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Mark Adams.
Introducing HOL Zero.
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Theorem provers are now playing an important role in two diverse fields: computer system verification and mathematics. In computer system verification, they are a key component in toolsets that rigorously establish the absence of errors in critical computer hardware and software, such as processor design and safetycritical software, where traditional testing techniques provide inadequate assurance. In mathematics, they are used to check the veracity of recent highprofile proofs, such as the Four Colour Theorem and the Kepler Conjecture, whose scale and complexity have pushed traditional peer review to its limits.

[6] 
Wolfgang Ahrendt, Bernhard Beckert, Reiner Hähnle, Wolfram Menzel, Wolfgang
Reif, Gerhard Schellhorn, and Peter H. Schmitt.
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In this chapter we present a project to integrate interactive and automated theorem proving. Its aim is to combine the advantages of the two paradigms. We focus on one particular application domain, which is deduction for the purpose of software verification. Some of the reported facts may not be valid in other domains. We report on the integration concepts and on the experimental results with a prototype implementation.

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According to one common view, information security comes down to technical measures. Given better access control policy models, formal proofs of cryptographic protocols, approved firewalls, better ways of detecting intrusions and malicious code, and better tools for system evaluation and assurance, the problems can be solved. The author puts forward a contrary view: information insecurity is at least as much due to perverse incentives. Many of the problems can be explained more clearly and convincingly using the language of microeconomics: network externalities, asymmetric information, moral hazard, adverse selection, liability dumping and the tragedy of the commons.

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The cut elimination theorem for predicate calculus states that every proof may be replaced by one which does not involve use of the cut rule. This theorem no longer holds when the system is extended with Peano's axioms to give a formalisation for arithmetic. The problem of generalisation results, since arbitrary formulae can be cut in. This makes theoremproving very difficult  one solution is to embed arithmetic in a stronger system, where cut elimination holds. This paper describes a system based on the omega rule, and shows that certain statements are provable in this system which are not provable in Peano arithmetic without cut. Moreover, an important application is presented in the form of a new method of generalisation by means of "guiding proofs" in the stronger system, which sometimes succeeds in producing proofs in the original system when other methods fail. The implementation of such a system is also described.

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Symbolic model checking has become a successful technique for verifying large finite state systems up to more than 10^{2}0 states. The key idea of this method is that extremely large sets can often be efficiently represented with propositional formulas. Most tools implement these formulas by means of binary decision diagrams (BDDs), which have therefore become a key data structure in modern VLSI CAD systems.

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General purpose theorem provers provide advanced facilities for proving properties about specifications, and may therefore be a valuable tool in formal program development. However, these provers generally lack many of the useful structuring mechanisms found in functional programming or specification languages. This paper presents a constructive approach to adding theory morphisms and parametrisation to theorem provers, while preserving the proof support and consistency of the prover. The approach is implemented in Isabelle and illustrated by examples of an algorithm design rule and of the modular development of computational effects for imperative language features based on monads.

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Abstract interpretation is a formal method that enables the static and automatic determination of runtime properties of programs. This method uses a characterization of program invariants as least and greatest fixed points of continuous functions over complete lattices of program properties. In this paper, we study precise and efficient chaotic iteration strategies for computing such fixed points when lattices are of infinite height and speedup techniques, known as widening and narrowing, have to be used. These strategies are based on a weak topological ordering of the dependency graph of the system of semantic equations associated with the program and minimize the loss in precision due to the use of widening operators. We discuss complexity and implementation issues and give precise upper bounds on the complexity of the intraprocedural and interprocedural abstract interpretation of higherorder programs based on the structure of their control flow graph.

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Simple type theory, also known as higherorder logic, is a natural extension of firstorder logic which is simple, elegant, highly expressive, and practical. This paper surveys the virtues of simple type theory and attempts to show that simple type theory is an attractive alternative to firstorder logic for practicalminded scientists, engineers, and mathematicians. It recommends that simple type theory be incorporated into introductory logic courses offered by mathematics departments and into the undergraduate curricula for computer science and software engineering students.

