M. Gordon, R. Milner, and C. Wadsworth. Edinburgh LCF: A mechanised logic of computation. In Lecture Notes in Computer Science Number 78. Springer-Verlag, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as its only datatype. A second version of SASL was later developed including pattern matching and lazy constructors. Hope [BMS80] is a pure functional language which provides strong typing and a polymorphic higher order type system. ML (Meta Language) was developed as a meta language for the LCF [GMW79b] proof system but later matured into SML (Standard page 34 20 ML) MTH90] a language in its own right. SML is actually an imperative language supporting assignment and iteration. However SML contains a widely used pure functional subset including modules for creating large systems. Miranda ....

M. J. Gordon, A. J. Milner, and C. P. Wadsworth. Edinburgh LCF - A mechanised logic of computation, volume 78 of Lecture Notes in Computer Science. Springer-Verlag, 1979. 232

....can be found, re ne fails. infer is dual to re ne and does one forward reasoning step. The use of explicit proof trees is a major feature of the KIV System. Most other tactical theorem provers represent proofs either as functions which given the premises as theorems yield the conclusion ( GMW79] Gor88] or as meta assertions of the form premises imply conclusion ( Pau90] An exceptions is NUPRL ( CAB 86] The main reasons to reject the use of proof trees seem to be problems of eciency and storage requirements ( Pau89] p. 394) Indeed the storage requirements for average proof ....

M. Gordon, R. Milner, and C. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation. Springer LNCS 78, 1979.

....[17] They present the proof in detail so that it can be mechanized. The proof, which also constitutes a verification of the unification algorithm, relies on a substantial theory of substitutions, consisting of twenty three propositions and corollaries. Using the interactive theorem prover LCF [12], I have verified both the unification algorithm and the theory of substitutions. The project has grown too large to describe in a single paper. This paper is a survey, discussing the main aspects and mentioning papers where you can find more details. The proof is not entirely beautiful. A ....

.... existence proofs for inclusive predicates, a highly technical aspect of compiler verification [20] Gordon has extended Cambridge LCF for reasoning about hardware, and proved the correctness of a small computer [14] This section introduces the principles; tutorials have appeared elsewhere [12, 13]. The unification proof uses Cambridge LCF. 3.1 The logic PPLAMBDA Theorems are proved in the logic PPLAMBDA, which provides the usual predicate calculus formulas, such as conjunctions P disjunctions P existentials #x.P and logical equivalences P Q [22] Theorems are proved via ....

M. J. C. Gordon, R. Milner, and C. Wadsworth, Edinburgh LCF: A Mechanised Logic of Computation (Springer, 1979).

....processor was verified in [15] using Maude [5] The ground work for this report can be found in the thesis [7] which also explains how models with input and output can be supported within this framework. HOL is founded on Church s theory of simple types [4] and has its origins in Edinburgh LCF [13] and Cambridge LCF [24] The version of HOL used in the production of this report is HOL98 Taupo5, which is written in Standard ML (specifically MoscowML) The current HOL distribution, and additional information, may be found at www.cl.cam.ac.uk Research HVG HOL. The source for the HOL theories ....

Mike J. C. Gordon, Arthur J. Milner, and Christopher P. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation, volume 78 of Lecture Notes in Computer Science. Springer-Verlag, 1979.

....specify a program to produce a KWIC index [17] Its potential for the systematic development for correct programs is attracting the interest of computer scientists [1, 3] Many people have used the interactive theorem prover LCF to verify functional programs by computer. Most LCF work until now [10, 18] has been conducted in a logic for domain theory. Petersson [21] has already implemented a version of LCF for Intuitionistic Type Theory (henceforth Type Theory) Constable s group has implemented an elaborate theorem prover, called Nuprl, for a version of Type Theory [6] I am also experimenting ....

M. J. C. Gordon, R. Milner, and C. Wadsworth, Edinburgh LCF: A Mechanised Logic of Computation (Springer, 1979).

....U UV UW (U UV UW)Y (U UV UW)Z = U UV UW UY UVY UWY UZ UVZ UWZ . Cancelling common terms, this case terminates because UY UZ 0. 3 A proof in the Logic of Computable Functions Jacek Leszczy#lowski [4] has proved the termination of norm using the theorem prover Edinburgh LCF [3]. LCF s logic, a formalization of domain theory, allows reasoning about partial functions. Leszczy#lowski s proof uses a lemma that the termination of norm in particular cases implies termination in other cases. Each domain contains an undefined element representing the result of a divergent ....

