paradigm. Nowadays we have a very good choice of generic proof development environments based on type theory, e.g. NuPRL, Alf, Lego, Coq, Isabelle, which implement respectively Martin Lof Type Theory, the Calculus of (Co)Inductive Constructions, and Higher Order Logic [31, 28, 2, 27]. Following this line of research, our interactive proof assistant for the # calculus, is obtained directly from Coq [2] which is the full fledged computerized proof environment for CC (Co)Ind , developed at INRIA. The crucial step is the specification in CC (Co)Ind of an adequate encoding ....

....interesting set of properties about the operator and the encoding of # calculus into the # calculus, using Sangiorgi s theory of progressions and up to context techniques, in an inductive setting. Formalized metatheoretic reasoning on systems in HOAS has been explored by other authors, e.g. [3, 4, 14, 16, 28]. But apart from [16] none of their approaches, in their current state of development, appears to be general enough to sustain the metatheoric investigations of the # calculus presented in this paper. Current and future work The encoding in CC (Co)Ind of the # calculus presented in this paper ....

B. Nordstrom, K. Petersson, and J. M. Smith. Programming in MartinL of's Type Theory: An Introduction. OUP, 1990.

....notions to provide schematic mechanisms for specification of type theories and their use in practice. The framework presented here, PAL , is a logical framework for specification and implementation of type theories, such as Martin Lof s type theory or UTT. As in Martin Lof s logical framework [NPS90], computational rules can be introduced and are used to give meanings to the declared constants. However, PAL only allows one to talk about the concepts that are intuitively in the object type theories: types and their objects, and families of types and families of objects of types. In ....

.... logical frameworks include: ffl Parametric constants definitions and the associated operations of instantiation (substitution or cut) are more basic and arguably simpler notions mechanisms than that of lambda abstraction as found in other logical frameworks such as Martin Lof s logical framework [NPS90]. A parametrically defined entity represents a family of entities, rather than a functional operation. ffl The user of a proof system based on a lambda free framework does not have to understand the meta level lambda abstraction that can be used to represent concepts such as families of families ....

[Article contains additional citation context not shown here]

B. Nordstrom, K. Petersson, and J. Smith. Programming in MartinL of's Type Theory: An Introduction. Oxford University Press, 1990.

.... terms as proofs paradigm. Nowadays we have a very good choice of generic proof development environments based on type theory, e.g. NuPRL, Alf, Lego, Coq, Isabelle, which implement respectively Martin Lof Type Theory, the Calculus of (Co)Inductive Constructions, and Higher Order Logic [31, 28, 2, 27]. Following this line of research, our interactive proof assistant for the calculus, is obtained directly from Coq [2] which is the full fledged computerized proof environment for CC (Co)Ind , developed at INRIA. The crucial step is the specification in CC (Co)Ind of an adequate encoding ....

....interesting set of properties about the operator and the encoding of calculus into the calculus, using Sangiorgi s theory of progressions and up to context techniques, in an inductive setting. Formalized metatheoretic reasoning on systems in HOAS has been explored by other authors, e.g. [3, 4, 14, 16, 28]. But apart from [16] none of their approaches, in their current state of development, appears to be general enough to sustain the metatheoric investigations of the calculus presented in this paper. Current and future work The encoding in CC (Co)Ind of the calculus presented in this paper is ....

B. Nordstrom, K. Petersson, and J. M. Smith. Programming in MartinL of 's Type Theory: An Introduction. OUP, 1990.

....consist of a simple realization of the required behaviour. Confusingly because the theoretical basis of algebraic specification is largely in terms of constructions on algebraic models it is at the same time much more model oriented than approaches such as those based on type theory (see e.g. [35]) where the emphasis is almost entirely on syntax and formal systems of rules and semantic models are absent or regarded as of secondary importance. The past 25 years has seen a great deal of research on the theory and practice of algebraic specification. Overviews of this material include ....

Bengt Nordstrom, Kent Petersson, and Jan Smith. Programming in MartinL of 's Type Theory: An Introduction. Oxford Univ. Press, 1990.

