M. Gordon, R. Milner, and C. P. Wadsworth. Edinburgh LCF, LNCS 78. 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of recursive types and recursive functions, which are not available in classical higher order logic. The tradional 2 way of dealing with these features is Scott Strachey denotational semantics [39] which needs to talk about things like complete partial orders and least xed points. The logic LCF [14] has been designed to reason about this. What we want to avoid is to put classical higher order logic and denotational semantics side by side, such as in HOLCF [31] a speci cation of LCF within higher order logic) or Spectrum [15] which has distinguished domains with a three valued logic) ....

M. Gordon, R. Milner, and C. Wadsworth, Edinburgh LCF, LNCS, vol. 78, Springer, 1979.

....are undecidable, including first order logic. Due to this limitation, complete automated theorem prover is infeasible. However, there are user guided automated theorem provers, such as ACL2 [KM95] Eves [CKM 88] LP [GJ88] Nqthm [BM79] Interactive proof checkers include HOL [Gor88] LCF [GMW79] PVS [ORN92, OSR93] etc. Theorem provers suffer two weak points: the inability of finding counter examples and the necessity of user interaction. However, theorem provers can handle infinite state models using structural induction. 2.1.2 Model Checker Model checking is a technique building a ....

M.J. Gordon, A.J. Milner, and C.P. Wadsworth. Edinburgh lcf. In LNCS 78, 1979.

....convenient to write specifications. However, any non trivial specification would still be much too large and obscure to be practical. What is needed is some way of structuring the specification. In the prototype tools we have developed [11, 32] this is 32 accomplished by using a meta language [20]. In other words, we use a general purpose language to build up the various constructs that our specification language contain. In our original prototype system [11] we used a dialect of Lisp as meta language. When the Lisp program was run, it wrote to a file the verification conditions expressed ....

M. Gordon, R. Milner, and C. Wadsworth, "Edinburgh LCF", Lecture Notes in Computer Science, No. 78, Springer Verlag, 1979.

....and discrimination code. Because of data type definitions, the compiler can optimize data representations somewhat; for example, on totally general principles the abstract ML 1 A superficial knowledge of ML is assumed. 2 A compiler for ML was already available as part of the LCF system [Gordon 79] 3 Except some used by the garbage collector, which does not affect normal execution speed; this can be encoded in the address of a datum so that no extra space is used. Page 2 June 4, 1995 10:48 AM definition of list type (which seems to require extra pointers) produces the normal ....

.... list of non generic variables for the scope of the declaration) Then all the t i s are typechecked in that environment, and their types T i are matched again against the a i (or their instantiations) Side effects do not interact nicely with polymorphism, and require a special treatment (see [Gordon 79] page 52, for a discussion of the problems) The simplest solution consists in requiring all the assignments to act on monomorphic values; this is the position taken in Standard ML. The ML compiler uses a more flexible discipline, devised by Luis Damas, which however will not be discussed here. ....

M.Gordon, R.Milner, C.Wadsworth. "Edinburgh LCF", Lecture Notes in Computer Science, No. 78, Springer-Verlag, 1979.

....by means of theories. Gentzen like proof systems are found suitable for proving tautologies. 8 Temporal logic of programs: standard approach They can be implemented directly, or at least can suggest such an implementation. For a more comprehensive study of implementing proof systems see e.g. [11,17]. Although very important, tautologies can only support reasoning about validities. On the other hand, when analyzing properties of programs, one always deals with some speci c theories describing certain interpretations, such as natural numbers, stacks, trees or lists. Hilbert like axiomatic ....

Gordon, M. J., A. J. Milner & C. P Wadsworth 1979. Edinburgh LCF. Lecture Notes in Computer Science 130.

....upon, after extensive testing on benchmarks of user programs, the systematic development of further tools (compilers, debuggers, proof assistants) can proceed mechanically from the official CAML definition. Designing compiler generators from denotational definitions is an active research area [30, 26, 4, 11]. References [1] The ML Handbook, Version 6.1. Internal document, Projet Formel, Inria (July 1985) 2] The CAML Primer, Version 2.2. Internal document, Projet Formel, Inria (March 1987) 3] H. Abelson and G. J. Sussman. Structure and Interpretation of Computer Programs. MIT Press (1985) ....

....References [1] The ML Handbook, Version 6.1. Internal document, Projet Formel, Inria (July 1985) 2] The CAML Primer, Version 2.2. Internal document, Projet Formel, Inria (March 1987) 3] H. Abelson and G. J. Sussman. Structure and Interpretation of Computer Programs. MIT Press (1985) [4] A. W. Appel. Semantics Directed Code Generation. Twelfth ACM POPL Symposium, NewOrleans (Jan. 1985) 315 324. 5] R. Bird. An Introduction to the Theory of Lists. Course Notes, International Summer School on Logic of Programming and Calculi of Discrete Design, Marktoberdorf, Aug. 86. 6] W. ....

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M. Gordon, R. Milner and C. Wadsworth. "Edinburgh LCF." Springer-Verlag LNCS 78 (1979).

