L.C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge University Press, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... functions, and if partial functions are admitted in logical formulae, a programming logic is needed that handles partial functions and undefined expressions [137] Two valued logics with undefined handles partial functions by introducing a constant (e.g. #) to represent an undefined value [122]. Three valued logics with undefined allows the logical formulae to have the undefined value [11] 14] 68] 79] Some logics introduce even more than one special value, thus, leading to many valued logics [9] 10] Gries and Schneider modeled partial functions by under specified total functions ....

Lawrence C. Paulson. Logic and Computations: Interactive Proof with Cambridge LCF. Cambridge Tracts in Theoretical Computer Science, Volume 2. Cambridge University Press, 1987.

....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 developed in this report ....

Larry Paulson. Logic and Computation: Interactive Proof with Cambridge LCF, volume Cambridge Tracts in Theoretical Computer Science 2. Cambridge University Press, 1987.

.... 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 modelling the ....

Larry Paulson. Logic and Computation: Interactive Proof with Cambridge LCF, volume Cambridge Tracts in Theoretical Computer Science 2. Cambridge University Press, 1987.

....commutativity as a rewrite rule, or we would get non terminating reductions. The above two proofs show that we are entitled to use associative commutative rewriting for , and we do so below. It is interesting to contrast the above proofs with corresponding proofs due to Paulson in Cambridge LCF [129]. The LCF proofs are much more complex, in part because LCF functions are partial, and therefore must be proved total, whereas functions are automatically total (on their domain) in equational logic. C.4.3 Formula for 1 : n We now give a standard inductive proof over the natural numbers, ....

Lawrence Paulson. Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge, 1987. Cambridge Tracts in Theoretical Computer Science, Volume 2.

....both in academia and industry. It is free, comes with extensive documentation, libraries, an interactive help system, and myriad web sites providing information and a dynamic search engine for HOL information . HOL is a direct descendant of the innovative LCF (Logic of Computable Functions) [Pau87] theorem prover developed by Robin Milner in the early 1970s, and is an implementation of a version of Church s simple theory of types, a formalism dating back more than 50 years. HOL is an acronym of Higher Order Logic, the logic used by the HOL system. Basically, this logic is first order ....

Lawrence C. Paulson. Logic and Computation:Interactive Proof with Cambridge LCF. Cambridge University Press, 1987.

....addition, the Transputer microprocessor developed by Inmos [37] provides a platform for the implementation of Occam programs. The HOL (Higher Order Logic) 20,18,52] theorem proving system was used to perform machine checked proofs. The HOL system provides an LCF style theorem proving environment [21, 45] and supports a version of classical higher order logic based on Church s formulation of simple type theory [2,16] HOL includes ML as a metalanguage: the ML language was originally developed as part of LCF, but is now an independent programming language in its own right [40] It is an ....

L.C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF,volume 2 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1987.

....a balance between allowing the programmer flexibility, and providing a useful conceptual discipline which filters out many errors at an early stage. On the other hand, stronger correctness properties must be guaranteed by a separate phase of explicit verification, using some logic such as LCF [28, 53] or one of its many descendants. More generally, there is a spectrum of type disciplines : ML : System F : Intuitionistic Type Theory : 46 Thus, a System F typing f : list nat list nat guarantees that f terminates, but not, say, that it sorts its input; while a type can be ....

L. C. Paulson. Logic and Computation : Interactive proof with Cambridge LCF, volume 2 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1987.

....illustrate reasoning about infinite values and nonterminating functions and show how domain and set theoretic reasoning can be mixed to advantage. An example presents a proof of correctness of a recursive unification algorithm using well founded induction. 1 Introduction The LCF system [GMW79, Pa87] is a theorem prover based on a version of Scott s Logic of Computable Functions (a first order logic of domain theory) It provides the concepts and techniques of fixed point theory to reason about nontermination and arbitrary recursive (computable) functions. For instance, it has been ....

....total functions in set theory (higher order logic) before turning to domain theory. The examples have already been done in LCF by Paulson which makes a comparison of the two systems possible. The first two examples, on natural numbers and lazy sequences, are described in chapter 10 of the LCF book [Pa87] and the third example is based on Paulson s version of a correctness proof of a unification algorithm by Manna and Waldinger [MW81, Pa85] The unification algorithm is defined as a fixed point and proved total afterwards. Termination is non trivial and proved by well founded induction [Ag91] ....

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L.C. Paulson, Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge Tracts in Theoretical Computing 2, Cambridge University Press, 1987.

....on categorical methods using embedding projection pairs, see e.g. 29, 25] This was suggested by Plotkin as a generalisation of Scott s original inverse limit construction of a model of the calculus in the late 60 s. The formalisation is based on Paulson s accessible presentation in the book [20] but Plotkin s [25] was also used in part (in fact, Paulson based his presentation on this) 3.1 Basic Concepts of Domain Theory Domain theory is the study of complete partial orders (cpos) and continuous functions between cpos. This section very briefly introduces the semantic definitions of ....

L. C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge Tracts in Theoretical Computing 2, Cambridge University Press, 1987.

....tacti language. Ini886#[ i wasi ntended to support proofs i thegoal di[Iz# style, thatia provi8S# a framework for the composi4 ## ofpri #6B ei nference rulesi n theconstructi6 of backwards proofs, but i turns out to be more general than thi[ Term rewriB#[I for example, whi h i CambriBz LCF [Pau87]i di] S6S by a separate set of operators from those whi hdescri etacti8[ could also bedescri edusi8 Angel. Because Angeli a small language,in semanti descriB##[ i quic clean and easy to reason about. Neverthelessi ti s able todescri e a large class of useful algori ##z# The languagei named Angel ....

