Bruijn, N.G. de, A survey of the project AUTOMATH, in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, ed. Hindley, J.R., and Seldin, J.P., Academic Press, 29-61, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... ) x : t 1 : 2 u : t 2 : 2 (2; 2) x : t 1 : 2 The system with only ( for (s 1 ; s 2 ) is known as Church or (this is essentially the Automath system AUT 68) The addition of ( 2) gives P , which is a system that is rather close to another variant of the Automath family, AUT QE (see [deB80] The addition of (2; to ( gives the second order typed lambda calculus, also called 2. Adding (2; 2) to ( we obtain . 38 There are three systems that are defined by adding a combination of two of the three last mentioned possibilities to ( When all mentioned (s 1 ; s 2 ) ....

Bruijn, N.G. de, A survey of the project AUTOMATH, in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Eds. J.R. Hindley and J.P. Seldin, Academic Press, New York/London, pp. 29-61, 1980.

....Heim (1982) and Kamp (1981) The work has become known as File Change Semantics and Discourse Representation Theory (DRT) respectively. Interestingly, the formal frameworks set up by Kamp and Heim are similar to a class of formal systems, Constructive Type Theories (e.g. Martin Lf, 1984; De Bruijn, 1984; Nederpelt, Geuvers De Vrijer, 1994) which predate them. Sundholm (1986) was the first to use Constructive Type Theory (CTT) to model some of the data that are also covered by DRT. 112 Other researchers who have employed CTT for modelling natural language are, notably, Ranta (1994) and ....

Bruijn, N.G. de (1984). A survey of the project Automath. In: J.R. Seldin and J.P. Hindley (Eds.): To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalisms, 589-606. London: Academic Press.

.... lambda terms provide a convenient means for representing objects whose structures incorporate the notion of binding [Chu40] and are used for this purpose in a number of computer systems and programming languages that support the manipulation of formulas, programs, proofs and other similar objects [Bru80, CAB 86, CH88, GMW79, HHP93, NM88, Pau90, Pfe89] In a sense specifically pertinent to this paper, objects are represented directly by lambda terms in systems like L [Mil91] Prolog [NM88] Isabelle [Pau90] and Elf [Pfe89] to be manipulated by some form of higher order unification [Hue75, ....

N. de Bruijn. A survey of the project AUTOMATH. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 579--606. Academic Press, 1980.

....For the proof verification we resort to type theory. Type theory (or typed lambda calculus, see for example Barendregt [3] provides a very expressive formalism in which CRL can conveniently be embedded. Also, it has a long standing tradition of automated verification of proofs (AUTOMATH [7], LCF [13] Nuprl [9] LEGO [17] Without specific reason we chose the Calculus of Constructions of Coquand and Huet [10] on which the interactive proof construction and verification system Coq [11] has been based. There are numerous examples of verifications of proofs of the correctness of ....

N.G. de Bruijn. A survey of the project AUTOMATH. In J.R. Hindley and J.P. Seldin, editors, Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 580--606. Academic Press, London, 1980.

....signatures. I then extend the semantics to modules using phase splitting. 1 Introduction Type theory has become a popular framework for formal reasoning in computer science [7, 38, 21] and has formed the basis for a number of automated deduction systems, including Automath, Nuprl, HOL and Coq [18, 5, 22, 2], among others. In addition to formalizing mathematics, these systems are widely used for the analysis and verification of computer programs. To do this, one must draw a connection between the programming language used and the language of type theory; however, these connections have typically been ....

N. G. de Bruijn. A survey of the project Automath. In J. P. Seldin and J. R. Hindley, editors, To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pages 579--606. Academic Press, 1980.

....Such derivations easily become long and tedious and, hence, error prone; so, it is essential to formalize the proofs and to have computerized tools to check them. There are several examples of computer implementations of proof checkers for formal logics. An early example is the AUTOMATH system [11, 12] which was designed by de Bruijn to check proofs of mathematical theorems. Quite large proofs were checked by the system, for example the proofs in Landau s book Grundlagen der Analysis [24] Another system, which is more intended as a proof assistant, is the Edinburgh (Cambridge) LCF system [19, ....

N. G. de Bruijn. A survey of the project AUTOMATH. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism, pages 589606, New York, 1980. Academic Press.

....in LF . In section 4, we show that our syntactic definitions of representation give rise to indexed isomorphisms. 2 The Logical Framework LF The framework LF is based on the Edinburgh Logical Framework (LF) of Harper, Honsell and Plotkin [HHP87] Influenced by various AUTOMATH languages [Bru80] and by Martin Lof s work on the foundations of intuitionistic logic [Mar85] LF constitutes an important advance in the study of logical frameworks. It is not possible, however, to provide general definitions of correct representation using LF. These definitions are possible using LF . A ....

....as part of the machinery of the meta theory, rather than as having a direct correspondence in the object logic, since the aim is to capture a wide variety of logics. In contrast, various predicative intuitionistic logics can be presented as type theories using the propositions as types paradigm [CF58, Bru80, How80], by equating Pi abstraction with universal quantification. Alternatives to this choice of rules are discussed in [Gar92] For example, it seems reasonable to assume that the syntax of a logic does not depend on the derivations of the logic; for this reason we have omitted the rules (Judge; Sort ....

N.G. de Bruijn. A Survey of the Project Automath, in [SH80], pp 589--606, 1980.

....Types and Impredicativity Russell s initial description of a type theory was naturally higher order. There are types of objects and types of types (sets of sets) and so on. The utility of higher order types is partially dependent on what one wants to say and how one wants to say it. Automath [8] is first order; de Bruijn justifies this restriction to first order by referring to additional axioms (extra logical notions) that extend the power of the language. Under the proviso that one is willing to assume certain propositions as proven, a first order language can be extended to encode ....

....algorithm refinement, and program realization) could be used together 122 to provide support for the different aspects of program development: algorithm design, refinement, implementation, and verification. I will look at systems that have provided similar or overlapping capabilities: Automath [8], CLEAR [11] Institutions [21] 41] consequence relations in LF [24] ELF [37] Extended ML [42] a system of Deliverables [9] and Specware [49] Each of these formalizes a particular activity associated with program or theory development, with the Specware system being the closest in terms ....

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Bruijn, N. G. d. A survey of the project AUTOMATH. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. P. Seldin and J. R. Hindley, Eds. Academic Press, 1980, pp. 579--606.

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Bruijn, N.G. de, A survey of the project AUTOMATH, in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, ed. Hindley, J.R., and Seldin, J.P., Academic Press, 29-61, 1980.

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