Douglas J. Howe. Semantic foundations for embedding HOL in NuPRL. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85--101. SpringerVerlag, Berlin, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....world wide using different provers. The bigger these libraries become the more obvious it is that translating formal results from one system into another may be less time consuming than re proving almost identical theorems from scratch. One of the first works in this area has been done by Howe in [8] and [7] He developed semantical foundations and semiautomated procedure for importing Mathematics from HOL [6] into NuPRL [4] Later, in a joint work with Felty [5] he used this translation to conduct proofs in a hybrid HOL NuPRL system. This paper presents an attempt to develop a general ....

D.J Howe. Semantics foundation for embedding HOL in Nuprl. In M. Wirsing and A. Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85--101, Berlin, 1996. Springer-Verlag.

....a useful tool to have on its own. See Section 7.1 for more on extensional and intensional quotient types. Where B is the type of booleans and tt ( true ) and ff ( false ) are its two member s. At first, we considered adding the nd operation similar to arab in [21] and to the approach used in [17]. The idea was to have nd t; t2 which can be either t or t2 nondeterministically. Then we were going to say that the expression that contains nd operators is well formed iff its meaning does not depend on choosing which of nd s arguments to use. The problem with such an approach is that we need ....

Douglas J. Howe. Semantic foundations for embedding HOL in NuPRL. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85 101. SpringerVerlag, Berlin, 1996.

....explicit nondeterminicity rst came up as a way to be able to express the elimination rules for quotient types in a more natural way, but it seems that it is also a useful tool to have on its own. At rst, we considered adding the ndfg operation similar to amb in [18] and to approach used in [14]. The idea was to have ndft 1 ; t 2 g which can be either t 1 or t 2 nondeterministically. Then we were going to say that the expression that contains ndfg operators is well formed i its meaning does not depend on choosing which of ndfg s arguments to use. Unfortunately, the problem with such an ....

Douglas J. Howe. Semantic foundations for embedding HOL in NuPRL. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85-101. SpringerVerlag, Berlin, 1996.

....The HOL theorem proving system [29] has a rich library of theories that can save a lot of effort by not having to specify from scratch many commonly encountered theories. Potentially, this is a very useful resource not only for HOL, but for other theorem proving systems based on other logics. Howe [32] defined a map of logics mapping the HOL logic into the logic of Nuprl [17] and implemented such a mapping to make possible the translation from HOL theories to Nuprl theories. In this way, the practical goal of relating both systems and making the HOL libraries available to Nuprl was achieved. ....

D. Howe. Semantic foundations for embedding HOL in Nuprl. Manuscript; Bell Labs, 1996.

....world wide using different provers. The bigger these libraries become the more obvious it is that translating formal results from one system into another may be less time consuming than re proving almost identical theorems from scratch. One of the first works in this area has been done by Howe in [8] and [7] He developed semantical foundations and semiautomated procedure for importing Mathematics from HOL [6] into NuPRL [4] Later, in joint work with Felty [5] he used this translation to conduct proofs in a hybrid HOL NuPRL system. This paper presents an attempt to develop a general ....

D.J Howe. Semantics foundation for embedding HOL in Nuprl. In M. Wirsing and A. Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85--101, Berlin, 1996. Springer-Verlag.

....concrete situation. ffl A preliminary design according to our general methodology of a translation from HOL to NuPrl. The benefit of this is making available to NuPrl a large library of HOL theories and specifications. Although a translation from HOL to NuPrl has already been proposed by Howe [78] who stresses the advantages of making in this way avalilable to NuPrl a large body of HOL specifications it is not clear at present how such a translation can be seen as a particular instance of our proposed general methodology for translations between formal systems. This therefore will be ....

D.J. Howe. Semantic foundations for embedding HOL in Nuprl. Manuscript; Bell Labs, 1996.

....rules and semantics for the new function type, ffl a correspondence between abstract data types and theories, and between objects and proofs. There are several reasons to develop an account of objects within the type theory. The type theory already has several semantics, including set theoretic [16], PER models [2, 21] denotational models [25] and others. By developing our account within the type theory, we provide a solid mathematical foundation for objects. In addition, we can draw on mathematical techniques, especially those of abstract algebra, to contribute toward the understanding ....

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In AMAST '96, 1996.

....through grant N00014 92 J 1764, from the National Science Foundation through grant CCR 9244739, from DARPA through grant 93 11 271, and from AASERT through grant N00014 95 1 0985. ticular, the Nuprl type theory that we use in this paper has multiple mathematical models, notably set theoretic [17], PER models [3, 20] denotational models [23] and others. In addition, our interpretation is extendable to foundational formal objects that encode proofs [15, 11] a critical step toward our goal of providing a modular and object oriented logical framework [12] The object interpretation is ....