Lawrence C. Paulson. Logic and Computation---Interactive Proof with Cambridge LCF. CambridgeUniv ersity Press, 1987.

....is well founded, that is has no infinite descending chains (like the chain Gamma1 Gamma2 : Gamman : over the integers. A recursion will terminate precisely when it can be shown to follow a well founded ordering. Further details about denotational semantics can be found in [23, 13]. We also refer back to denotational semantics at the end of section 8.3 8.2 Operational semantics The structured ( SOS ) style of operational semantics pioneered by Plotkin describes a programming language by means of deduction rules which explain how expressions are evaluated. This style has ....

Laurence C. Paulson. Logic and Computation --- Interactive proof with Cambridge LCF. Cambridge University Press, 1987.

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L. C. Paulson, Logic and Computation: Interactive Proof with Cambridge LCF (Cambridge University Press, 1987).

....may not be iterated in general, but may use recursion over the built in list type. The earliest use of least fixedpoints is probably Robin Milner s. Brian Monahan extended this package considerably [19] as did I in unpublished work. lcf The datatype package described in my lcf book [23] does not make definitions, but merely asserts axioms. 22 is a first order logic of domain theory; the relevant fixedpoint theorem is not Knaster Tarski but concerns fixedpoints of continuous functions over domains. lcf is too weak to express recursive predicates. The Isabelle package might be ....

Paulson, L. C., Logic and Computation: Interactive proof with Cambridge LCF, Cambridge Univ. Press, 1987

....expresses definitions. The meta logic includes the typed # calculus, which is convenient for formalizing the syntax of object logics, particularly variable binding. Provisos of quantifier rules (of the sort x not free in the assumptions ) are enforced by meta level quantification. Like in lcf [29], backwards proofs are developed using tactics and tacticals, which are implemented using Standard ml. But an inference rule in lcf is a function from the premises to the conclusion, while in Isabelle it is an axiom in the meta logic stating that the premises imply the conclusion. Since Isabelle ....

.... types of Martin Lof s Type Theory are general transfinite trees [22] The Nuprl system, although largely based on Martin Lof, uses positive recursive type definitions [7] Boyer and Moore s shell principle introduces recursive structures [4] lcf can define recursive types using domain theory [29]. Recursive types can also be constructed in simple type theory. The natural numbers can be constructed in various ways, assuming an Axiom of Infinity. In Principia, the number 2 is the class of all pairs of some type #. In Church, 2is#f : # #.#x : #.f(fx) Both definitions are cumbersome and ....

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Lawrence C. Paulson. Logic and Computation: Interactive proof with Cambridge LCF. Cambridge University Press, 1987.

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L. C. Paulson, Logic and Computation: Interactive Proof with Cambridge (Cambridge University Press, 1987). 79

....For instance, Isabelle represents the inference rule PQ P Q by the following axiom in the meta logic: P . Q . P =# (Q =# P Q) The structure of rules generalizes Prolog s Horn clauses; proof procedures can exploit logic programming techniques. Isabelle borrows ideas from lcf [8]. Formulae are manipulated through the meta language Standard ML; proofs can be developed in the backwards direction via tactics and tacticals. The key di#erence is that lcf represents rules by functions, 1.1. Overview of Isabelle 7 not by axioms. In lcf, the above rule is a function that maps the ....

L. C. Paulson, Logic and Computation: Interactive Proof with Cambridge LCF (Cambridge University Press, 1987).

....should o#er some evidence that its proofs are valid. The Boyer and Moore [1979] theorem prover prints an English summary of its reasoning, while Folderol prints a trace of the rules. Most LCF style systems o#er no evidence of correctness other than an obstinate insistence on playing by the rules [Paulson 1987]. If correctness is a matter of life and death, then a prover can be designed to output its proof for checking by a separate program. Absolute correctness can never be obtained, even with a computer checked proof. There are fundamental reasons for this. Any program may contain errors, including ....

....ML s strict semantics. universally quantified assumptions by matching. Soko#lowski [1987] wrote a set of Edinburgh LCF tactics permitting unification. They maintain an environment of variable instantiations. My recent book on LCF describes rules and tactics, as well as some theory and applications [Paulson 1987]. The Higher Order Logic (HOL) prover, which is based on LCF, is coming into widespread use for hardware verification [Gordon 1988] Several complicated chips have been verified using HOL. One of the largest HOL proofs concerns the Viper microprocessor [Cohn 1989a] Nuprl, which is another ....

Lawrence C. Paulson. Logic and Computation: Interactive proof with Cambridge LCF. Cambridge University Press.

....Demo2. The user gains immediate access to the existing tools in that theorem prover for rewriting, checking tautologies, etc. as well as the ability to move between window inference and the other styles of reasoning supported by the host system. Implementing window inference in an LCF style [Pau87] system like HOL, where theorems are represented by a secure data type, brings the further advantage that inconsistent results cannot be obtained through errors in the interface. 2.8 Related Work The concepts of window inference were first described by John Staples and Peter Robinson [RS93] ....

Lawrence C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF, volume 2 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, England, 1987.

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L.C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge University Press, 1987.

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L.Paulson. Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge University Press, 1987. Cambridge Tracts in Theoretical Computer Science 2.

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L. C.Paulson. Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge University Press, Cambridge, England, 1987.

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L. Paulson, Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge Tracts in Theoretical Computer Science Number 2, Cambridge University Press, 1987.

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L. C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge University Press, 1987.

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L.C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF. Cambridge University Press, 1987.

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L. C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF, volume 2 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1987.

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