[112]  V. A. Feldman and D. Harel. A probabilistic dynamic logic. Journal of Computer and System Sciences, 28(2):193–215, 1984. [ bib ] 
[113] 
Warren E. Ferguson, Jesse Bingham, Levent Erkok, John R. Harrison, and Joe
LeslieHurd.
Digit serial methods with applications to division and square root.
IEEE Transactions on Computers, 67(3):449–456, March 2018.
[ bib 
DOI 
http ]
We present a generic digit serial method (DSM) to compute the digits of a real number V. Bounds on these digits, and on the errors in the associated estimates of V formed from these digits, are derived. To illustrate our results, we derive bounds for a parameterized family of highradix algorithms for division and square root. These bounds enable hardware designers to determine, for example, whether a given choice of DSM parameters allows rapid formation and rounding of approximations to V.

[114] 
James H. Fetzer.
Program verification: the very idea.
Communications of the ACM, 31(9):1048–1063, 1988.
[ bib 
DOI ]
The notion of program verification appears to trade upon an equivocation. Algorithms, as logical structures, are appropriate subjects for deductive verification. Programs, as causal models of those structures, are not. The success of program verification as a generally applicable and completely reliable method for guaranteeing program performance is not even a theoretical possibility.

[115]  FIDE. The FIDE Handbook, chapter E.I. The Laws of Chess. FIDE, 2004. Available for download from the FIDE website. [ bib  http ] 
[116]  C. Fidge and C. Shankland. But what if I don't want to wait forever? Formal Aspects of Computing, to appear in 2003. [ bib ] 
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[118]  George S. Fishman. Monte Carlo: Concepts, Algorithms and Applications. Springer, 1996. [ bib ] 
[119]  J. D. Fleuriot. A Combination of Geometry Theorem Proving and Nonstandard Analysis, with Application to Newton's Principia. Distinguished Dissertations. Springer, 2001. [ bib ] 
[120]  Anthony Fox. Formal specification and verification of ARM6. In Basin and Wolff [591], pages 25–40. [ bib ] 
[121]  Anthony Fox. An algebraic framework for verifying the correctness of hardware with input and output: A formalization in HOL. In et al. [516], pages 157–174. [ bib ] 
[122] 
Adam Fuchs, Avik Chaudhuri, and Jeffrey Foster.
SCanDroid: Automated security certification of Android
applications.
Available from the second author's website, 2009.
[ bib 
http ]
Android is a popular mobiledevice platform developed by Google. Android's application model is designed to encourage applications to share their code and data with other applications. While such sharing can be tightly controlled with permissions, in general users cannot determine what applications will do with their data, and thereby cannot decide what permissions such applications should run with. In this paper we present SCanDroid, a tool for reasoning automatically about the security of Android applications. SCanDroid's analysis is modular to allow incremental checking of applications as they are installed on an Android device. It extracts security specifications from manifests that accompany such applications, and checks whether data flows through those applications are consistent with those specifications. To our knowledge, SCanDroid is the first program analysis tool for Android, and we expect it to be useful for automated security certification of Android applications.

[123] 
Dov M. Gabbay and Hans Jürgen Ohlbach.
Quantifier elimination in second–order predicate logic.
In Bernhard Nebel, Charles Rich, and William Swartout, editors,
Principles of Knowledge Representation and Reasoning (KR92), pages 425–435.
Morgan Kaufmann, 1992.
[ bib 
http ]
An algorithm is presented which eliminates second–order quantifiers over predicate variables in formulae of type exists P1,...,PnF where F is an arbitrary formula of first–order predicate logic. The resulting formula is equivalent to the original formula – if the algorithm terminates. The algorithm can for example be applied to do interpolation, to eliminate the second–order quantifiers in circumscription, to compute the correlations between structures and power structures, to compute semantic properties corresponding to Hilbert axioms in non classical logics and to compute model theoretic semantics for new logics.