....axiom exp DEFINED ; let EXP TAC = STRUCT TAC exp [ The Lemma is proved by induction followed by rewriting via the equations for norm and the totality of the constructors At and If. I have tweaked the statement of the Lemma to circumvent an annoyance involving admissibility of induction [3]. let NORM LEMMA = prove thm ( NORM LEMMA , x y z. y= UU z= UU NORM(y) UU NORM(z) UU = x= UU = NORM(IF x y z) UU : exp , EXP TAC x THEN ASM REWRITE TAC [NORM CLAUSES; EXP DEFINED] The proof that norm is total resembles the proof of the Lemma. let ....

M. J. C. Gordon, R. Milner, and C. Wadsworth, Edinburgh LCF: A Mechanised Logic of Computation (Springer, 1979).

....large pieces of mathematics: the Schroeder Bernstein Theorem, to name one example. The Calculus is also being applied towards the synthesis of correct programs by proof. 6. 3 LCF, a programmable theorem prover Edinburgh LCF was developed during the 1970 s by Robin Milner and his colleagues [Gordon, Milner, and Wadsworth 1979]. Its techniques have been adopted in numerous theorem provers. In order to be extensible, Edinburgh LCF was programmable. Its Meta Language, called ML, was the predecessor of Standard ML. The user could write ML functions to process terms, formulae, and theorems. Theorems were not simply ....

Michael J. C. Gordon, Robin Milner, and Christopher P. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation. Springer-Verlag LNCS 78.

....with appropriate tactics. The types of many constructions can be inferred automatically; simple functions can be derived interactively. 2 The LCF interactive theorem prover Edinburgh LCF introduced a new approach to theorem proving: embed the formal logic in a programmable meta language, ML [15]. Terms and formulas are values in ML: they have an explicit tree structure and can be decomposed and built up by ML functions. Theorems are values of an abstract data type thm. In place of arbitrary constructor functions, inference rules map theorems to theorems. This is forwards proof. For ....

....Milner [27] explains rules and tactics, while Gordon [12] works out the representation of a simple logic in ML. Recent LCF proofs involve denotational semantics, verification of functional programs, and verification of digital circuits [30] Logics include two versions of PP# (for domain theory) [15, 31], a Logic for Sequential Machines, two higher order logics [8, 13] and two constructive type theories [9, 32] Implementing a logic is a major undertaking: choosing a representation of formulas, implementing several dozen inference rules and tactics, implementing many more derived rules and ....

M. J. C. Gordon, R. Milner, and C. P. Wadsworth, Edinburgh LCF: A Mechanised Logic of Computation, Springer LNCS 78 (1979).

....3. 4) The o#cial arm architecture reference is [10] Furber s book is also a useful introductory text [5] The specification presented in this report was influenced work at Leeds using sml, see [8] hol is founded on Church s theory of simple types [1] and has its origins in Edinburgh lcf [6] and Cambridge lcf [9] The version of hol used in the production of this report is hol98 Taupo 6, which is written in Standard ML (specifically MoscowML) The current hol distribution may be found at www.cl.cam.ac.uk Research HVG HOL. Section 2 gives a brief overview of the approach taken in ....

Mike J. C. Gordon, Arthur J. Milner, and Christopher P. Wadsworth. Edinburgh lcf: A Mechanised Logic of Computation, volume 78 of Lecture Notes in Computer Science. Springer-Verlag, 1979.

....the overall nature of our machine and o ers some analysis of the scheme developed. Section 7 concludes the paper. 2 A Typed Version of Prolog The language of interest to us is based on a variant of Horn clauses [25] that incorporates a type system similar in some ways to the one used in ML [6]. There are two components of this language that need to be described: the types and the terms and programs. The types of the language are constructed from a set S of sorts, a set C of type constructors, each member of which is speci ed with a xed arity, and an in nite supply of type variables. ....

Michael J. Gordon, Arthur J. Milner, and Christopher P. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation, volume 78 of Lecture Notes in Computer Science. SpringerVerlag, 1979.