....concerning this general topic [567] 569] 568] 570] A good deal of work was and is being done within artificial intelligence. Here there are two major lines of research and development: ffl building theorem checkers (and program synthesizers) which are able to accept human guidance [236] [425] ffl searching for special purpose automatic theorem proving methods. See Section 6.3 for more on the topic of theorem proving. The pioneering work on deductive program synthesis is that of Manna and Waldinger [649] They built a formal system within which one can easily and in a natural way ....

M. J. C. Gordon, A. J. R. G. Milner, and C. P. Wadsworth. Edinburgh LCF. Springer LNCS 78, 1979.

.... ML s polymorphic type system and related type systems, beginning with [Mil 78] Typetheoretic studies of the Standard ML module system include [MacQ 86b] and [MH 88] the latter has been reformulated in category theoretic terms and modified in [HMM 90] The theorem proving systems Edinburgh LCF [GMW 79] and Cambridge LCF [Pau 87] implement versions of the logic PP (polymorphic predicate calculus) which can be used for reasoning about programs written in a subset of the Standard ML core language. A number of good implementations of SML exist, see e.g. Standard ML of New Jersey [AM 87] Although ....

.... the soundness of a proof system will not be an easy task (completeness is unachievable since datatype definitions correspond to data constraints see [MS 85] Although the way that we have dealt with polymorphism is somewhat unusual, the inference rules for type instantiation in PP [GMW 79] Pau 87] seem to remain sound. We have not yet thought about inference rules for the version of equality discussed above, and the impact of higher order functions and exceptions on the rest of the logic is not clear. It might be necessary to revise our decisions concerning the interpretation of ....

M. Gordon, R. Milner and C. Wadsworth. Edinburgh LCF. LNCS 78 (1979).

....is provided with a logical semantics by interpretation in the language of MPL . The approach to this interpretation, which is actually an interpretation of the language of LPF in the language of MPL , is connected with the layered approach to handle partial functions adopted in the logic PP [25]. It should be remarked that LPF can be reduced to MPL in the following sense: formulae of LPF can be translated to formulae of MPL and what can be proved remains the same after translation. This demonstrates that three valued logics such as LPF are not necessary to deal with partial ....

GORDON, M.J.C., MILNER, R., and WADSWORTH, C.: `Edinburgh LCF' (Springer Verlag, LNCS 78, 1979)

....(such as calling Nqthm) As with a variety of proof checking systems, PC Nqthm is goal directed; a proof is completed when the main goal and all subgoals have been proved. A notion of macro commands lets the user create compound commands, in the same spirit of the tactics and tacticals of LCF [20]. An interactive proof is complete when all goals have been proved. It is PC Nqthm s low level features that help us understand when and why a goal fails. 5 This is an extension to Nqthm by Matt Kaufmann, which is not documented in [9] See [29] for details. Chapter 3 The MC68020 Instruction ....

M. Gordon, R. Milner, and C. Wadsworth. Edinburgh LCF. SpringerVerlag, New York, 1979.

....programmer write generic functions that work uniformly on arguments of different types and it thus avoids the maintenance problem that results from duplicating similar program code at different types. The first programming language to use polymorphic type inference was the functional language ML [GMW79, Mil85]. Due to its usefulness, many of the aspects of ML have been subsequently incorporated in other languages (e.g. Miranda [Tur85] Haskell [HW88] ML shares with Algol 68 properties of compile time type checking, strong typing and higher order functions while also providing automatic type inference ....

M. J. Gordon, R. Milner, and C. P. Wadsworth. Edinburgh LCF, vol. 78 of LNCS. SpringerVerlag, 1979.

....Furthermore, there may be multiple choices for manipulating any one formula in the sequent. provides a programming language for specifying how these choices should be organized and executed. This programming language is an extension to the LCF tactic and tactical approach to theorem proving [6]. A primitive tactic is a (partial) function that takes a sequent, and checks if it can be the conclusion of a particular inference rule. If the sequent can be such a conclusion, the premise(s) of that instance of the rule are returned. If that particular inference rule can not be used, the tactic ....

Michael J. Gordon, Arthur J. Milner, and Christopher P. Wadsworth, "Edinburgh LCF," Lecture Notes in Computer Science, No. 78, Springer-Verlag, 1979.

....ADU SPS and ANU SPS are studied using the functional paradigm and as with dataflow languages the work of G Kahn has been widely adopted as a semantic approach for functional stream processing. However, other (sometimes related) approaches are also used including greatest fixed points (see [61] and [82]) and Aczel s logical theory of constructions (see [2] 71] and [72] In addition, the work of [78] and [100] on lazy evaluation has provided an implementation technique for functional stream processing that has been widely adopted. 4.2.2 Applications. The verification of functionally specified ....

M Gordon, R Milner, and C Wadsworth. Edinburgh LCF. In Semantics of Concurrent Computation, number 70 in LNCS. Springer-Verlag, 1979.