[124]  Chris Gathercole and Peter Ross. Small populations over many generations can beat large populations over few generations in genetic programming. http://www.dai.ed.ac.uk/students/chrisg/, 1997. [ bib ] 
[125]  J. T. Gill. Computational complexity of probabilistic Turing machines. SIAM Journal on Computing, 6(4):675–695, December 1977. [ bib ] 
[126]  P. C. Gilmore. A proof method for quantification theory: Its justification and realization. IBM Journal of Research and Development, 4:28–35, 1960. [ bib ] 
[127]  JeanYves Girard. Proofs and types, volume 7 of Cambridge tracts in theoretical computer science. Cambridge University Press, 1989. [ bib ] 
[128]  K. Gödel. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. Monatshefte für Mathematik und Physik, 38:173–198, 1931. [ bib ] 
[129]  K. Gödel. On Formally Undecidable Propositions of Principia Mathematica and Related Systems. Oliver and Boyd, London, 1962. Translated by B. Meltzer. [ bib ] 
[130]  J. Goguen and G. Malcolm. Algebraic Semantics of Imperative Programs. MIT Press, Cambridge, Mass., 1st edition, 1996. [ bib ] 
[131]  E. Goldberg and Y. Novikov. BerkMin: A fast and robust SATsolver. In Design, Automation, and Test in Europe (DATE '02), pages 142–149, March 2002. [ bib  .pdf ] 
[132]  Charles M. Goldie and Richard G. E. Pinch. Communication theory, volume 20 of LMS Student Texts. Cambridge University Press, 1991. [ bib ] 
[133] 
Georges Gonthier.
A computerchecked proof of the four colour theorem.
Available for download at the author's website, 2004.
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.pdf ]
This report gives an account of a successful formalization of the proof of the Four Colour Theorem, which was fully checked by the Coq v7.3.1 proof assistant. This proof is largely based on the mixed mathematics/computer proof of Robertson et al, but contains original contributions as well. This document is organized as follows: section 1 gives a historical introduction to the problem and positions our work in this setting; section 2 defines more precisely what was proved; section 3 explains the broad outline of the proof; section 4 explains how we exploited the features of the Coq assistant to conduct the proof, and gives a brief description of the tactic shell that we used to write our proof scripts; section 5 is a detailed account of the formal proof (for even more details the actual scripts can be consulted); section 6 is a chronological account of how the formal proof was developed; finally, we draw some general conclusions in section 7.

[134]  M. Gordon, R. Milner, L. Morris, M. Newey, and C. Wadsworth. A Metalanguage for interactive proof in LCF. In POPL1978 [565], pages 119–130. [ bib ] 
[135]  M. Gordon, R. Milner, and C. Wadsworth. Edinburgh LCF, volume 78 of Lecture Notes in Computer Science. Springer, 1979. [ bib ] 
[136]  Mike Gordon. Why HigherOrder Logic is a good formalism for specifying and verifying hardware. Technical Report 77, Computer Laboratory, The University of Cambridge, 1985. [ bib ] 
[137]  M. J. C. Gordon. Programming Language Theory and its Implementation. Prentice Hall, 1988. [ bib ] 
[138]  Michael J. C. Gordon. HOL: A proof generating system for higherorder logic. In Graham Birtwistle and P. A. Subrahmanyam, editors, VLSI Specification, Verification and Synthesis, pages 73–128. Kluwer Academic Publishers, Boston, 1988. [ bib ] 
[139]  M. J. C. Gordon. Mechanizing programming logics in higher order logic. In G. Birtwistle and P. A. Subrahmanyam, editors, Current Trends in Hardware Verification and Automated Theorem Proving, pages 387–439. SpringerVerlag, 1989. [ bib  .dvi.gz ] 
[140]  M. J. C. Gordon and T. F. Melham, editors. Introduction to HOL (A theoremproving environment for higher order logic). Cambridge University Press, 1993. [ bib ] 
[141] 
Mike Gordon.
Merging HOL with set theory: preliminary experiments.
Technical Report 353, University of Cambridge Computer Laboratory,
November 1994.
[ bib 
http ]
Set theory is the standard foundation for mathematics, but the majority of general purpose mechanized proof assistants support versions of type theory (higher order logic). Examples include Alf, Automath, Coq, Ehdm, HOL, IMPS, Lambda, LEGO, Nuprl, PVS and Veritas. For many applications type theory works well and provides, for specification, the benefits of typechecking that are wellknown in programming. However, there are areas where types get in the way or seem unmotivated. Furthermore, most people with a scientific or engineering background already know set theory, whereas type theory may appear inaccessable and so be an obstacle to the uptake of proof assistants based on it. This paper describes some experiments (using HOL) in combining set theory and type theory; the aim is to get the best of both worlds in a single system. Three approaches have been tried, all based on an axiomatically specified type V of ZFlike sets: (i) HOL is used without any additions besides V; (ii) an embedding of the HOL logic into V is provided; (iii) HOL axiomatic theories are automatically translated into settheoretic definitional theories. These approaches are illustrated with two examples: the construction of lists and a simple lemma in group theory.