....: A This type imposes a strong (or full) extensionality which allows us to assert a judgemental equality using the induction principle. However, there is a known problem in using W types to represent inductive types in an intensional type theory (for example, Martin Lof s intensional type theory [NPS90]) the W representations of inductive types do not give the expected induction principle, because there is junk in the representations (see chapter 15 of [NPS90] and explanations in section 2.2) In other words, W types may not be used in an To appear in: Logical Frameworks, G. Huet and G. ....

.... in using W types to represent inductive types in an intensional type theory (for example, Martin Lof s intensional type theory [NPS90] the W representations of inductive types do not give the expected induction principle, because there is junk in the representations (see chapter 15 of [NPS90] and explanations in section 2.2) In other words, W types may not be used in an To appear in: Logical Frameworks, G. Huet and G. Plotkin eds. Cambridge University Press, 1992 1 intensional type theory to represent inductive data types faithfully some extensionality is required. A basic ....

[Article contains additional citation context not shown here]

B. Nordstrom, K. Petersson, and J. Smith. Programming in MartinL of's Type Theory: an introduction. Oxford University Press, 1990.

....all definable terms. For example, in [LMS95, Tiu95, Che96] coercions are almost identical maps. They are characterized formally as terms that become j equal in the untyped calculus after all type information is erased. Data types in dependent type theories such as Martin Lof s type theory [NPS90] and the type theory UTT [Luo94] can in general be considered as inductive in the sense that they consist of their canonical objects. This is rather different from the traditional views when one studies type systems of programming languages and most of the work about subtyping, where objects ....

....kinds. K)K 0 denotes dependent product (x:K)K 0 when x doesn t occur free in K 0 and is sometimes denoted by K K 0 . We consider a type theory T specified in LF and study its extensions with coercive subtyping. Examples of such type theories include Martin Lof s intensional type theory [NPS90], UTT [Luo94] and many others. These theories have in particular the means to define inductive data types. For example, in UTT, given any collection of inductive schemata Theta in a context Gamma, one may declare the constant expressions M[ Theta] Type (inductive type itself) i : Theta i ....

B. Nordstrom, K. Petersson, and J. Smith. Programming in MartinL of's Type Theory: An Introduction. Oxford University Press, 1990.

....of type theories and a definitional mechanism for their use in practice. The framework presented here, PAL , is in the spirit of de Bruijn s PAL for Automath [dB80] PAL is a logical framework for specification and implementation of type theories. As in Martin Lof s logical framework [NPS90], computational rules can be introduced and are used to give meanings to the declared constants. However, PAL only allows one to talk about the concepts that are intuitively in the object type theories: types and their objects, and families of types and families of objects of types. In ....

....GR K79130 and GR M75518. 1 ffl Parametric constants definitions and the associated operations of instantiation (substitution or cut) are more basic and arguably simpler notions mechanisms than that of lambda abstraction as found in other logical frameworks such as Martin Lof s logical framework [NPS90]. A parametrically defined entity represents a family of entities, rather than a functional operation. ffl The user of a proof system based on a lambda free framework does not have to understand the meta level lambda abstraction that can be used to represent concepts such as families of families ....

[Article contains additional citation context not shown here]

B. Nordstrom, K. Petersson, and J. Smith. Programming in MartinL of's Type Theory: An Introduction. Oxford University Press, 1990.

....Appendix A gives a formal description of S2 and defines auxiliary notation and notational conventions used in the paper. Related work. The language S2 borrows the idea of built in phase distinction from HML (see [17] and few inductive types at compile time from type theory (e.g. see [19, 5]) There are also analogies with work on partial evaluation, in particular 2 level lambda calculi for binding time analysis (see [18, 6] None of these calculi make use of dependent types. Also [16] deals with shape checking of array programs, but without going through an intermediate language. ....

....The mixed translation can only cope with shapely operations (see [14] where the shape of the result is determined uniquely by the shape of the arguments. However, in S2 one can give a type to non shapely operations, for instance op: D;D (9x: 1 1:1) 5 Conclusions and further research [19] advocates the use of Martin Lof Type Theory for program construction. This paper advocates the use of Martin Lof Type Theory as part of an intermediate language S2. This avoids two major problems: a programmer does not have to deal with dependent type directly, decidability of type checking in S2 ....