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In AMAST '96, 1996.

....should be constructive or not. We do not make any assertion about this, but note, that it is simple to extend the hierarchy of representation languages by providing a logical morphism to intuitiunistic set theory that basically introduces oracles for the law of the excluded middle; see e.g. [How96]. In Figure 1, we have marked the intuitionistic logical systems with an i and the oracle morphisms with dotted lines. Draft 16 1 1999 Axiomatic set theories like ZF only come with a basic type fl of set and with the logical relation constant 2 for elementhood. The axiomatic method is used ....

Douglas Howe. Semantic foundations for embedding hol in nuprl. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodolgy and Software Technology, volume 1101 of LNCS, pages 85--101. Springer Verlag, 1996.

....a formal semantic connection between classical and constructive type theories but instead focus on the practical application of these ideas. Some steps have been taken towards such a connection. Howe has developed a modified semantics for Nuprl to allow HOL theorems to be used within Nuprl proofs [6]. Coq s Program tactic, described in [11, 12] provides automatic assistance for program verification. The approach described there can be viewed as the inverse of the program extraction process and is related to the discussion below in section 3.1. Another comparison of a constructive proof with ....

D. J. Howe. Semantic foundations for embedding HOL in Nuprl. In Fifth International Conference on Algebraic Methodology and Software Technology AMAST '96, July 1996. To appear as a volume in LNCS.

....This semantics is also useful even if one ultimately desires a semantics in some framework other than type theory. Martin Lof type theory is closely tied to a structured operational semantics and has denotational models in many frameworks including partial equivalence relations [5, 40] set theory [54] and domain theory [87, 81, 80] Thus, foundational type theory may be used as a semantic intermediate language. 3.1 A Computational Denotational Semantics My main motivation for a type theoretic semantics has been to draw formal connections between programming languages and type theory, ....

....is extracted a witness b, possibly containing free occurrences of the variable x, and that witness is used to construct the extract, x:b, of the original sequent. In fact, since the variable x is only used 3 In some alternative semantics for type theory (such as Howe s set theoretic semantics [54]) equality in a type may be collapsed to actual object equality, trivially making the membership and functionality interpretations the same. Extending the semantics of Howe to account for all the devices in this type theory is an open problem, but one that shows some promise of feasibility. in ....

[Article contains additional citation context not shown here]

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. Technical report, Bell Labs, 1996.

....algorithms and procedures, but not in a systematic way. The first thing we show is how to treat computation systematically and foundationally with minor changes in their text. Our presentation then enables a person to imagine that all of the mathematics is classical, as Howe s work illustrates [19]. It also allows the interpretation of recursive mathematics that all functions are given by Turing machines or Lisp programs. It also allows an Intuitionistic interpretation. One way to describe this style is to relate it to the work of Bishop [3] who showed that real, complex, and abstract ....

....to formalize Chapters 1 9 with our four person team in about eighteen months. The collaboration methods we have learned would extend to larger teams. It would be especially interesting to collaborate with other theorem proving systems as Howe and his colleagues are doing with HOL and Nuprl [19, 18]. Much of a classical treatment of languages can easily be re interpreted constructively. It would be especially fruitful to collaborate with other constructive provers such as Alf, Coq and Lego or with Isabelle which has formalized Martin L of type theory. Although these provers are based on ....

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of LNCS, pages 85--101. Springer-Verlag, Berlin, 1996.

....a solution to the proliferation of logics; they also propose object oriented theories. Guinchiglia et al. 6] are also exploring a general architecture where provers can be combined in a plug and play manner. As this task proceeds, we need a semantic basis for relating theories, as Howe [16] provides in his semantics for HOL [7] and Nuprl. 6 Conclusion We have described the Nuprl Light framework, which extends the results of generic theorem provers by adding formal, first class theories. First class theories enable a new style of reasoning where multiple type theories and theorem ....

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In AMAST '96, 1996.

....useful even if one ultimately desires a semantics in some framework other than type theory. Martin Lof type theory is closely tied to a structured operational semantics and has denotational models in many frameworks including partial equivalence relations (Allen, 1987; Harper, 1992) set theory (Howe, 1996) and domain theory (Rezus, 1985; Palmgren and Stoltenberg Hansen, 1989) Thus, foundational type theory may be used as a semantic intermediate language. The paper is organized as follows: Section 2 presents the paper s object language, K . This object language is a small programming calculus, ....

Howe, D. J. (1996) Semantic foundations for embedding HOL in Nuprl. Technical report, Bell Labs.

....This semantics is also useful even if one ultimately desires a semantics in some framework other than type theory. Martin Lof type theory is closely tied to a structured operational semantics and has denotational models in many frameworks including partial equivalence relations [1, 23] set theory [33] and domain theory [49, 46, 45] Thus, foundational type theory may be used as a semantic intermediate language. The paper is organized as follows: Section 2 presents the paper s object language, K . This object language is a small programming calculus, not a practical programming language, ....

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. Technical report, Bell Labs, 1996.

....weak that non classical results in recursive mathematics, such as that every function from R to R is continuous, are not provable. Howe has given a set theoretic interpretation of Nuprl s type theory that shows that every theorem provable in Nuprl can be read as a theorem of classical mathematics [19]. This interpretation includes all non computational set theoretic functions in the denotation of function types. Having both classical and recursive interpretations makes Nuprl a suitable tool for formalizing constructive mathematics in the style advocated by Bishop [2] 1.5 Electronic Access to ....

....to formalize Chapters 1 9 with our four person team in about eighteen months. The collaboration methods we have learned would extend to larger teams. It would be especially interesting to collaborate with other theorem proving systems as Howe and his colleagues are doing with HOL and Nuprl [19, 18]. Much of a classical treatment of languages can easily be re interpreted constructively. It would be especially fruitful to collaborate with teams using other constructive provers such as Alf, Coq, Lego, or Isabelle with its Martin Lof type theory object logic. Although these provers are based on ....

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of LNCS, pages 85--101. Springer-Verlag, Berlin, 1996.

....discrete mathematics. The work is part of a research program to explore the foundations of computational mathematics. An attempt to formalize the undecidability results of Hopcroft and Ullman s Chapter 3 in classical set theory (say as in Mizar [46] or in classical type theory (say HOL [34, 43, 42] or PVS [60] would encounter severe difficulties. It could not remain close to the style of the book. It could not capture the computational intuitions that motivate and permeate the whole account. But in constructive set theory the Hopcroft and Ullman account can be followed essentially ....

....presuppose the very subject they are introducing. So they seem to have in mind a foundation that includes Set Theory Effective Procedures. Alas, there was no careful account of such a theory for them to rely on in 1969 as there are now: Alf, Coq [4, 28, 51] IZF [6, 32, 33, 54, 67] Nuprl [30, 3, 45, 43]. Hopcroft and Ullman also intended that there be a way to provide data to algorithms explicitly. 4 2 This theory is also called type theory to stress its difference from classical set theory and its similarity to the notion of type in programming languages. Martin L of also refers to it as a ....

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85--101. Springer-Verlag, Berlin, 1996.

....part is a more detailed discussion of the problems we encountered and how we solved them. The hardest problem was one of semantics. To accommodate the embedding, we extended the Nuprl semantics to incorporate an operator similar to HOL s select. This extension is the subject of another paper [5], and will only be described briefly here. The other problems, though numerous, were not so difficult to deal with. This section also discusses some problems which we have not solved. The work described in this paper is somewhat in progress . The basic connection between the systems has been ....

....Note that non emptiness of a type is taken to represent truth of the corresponding proposition. T fl 8 ff 2 fl: P [ff=x] fl ff ff 0 2 fl of minimum rank such that fl ff 0 6= x2T: P ff 0 Showing that a sensible semantics can be built based on the ideas described above is the subject of [5]. In this semantics, the Nuprl types are exactly the programs that evaluate to some set fl. The members of such a type are the members of fl together with all terms approximated by some ff 2 fl. 3.2 Logic In HOL logic is given a Tarskian semantics. A proposition is either true or false. In ....

D. J. Howe. Semantics foundations for embedding hol in nuprl. Proceedings of AMAST'96, 1996. (to appear).

....the SCI cache coherence protocol, there is a 2 page summary of a prototype of this annotation scheme. 2 Nuprl: Syntax and Semantics This section gives a description of Nuprl, focusing on the syntax and semantics of the type theory. The semantics we use is the set theoretic one we developed in [6]. The main advantage of this semantics for our purposes here is that it has a much simpler account of hypothetical judgments than other semantics, in particular the one given in [2] A hypothetical judgment in our context is essentially a typing judgment, where a term is asserted to have a type ....

....x; the syntax does not specify that A is a type to be associated with x. At the level of syntax, there is no distinction between types and terms. There are only terms, and types will be those terms that can be proved to be types using Nuprl s inference rules. In the semantics for Nuprl given in [6], we augment the set the of terms with an infinite collection of constants representing objects from a universe of set theoretic objects, and extend the operational semantics to deal with these new objects. This allows us to give set theoretic meaning to types and members of types in the type ....

[Article contains additional citation context not shown here]

D. J. Howe. Semantics foundations for embedding HOL in Nuprl. In M. Wirsing and M. Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85--101, Berlin, 1996. Springer-Verlag.

....with a rich set of constructors. Because of the constructivity, programs can be extracted from proofs. Logic is encoded via the propositions as types principle, whereby a proposition is identified with the type of data that provides evidence for the proposition s truth. The version of Nuprl we use [8] also supports classical reasoning, which can be used in any part of a proof that does not affect the extracted program. Formal mathematics in Nuprl is organized in a single library, which is broken into files simulating a theory structure. Library objects can be definitions, display forms, ....

....protocol language is its lack of types. This language was developed inside Nuprl to support reasoning about particular protocols, and not for metareasoning about programs. The precise form of inductive definition mechanism implemented in standard Nuprl [4] is not valid in the classical extension [8] used here. It is not too difficult to adapt it, but we have not done so yet and hence have simply axiomatized the two inductive definitions we needed. We believe that our results can be extended to deal with temporal properties in the same way as programs. One difficulty is dealing with binding ....

D. J. Howe. Semantics foundations for embedding HOL in Nuprl. In Algebraic Methodology and Software Technology, pages 85--101. Springer-Verlag Lecture Notes in Computer Science, 1996.

....individual constants, axioms and theorems. An HOL theory is imported into Nuprl via interpretation, whereby particular Nuprl objects are associated with the theory constants, the axioms are proven true of these objects, and then the theorems from the theory are declared to be Nuprl theorems. In [10] we gave a new semantics for Nuprl that justifies an extension in which HOL s classical type theory (and other classical set type theories) can be directly embedded. The extended logic is classical, but proofs that use only constructive reasoning still yield executable programs. In [9] we ....

....parameterized by interpretations of all ancestor theories. The old semantics of Nuprl was a term model based on an extended untyped calculus, and the programs of this language are the basic objects one reasons about in Nuprl. Nevertheless, we have been able to construct a set theoretic semantics [10] in which the meaning of each type is an ordinary set, and any program which is a member of a type can be given a meaning as a member of the set. In particular, the meaning of a function type A B in Nuprl is the set of all set theoretic functions from A to B. The collection of all sets that can ....

D. J. Howe. Semantics foundations for embedding HOL in Nuprl. In M. Wirsing and M. Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85--101, Berlin, 1996. SpringerVerlag.

....with a rich set of constructors. Because of the constructivity, programs can be extracted from proofs. Logic is encoded via the propositions as types principle, whereby a proposition is identified with the type of data that provides evidence for the proposition s truth. The version of Nuprl we use [8] also supports classical reasoning, which can be used in any part of a proof that does not affect the extracted program. Formal mathematics in Nuprl is organized in a single library, which is broken into files simulating a theory structure. Library objects can be definitions, display forms, ....

....protocol language is its lack of types. This language was developed inside Nuprl to support reasoning about particular protocols, and not for metareasoning about programs. The precise form of inductive definition mechanism implemented in standard Nuprl [4] is not valid in the classical extension [8] used here. It is not too difficult to adapt it, but we have not done so yet and hence have simply axiomatized the two inductive definitions we needed. We believe that our results can be extended to deal with temporal properties in the same way as programs. One difficulty is dealing with binding ....

D. J. Howe. Semantics foundations for embedding HOL in Nuprl. In Algebraic Methodology and Software Technology, pages 85--101. Springer-Verlag Lecture Notes in Computer Science, 1996.

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Douglas J. Howe. Semantic foundations for embedding HOL in NuPRL. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85--101. SpringerVerlag, Berlin, 1996.

No context found.

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85--101. Springer-Verlag, Berlin, 1996.

No context found.

Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodology and Software Technology, volume 1101 of Lecture Notes in Computer Science, pages 85--101. Springer-Verlag, Berlin, 1996.

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Douglas Howe. Semantic foundations for embedding hol in nuprl. In Martin Wirsing and Maurice Nivat, editors, Algebraic Methodolgy and Software Technology, volume 1101 of LNCS, pages 85-101. Springer Verlag, 1996.

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Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In Proceedings of AMAST'96, 1996. To appear.

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Douglas J. Howe. Semantic foundations for embedding HOL in Nuprl. In Proceedings of AMAST'96, 1996. To appear.

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