[142] 
M. J. C. Gordon.
Notes on PVS from a HOL perspective.
Available from the author's web page, 1996.
[ bib 
.html ]
During the first week of July 1995 I visited SRI Menlo Park to find out more about PVS (Prototype Verification System). The preceding week I attended LICS95, where I had several talks with Natarajan Shankar accompanied by demos of the system on his SparcBook laptop. This note consists of a somewhat rambling and selective report on some of the things I learned, together with a discussion of their implications for the evolution of the HOL system.

[143] 
Michael J. C. Gordon.
Reachability programming in HOL98 using BDDs.
In Aagaard and Harrison [586], pages 179–196.
[ bib 
http ]
Two methods of programming BDDbased symbolic algorithms in the Hol98 proof assistant are presented. The goal is to provide a platform for implementing intimate combinations of deduction and algorithmic verification, like model checking. The first programming method uses a small kernel of ML functions to convert between BDDs, terms and theorems. It is easy to use and is suitable for rapid prototying experiments. The second method requires lowerlevel programming but can support more efficient calculations. It is based on an LCFlike use of an abstract type to encapsulate rules for manipulating judgements r t > b meaning “logical term t is represented by BDD b with respect to variable order r”. The two methods are illustrated by showing how to perform the standard fixedpoint calculation of the BDD of the set of reachable states of a finite state machine.

[144]  Michael J. C. Gordon. Proof, Language, and Interaction: Essays in Honour of Robin Milner, chapter 6. From LCF to HOL: A Short History. MIT Press, May 2000. [ bib ] 
[145] 
Michael J. C. Gordon.
Programming combinations of deduction and BDDbased symbolic
calculation.
LMS Journal of Computation and Mathematics, 5:56–76, August
2002.
[ bib 
http ]
A generalisation of Milner's `LCF approach' is described. This allows algorithms based on binary decision diagrams (BDDs) to be programmed as derived proof rules in a calculus of representation judgements. The derivation of representation judgements becomes an LCFstyle proof by defining an abstract type for judgements analogous to the LCF type of theorems. The primitive inference rules for representation judgements correspond to the operations provided by an efficient BDD package coded in C (BuDDy). Proof can combine traditional inference with steps inferring representation judgements. The resulting system provides a platform to support a tight and principled integration of theorem proving and model checking. The methods are illustrated by using them to solve all instances of a generalised Missionaries and Cannibals problem.

[146]  Michael J. C. Gordon. PuzzleTool : An example of programming computation and deduction. In Carreño et al. [590], pages 214–229. [ bib ] 
[147] 
Mike Gordon, Joe Hurd, and Konrad Slind.
Executing the formal semantics of the Accellera Property
Specification Language by mechanised theorem proving.
In Geist and Tronci [520], pages 200–215.
[ bib 
http ]
The Accellera Property Specification Language (PSL) is designed for the formal specification of hardware. The Reference Manual contains a formal semantics, which we previously encoded in a machine readable version of higher order logic. In this paper we describe how to `execute' the formal semantics using proof scripts coded in the HOL theorem prover's metalanguage ML. The goal is to see if it is feasible to implement useful tools that work directly from the official semantics by mechanised proof. Such tools will have a high assurance of conforming to the standard. We have implemented two experimental tools: an interpreter that evaluates whether a finite trace w, which may be generated by a simulator, satisfies a PSL formula f (i.e. w ⊨f), and a compiler that converts PSL formulas to checkers in an intermediate format suitable for translation to HDL for inclusion in simulation testbenches. Although our tools use logical deduction and are thus slower than handcrafted implementations, they may be speedy enough for some applications. They can also provide a reference for more efficient implementations.