....to devise the appropriate space for the semantic model, and to then ensure that one s denotation function was both correct and compositional. The mathematical theory of domains required for the treatment of unbounded loops and recursion has been handled in theorem provers before (notably in LCF [GMW79]) but its complications don t seem to be as well served by the existing tools as the simpler demands made by operational semantics. An axiomatic semantics for a language presents a set of rules that allow one to conclude that particular syntactic phrases will satisfy a specification, given as a ....

....major error and one major omission in their proof. Using a theorem prover means that we are confident that all of the results we have proved are correct. Having used the theorem prover HOL [GM93] we are particularly confident, as this system, following the example of its ancestor system LCF [GMW79], uses the strong type system of ML to guarantee that values of type theorem are only produced in ways that are logically sound. This guarantee extends to HOL s use of conservative principles of definition, meaning that we are not forced to assert definitions as potentially inconsistent axioms. ....

M. J. C. Gordon, Robin Milner, and Christopher P. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation. Number 78 in Lecture Notes in Computer Science. Springer, 1979.

....rst an outline proof, or proof plan is generated. This proof plan is then converted to a tactic which will (usually) prove the conjecture in hand, giving an object level which can use heuristics to restrict their application. Recently, a language of methodicals [12] analogous to tacticals [5], has been developed and implemented in the Clam proof planner to allow methods to be written in a structured, modular way. Complex methods can be composed from smaller, simpler methods, with bene ts such as reusability and readability of methods (bene ts similar to those provided by ....

M. J. Gordon, A. J. Milner, and C. P. Wadsworth. Edinburgh LCF - A mechanised logic of computation, volume 78 of Lecture Notes in Computer Science. SpringerVerlag, 1979.

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M. Gordon, R. Milner, and C. Wadsworth. Edinburgh LCF: A mechanised logic of computation. In Lecture Notes in Computer Science Number 78. Springer-Verlag, 1979.

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M. J. Gordon, A. J. Milner, and C. P. Wadsworth. Edinburgh LCF - A mechanised logic of computation, volume 78 of Lecture Notes in Computer Science. Springer-Verlag, 1979.

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Gordon, M. J., Milner, A. J. & Wadsworth, C. P. (1979), Edinburgh LCF : A Mechanised Logic of Computation, Vol. 78 of Lecture Notes in Computer Science, Springer-Verlag.

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Michael J. C. Gordon, Robin Milner, and C.P. Wadsworth. Edinburgh LCF: a Mechanised Logic of Computation, volume 78 of Lect. Notes in Comp. Sci. Springer-Verlag, 1979.

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Gordon, M. J., Milner, A. J. and Wadsworth, C. P. (1979). Edinburgh LCF -- A mechanised logic of computation. Springer Verlag, LNCS 78.

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M.J. Gordon, A.J. Milner, and C.P. Wadsworth. Edinburgh LCF -- A Mechanised Logic of Computation. LNCS 78. Springer, 1979.

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Michael J.C. Gordon, Robin Milner, and Cristopher P. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation, volume 78 of Lecture Notes in Computer Science. Springer--Verlag, 1979.

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M.J.C. Gordon, R. Milner, and C.P. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation, volume 78 of LNCS. Springer Verlag, Berlin, Heidelberg, New York, 1979.

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M. J. C. Gordon, R. Milner, and C. P. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation, volume 78 of Lecture Notes in Computer Science. SpringerVerlag, 1979.

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M. Gordon, R. Milner, and C. P. Wadsworth. Edinburgh lcf: A Mechanised Logic of Computation, volume 78 of Lecture Notes in Computer Science. Springer-Verlag, 1979.

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M. Gordon, R. Milner, C. Wadsworth. Edinburgh LCF: A Mechanised Logic of Computation. Springer Verlag, LNCS 78. 1979

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M. J. Gordon, R. Milner, and C. P. Wadsworth. Edinburgh LCF: A mechanised logic of computation. Number 78 in Lecture Notes in Computer Science, 1979.

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M.J.C. Gordon, R. Milner and C. Wadsworth, "Edinburgh LCF: A Mechanised Logic of Computation". J. R. Harrison, "Floating-point verification in HOL light: the exponential function", Technical Report number 428, University of Cambridge Computer Laboratory. UK, June 1997.

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