....ADU SPS and ANU SPS are studied using the functional paradigm and as with dataflow languages the work of G Kahn has been widely adopted as a semantic approach for functional stream processing. However, other (sometimes related) approaches are also used including greatest fixed points (see [52] and [66]) and Aczel s logical theory of constructions (see [2] 57] and [58] In addition, the work of [63] and [78] on lazy evaluation has provided an implementation technique for functional stream processing that has been widely adopted. 5.2.2 Applications. The verification of functionally specified ....

M Gordon, R Milner, and C Wadsworth. Edinburgh LCF. In Semantics of Concurrent Computation, number 70 in LNCS. Springer-Verlag, 1979.

....Aside from their typing benefits, variables are also of interest because their implicit dereferencing is a syntactic convenience, and because they are at the core of mainstream imperative languages. The idea of including variables in a polymorphic language is not new. In fact, Edinburgh LCF ML [Gordon et al. 1979] had a letvar construct, which it called letref. But it did not have first class references, and according to Tofte [1990] its type system was never proved sound. 2. AN INFORMAL DESCRIPTION OF THE TYPE SYSTEM The language we consider is the core ML of Damas and Milner [1982] together with ....

....to give makestack a weak polymorphic type makes our system substantially better than Edinburgh LCF ML on cases of this kind. In LCF ML, a letvar bound identifier must be given a monotype (i.e. a type with no variables) if it is assigned to within a abstraction within its scope (restriction 2ib [Gordon et al. 1979, p. 49] Hence, since stk is assigned to within the push and pop operations, LCF ML requires stk to be annotated with a monotype. This forces makestack to be monomorphic. Finally, typings of purely functional programs in our system are preserved as they are in the type systems for Standard ML ....

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Gordon, M., Milner, R., and Wadsworth, C. 1979. Edinburgh LCF. Lecture Notes in Computer Science, vol. 78. Springer-Verlag, Berlin.

....and simplification. Nqthm, the Boyer Moore theorem prover, has been used to check a proof of Godel s first incompleteness theorem, and in a variety of large scale verification efforts. Proof checkers. Examples include Coq [Cornes et al. 1995] HOL [Gordon 1987] LEGO [Luo and Pollack 1992] LCF [Gordon et al. 1979], and Nuprl [Constable et al. 1986] They have been used to formalize and verify hard problems in mathematics and in program verification. Combination provers. Analytica [Clarke and Zhao 1993] which combines theorem proving with the symbolic algebra system Mathematica, has successfully proved ....

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

....examples which fit into this framework. The structure for relations we use is based on that used by O Hearn and Tennent (1993) in their work on applying relational parametricity to the study of local variable declarations. To it we add a treatment of the crucial property of relations that in LCF (Gordon, Milner and Wadsworth, 1979) is called admissibility (meaning admitting induction ) Definition 4.1 A relational structure, R, on a category C is specified by the following data: ffl For each C object D, a set R(D) of R relations on D ) ffl For each C morphism f : D Gamma E, a binary relation between elements R 2 ....

Gordon, M. G. C., Milner, R., and Wadsworth, C. P. (1979), "Edinburgh LCF", Lecture Notes in Computer Science, Vol. 78, Springer-Verlag, Berlin.

....in section 7. Before concluding in section 9, we show how our algorithm works on a few examples (section 8) 2 Related Work Our language is equivalent to Core ML [Mitchell] extended to allow references. The classical way of dealing with non referentially transparent constructs is described in [Gordon] where some ad hoc rules are introduced to avoid creating inconsistencies within the type system. Tofte] introduces a nicer imperative type discipline within which types are categorized between applicative and imperative types; only applicative types can be generalized in let bindings. An ....

Gordon, M. C. J., and Milner, R. Edinburgh LCF. In Lecture Notes in Computer Science, vol. 78. Springer Verlag, 1979.

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M. Gordon, R. Milner and C. Wadsworth, Edinburgh LCF, LNCS 78, SpringerVerlag, Berlin, 1978.

....of adequacy) Operational soundness (that is, if M N then M N) is a consequence of the Context Lemma, together with the Church Rosser Property. Now that we have defined the languages of interest, we introduce the syntax of a simple many sorted program logic J L , much in the spirit of LCF [5]. We believe that this is the kind of logic that would in principle be useful for specifying and proving properties of programs. The sorts of J L are the types of L; the expressions of sort oe in J L are precisely the terms of type oe in L; and the logical variables of sort oe are just the term ....

M. Gordon, R. Milner and C. Wadsworth, Edinburgh LCF, LNCS 78, SpringerVerlag, Berlin, 1978.

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M. Gordon, R. Milner, and C. P. Wadsworth. Edinburgh LCF, LNCS 78. 1979.

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M. Gordon, R. Milner, and C. Wadsworth, "Edinburgh LCF", Lecture Notes in Computer Science, No. 78, Springer Verlag, 1979.

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M. J. Gordon, A. J. Milner, C. P. Wadsworth. "Edinburgh LCF." Springer-Verlag LNCS 78 (1979).

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M. J. Gordon, A. J. Milner, C. P. Wadsworth. "Edinburgh LCF." Springer-Verlag LNCS 78 (1979).

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Gordon, M., Milner R. &Wadsworth C. (1978). Edinburgh LCF. LNCS 78, Springer-Verlag.

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