Bengt Nordstrom, Kent Petersson, and Jan M. Smith. Programming in MartinL of's type theory:an introduction. Number 7 in International series of monographs on computer science. Oxford University Press, New York, 1990.

....importance of coherence conditions for the conservativity result is also discussed. 1 Introduction Coercive subtyping, as introduced and studied in [Luo97, Luo98] is a new general approach to subtyping and inheritance for constructive type theories such as Martin Lof s intensional type theory [NPS90] and the type theory UTT [Luo94] Introducing a simple abbreviational mechanism into the logical framework (the meta language used to specify type theories) it provides a uniform framework that generalises the traditional approaches to subtyping that based on the notion of subset or inclusion ....

B. Nordstrom, K. Petersson, and J. Smith. Programming in MartinL of's Type Theory: An Introduction. Oxford University Press, 1990.

....a correct implementation of a specification like Sorting is just to find an object (proof) of the type (proposition) Sorting. Program development corresponds to proof development in type theory. Such a programming methodology is seriously taken by the Goteberg group based Martin Lof s type theory [NPS89] Particularly, they investigated how to extract programs from proofs, called program extraction. 7 In 7 Deductive approaches to program extraction or program synthesis have been considered by many others in different settings. Manna and Waldinger [MW71] considered an approach in classical ....

....thesis) the model in [Luo88a] is also extended to the type universes (using an idea of Hayashi) Burstall [Bur89a] developed an approach to program development, using the extended calculus of constructions and the system LEGO. He uses Sigma types to express specifications in a similar style as [NPS89] but uses Leibniz s equality to model computational equality and gives a categorical explanation of the approach. The formulation and proof of the equality reflection result in this thesis was motivated by Burstall s work on specifications. The proof development system LEGO is developed by ....

B. Nordstrom, K. Petersson and J. Smith, Programming in MartinL of's Type Theory: an introduction, book to appear.

.... terms as proofs paradigm. Nowadays we have a very good choice of generic proof development environments based on type theory, e.g. NuPRL, Alf, Lego, Coq, Isabelle, which implement respectively Martin Lof Type Theory, the Calculus of (Co)Inductive Constructions, and Higher Order Logic [25, 22, 3, 21]. Following this line of research, our interactive proof assistant for the calculus, is obtained directly from Coq [3] which is the full fledged computerized proof environment for CC (Co)Ind , developed at INRIA. The crucial step is the specification in CC (Co)Ind of an adequate encoding ....

B. Nordstrom, K. Petersson, and J. M. Smith. Programming in MartinL of 's Type Theory: An Introduction. OUP, 1990.

....of proofs. They can then be extracted from proofs and be certified to be correct. We focus on the inverse problem : is it possible to reconstruct proof obligations from a program and its specification The framework is the type theory where a proof can be represented as a typed term [Con86, NPS90] and particularly the Calculus of Inductive Constructions [Coq85] A notion of coherence is introduced between a specification and a program containing annotations as in the Hoare sense. This notion is based on the definition of an extraction function called the weak extraction. Such an annotated ....

B. Nordstrom, K. Petersson, and J. M. Smith. Programming in MartinL of's Type Theory : an introduction. Oxford Science Publications, 1990.

....language variables are represented by variables in the meta language, and variables bound in the object language are bound with corresponding scope in the meta language. This is the fundamental idea of higherorder abstract syntax [HHP93, PE88] which goes back to Martin Lof s system of arities [NPS90] Higher order abstract syntax identifies expressions that differ only in the names assigned to their bound variables through ff conversion in the meta language. It also supports substitution through fi reduction in the meta language: Since the representation expresses the scope of all ....

B. Nordstrom, K. Petersson, and J.M. Smith. Programming in MartinL of's Type Theory: An Introduction. Oxford University Press, 1990.

No context found.

B. Nordstrom, K. Petersson, and J. Smith. Programming in MartinL of's Type Theory: An Introduction. Oxford University Press, 1990.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC