M. Davis and J. Schwartz, Metamathematical extensibility for theorem verifiers and proof checkers, Computers and Mathematics with Applications, vol. 5 (1979), pp. 217--230.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....con l idea di metateoria procedurale) si intende una metateoria logica in cui il ragionamento viene effettuato per deduzione logica automatica (analogamente a quanto avviene nella teoria oggetto) 20 al lavoro di Alan Bundy (si vedano, per esmpio i lavori di M. Davis e T. Schwartz [DS79] R.Weyehrauch [Wey80, Wey82] in FOL, Boyer e Moore [BM81] nel BMTP e Howe, Constable e Knoblock [How88, KC86] in NuPrl) In questi lavori la metateoria semplicemente descrive la teoria oggetto ed e utilizzata per rappresentare le regole di inferenza della teoria oggetto, per costruirne di nuove ....

.... in FOL, Boyer e Moore [BM81] nel BMTP e Howe, Constable e Knoblock [How88, KC86] in NuPrl) In questi lavori la metateoria semplicemente descrive la teoria oggetto ed e utilizzata per rappresentare le regole di inferenza della teoria oggetto, per costruirne di nuove (per deduzione metateorica) DS79, Wey82, How88, KC86] o dimostrare la correttezza [BM81] di nuove regole derivate. Nell approccio di Alan Bundy, d altra parte, a run time, la fase di dimostrazione e preceduta (ma anche alternata) da una fase di pianificazione (proof planning usando la terminologia di Alan Bundy) il cui scopo ....

M. Davis and T. Schwartz. Metamathematical extensibility for theorem verifiers and proof-checkers. Computer and Matemathics with Applications, 5:217--230, 1979.

....and for type checking in programming languages [10] I think that this will be an important technique for modern provers that employ decision procedures. There are many other uses of reflection, and I recommend looking at articles by Boyer and Moore [8] Weyrauch [50] Howe [29] as well as [3, 6, 16, 37, 47, 48]. Acknowledgements I want to thank Kate Ricks for preparing this manuscript in L a T E X and for cheerfully tolerating such complexity in her first such project. I appreciate Stuart Allen s comments on an earlier draft. 32 ....

M. Davis and J. T. Schwartz. Metamathematical extensibility for theorem verifiers and proof checkers. Comp. Math. with Applications, 5:217--230, 1979.

....The joint logic of proofs and formal provability has been found in [90] 105] 32 SERGEI N. ARTEMOV 2. A recent application of explicit provability model: stability of verification. In the framework of formal provability the stability of verification systems is not internally provable ( 1] [33]) Rather the reflexive provability model provides a verification mechanism with provable stability ( 9] thus fixing a certain loophole in the foundations of verification. 3. The format t is a proof of F introduced by Godel in [41] along with the choice of the propositional logical language ....

M. Davis and J. Schwartz, Metamathematical extensibility for theorem verifiers and proof checkers, Computers and Mathematics with Applications, vol. 5 (1979), pp. 217--230.

....practical side, the paper gives speci c recommendations concerning the veri cation of inference rules and building a veri able re ection mechanism for a theorem proving system. 1 Introduction There is a large variety of theorem provers and proof checkers which can be used for veri cation (cf. [8], 1] 11] The mathematical counterparts of those systems range from rst order logic (e.g. in FOL) and certain fragments of rst order arithmetic to higher order logic (HOL) the systems with powerful principles sucient to accommodate most of the classical mathematics (Mizar) and most of the ....

....mathematical means. In this paper we will try to demonstrate the following three points: 1. Some form of the re ection rule is a necessary part of an extendable and stable veri cation system. This will emerge as a natural corollary of the soundness, extensibility, and stability assumptions (cf. [8]) about a veri cation system. Moreover, even the most basic proof checking scheme when V veri es a proof of F and then concludes that F itself holds requires re ection. 2. The traditional re ection based on the implicit provability predicate does not provide a satisfactory justi cation of formal ....

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Davis, M., Schwartz, J.: Metamathematical Extensibility for Theorem Veriers and Proof Checkers. Computers and Mathematics with Applications 5 (1979) 217-230

....combining correct program fragments into larger correct fragments were a key feature. Davis and Schwartz developed a formal way to enable a mechanical verification system to extend itself with new proof methods without violating soundness and without changing the set of provable statements; see [24]. Edith Schonberg later extended Davis and Schwartz s verification framework to support the computer assisted verification and application of program transformations; see [25] She presented one of the most difficult transformational proofs in the literature a formal derivation of Tarjan s ....

.... statements, i.e. assertions that must be verified by direct evaluation in Cantor. Examples of such statements are wp 1 in 1,2,3; wp 4 = 2 2; wp # = 0; wp [ a , b , c ] 1) a ; wp forall x in [ #x = 0; PROGRAM DERIVATION 43 An equivalent feature is described in [24] as part of what is called the verifier programming environment. 3. It is necessary to introduce new proof rules (new sequent patterns) The rules for predicate calculus are clearly sufficient. However, the specific rules for set theory and for dealing with SETL like program expressions may ....

Davis, M., and Schwartz, J., Metamathematical extensibility for theorem verifiers and proof checkers, Comp. Math. Appl. 5, 1979, pp. 217--230.

....up. The realization of these ideas in the GETFOL system is briefly described via the implementation of a simplified version of the Boyer and Moore theorem prover. 1 Introduction The idea of using metatheories in theorem proving has been extensively studied in the past, a not exhaustive list is [DS79, Wey82, BM81, BW81, Bun88, BSG 88, KC86, How88, GMW79] The goal of the research described in this paper is to get a better understanding of (mathematical) reasoning and to automate it on a machine (partially) by the use of metatheoretic theorem proving. The work is based on two main ideas: ....

....of extensibility, namely of how, by using metatheoretic capabilities, the inference mechanism could be improved without endangering the soundness of the system. Following a distinction first made in [BM81] all the attempts can be divided in two broad classes. In the first class (containing [BM81, DS79, KC86, Wey82] to preserve soundness, the procedures should be proved correct in a formalized metatheory, in the second (containing [CAB 86, GMW79] a mechanism is provided which guarantees that any added metatheoretic tactic, written in terms of primitive inference rules, will preserve ....

M. Davis and T. Schwartz. Metamathematical extensibility for theorem verifiers and proof-checkers. Computer and Matemathics with Applications, 5:217--230, 1979.

....this reasoning is done within the system itself. In order to reason about itself such a system must have an embedded declarative description of (a part of) itself. Therefore, reflective provers are introspective systems. However, though the reflective approach has been pursued several times [17, 51, 10, 32, 28, 37, 26], it appears that up to now it has not been used in significant applications. We believe that the reason for this is in the particular mechanisms taken so far: reflection requires to prove complex obligations, and so techniques are necessary to handle these tasks. Our approach reduces sound ....

....a specific tactic mechanism (very similar to the one proposed by Brown [11] In particular the inefficiency problem of tactic mechanisms appears in a similar way (cf. 13] We now concentrate on work concerning metatheoretical extensibility of proving systems. The pioneers are Davis Schwartz [17], Weyhrauch [51, 52] and Boyer Moore [10] Very close to the approach of Boyer Moore is the one taken by Howe in the Nuprl system [28] Another line of research at Cornell was that of Constable Knoblock [32] in which they formalized the structure of proofs within (an extension of) Nuprl. ....

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M. Davis and J. Schwartz. Metamathematical extensibility for theorem verifiers and proof-checkers. Computers and Mathematics with Applications, 5:217--230, 1979.

....than when a human answers the question. However, several approaches to verified verifiers are being pursued, ranging from running the output of the theorem 6 prover through a primitive but trusted proof checker to bootstrapping from such a proof checker to a verified automatic theorem prover [22, 55, 14, 8, 31]. The difficulties of verification notwithstanding, there is widespread interest in the field. The U.S. Government has already issued R.F.Q. s requiring various forms of mechanical verification. In addition, the recently established Department of Defense Computer Security Center has defined ....

M. Davis and J. Schwartz. Metamathematical Extensibility for Theorem Verifiers and Proof-checkers. Tech. Rept. 12, Courant Institute of Mathematical Sciences, 1977.

....at the time, must, for his incompleteness results, have been 2 At least for predicate logic see the previous footnote. the first to do this. The first suggestion that such a theory might be actually be used, as the basis of a mechanical proof development system, was by Davis and Schwartz [7], but their paper does not convincingly address the practical issues involved. More recently Basin and Constable, in [2] have suggested an approach that is in many ways similar to what is described here: the chief difference is that they define a logic in terms of an abstract data type, rather ....

Martin Davis and Jacob T. Schwartz. Metamathematical extensibility for theorem verifiers and proof-checkers. Computers and Mathematics with Applications, 5:217--230, 1979.

....516 5299, fax (410) 516 6134. Now, if a proof of B is desired and a proof of A is given, if A is of the appropriate form the above principle allows us to immediately conclude Provable L (dBe) whence B by the reflection axiom. Some systems incorporating this general paradigm are [ACHA90, BM81, DS79] Cornell researchers [ACHA90] have formulated such a reflected proof hierarchy for the Nuprl type theory [CAB 86] One advantage of studing reflected proof in such a context is there is a programming language built in to Nuprl, so it is possible to have the condition dAe is of appropriate ....

M. Davis and J. T. Schwartz. Metamathematical extensibility for theorem verifiers and proof-checkers. Computers and Mathematics with Applications, 5:217--230, 1979.

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Martin Davis and Jacob Theodore Schwartz. Metamathematical extensibility for theorem verifiers. Computer and Mathematics with Applications, 5:217--230, 1979.

....Alternatively, the model for an integrated verifier might be the SETL like NAP [126] system, itself implemented in the SETL derivative Cantor [127, 124] Verification of assertions in, say, Hoare logic [113] would increase confidence in automatically applied transformations. Davis and Schwartz [50] showed how mechanical verification systems could extend themselves with new proof methods without violating soundness or changing the set of statements that could be proved. The main existing impediment to the speed of APTS is the fact that its database of program property relationships, which ....

M. Davis and J. Schwartz. Metamathematical extensibility for theorem verifiers and proof checkers. Comp. Math. Appl., 5:217--230, 1979.

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M. Davis and J. Schwartz, Metamathematical extensibility for theorem verifiers and proof checkers, Computers and Mathematics with Applications, vol. 5 (1979), pp. 217--230.

No context found.

M. Davis and J. T. Schwartz. Metamathematical extensibility for theorem verifiers and proof checkers. Comp. Math. with Applications, 5:217--230, 1979.

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Martin Davis and Jacob Theodore Schwartz. Metamathematical extensibility for theorem verifiers. Computer and Mathematics with Applications, 5:217--230, 1979. 20

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