[148]  Mike Gordon, Juliano Iyoda, Scott Owens, and Konrad Slind. A proofproducing hardware compiler for a subset of higher order logic. In Hurd et al. [595], pages 59–75. [ bib  http ] 
[149]  M. J. C. Gordon. Specification and verification I, 2009. Course notes available from http://www.cl.cam.ac.uk/~mjcg/Teaching/SpecVer1/SpecVer1.html. [ bib ] 
[150]  M. J. C. Gordon. Specification and verification II, 2009. Course notes available from http://www.cl.cam.ac.uk/~mjcg/Teaching/SpecVer2/SpecVer2.html. [ bib ] 
[151]  Daniel Gorenstein, Richard Lyons, and Ronald Solomon. The Classification of the Finite Simple Groups. AMS, 1994. [ bib  http ] 
[152] 
Paul Graunke.
Verified safety and information flow of a block device.
In SSV2008 [578], pages 187–202.
[ bib 
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This work reports on the author's experience designing, implementing, and formally verifying a lowlevel piece of system software. The timing model and the adaptation of an existing information flow policy to a monadic framework are reasonably novel. Interactive compilation through equational rewriting worked well in practice. Finally, the project uncovered some potential areas for improving interactive theorem provers.

[153]  Penny Grubb and Armstrong A. Takang. Software Maintenance: Concepts and Practice. World Scientific Publishing Company, 2nd edition, July 2003. [ bib ] 
[154]  Elsa Gunter. Doing algebra in simple type theory. Technical Report MSCIS8938, Logic & Computation 09, Department of Computer and Information Science, University of Pennsylvania, 1989. [ bib ] 
[155]  Joshua D. Guttman, Amy L. Herzog, John D. Ramsdell, and Clement W. Skorupka. Verifying information flow goals in SecurityEnhanced Linux. Journal of Computer Security, 13(1):115–134, 2005. [ bib  .pdf ] 
[156] 
Florian Haftmann.
From higherorder logic to Haskell: There and back again.
In Gallagher and Voigtländer [563], pages 155–158.
[ bib 
.pdf ]
We present two tools which together allow reasoning about (a substantial subset of) Haskell programs. One is the code generator of the proof assistant Isabelle, which turns specifications formulated in Isabelle's higherorder logic into executable Haskell source text; the other is Haskabelle, a tool to translate programs written in Haskell into Isabelle specifications. The translation from Isabelle to Haskell directly benefits from the rigorous correctness approach of a proof assistant: generated Haskell programs are always partially correct w.r.t. to the specification from which they are generated.

[157] 
Petr Hájek.
Metamathematics of Fuzzy Logic, volume 4 of Trends in
Logic.
Kluwer Academic Publishers, Dordrecht, August 1998.
[ bib 
http ]
This book presents a systestematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. Some important systems of realvalued propositional and predicate calculus are defined and investigated. The aim is to show that fuzzy logic as a logic of imprecise (vague) propositions does have welldeveloped formal foundations and that most things usually named `fuzzy inference' can be naturally understood as logical deduction.

[158] 
Thomas C. Hales.
Introduction to the Flyspeck project.
In Coquand et al. [521].
[ bib 
http ]
This article gives an introduction to a longterm project called Flyspeck, whose purpose is to give a formal verification of the Kepler Conjecture. The Kepler Conjecture asserts that the density of a packing of equal radius balls in three dimensions cannot exceed pi/sqrt18. The original proof of the Kepler Conjecture, from 1998, relies extensively on computer calculations. Because the proof relies on relatively few external results, it is a natural choice for a formalization effort.

[159] 
Joseph Y. Halpern.
An analysis of firstorder logics of probability.
Artificial Intelligence, 1990.
[ bib 
http ]
We consider two approaches to giving semantics to firstorder logics of probability. The first approach puts a probability on the domain, and is appropriate for giving semantics to formulas involving statistical information such as “The probability that a randomly chosen bird flies is greater than .9.” The second approach puts a probability on possible worlds, and is appropriate for giving semantics to formulas describing degrees of belief, such as “The probability that Tweety (a particular bird) flies is greater than .9.” We show that the two approaches can be easily combined, allowing us to reason in a straightforward way about statistical information and degrees of belief. We then consider axiomatizing these logics. In general, it can be shown that no complete axiomatization is possible. We provide axiom systems that are sound and complete in cases where a complete axiomatization is possible, showing that they do allow us to capture a great deal of interesting reasoning about probability.

[160]  Darrel Hankerson, Alfred Menezes, and Scott Vanstone. Guide to Elliptic Curve Cryptography. Springer, 2003. [ bib  http ] 
[161] 
R. W. Hansell.
Borel measurable mappings for nonseparable metric spaces.
Transactions of the American Mathematical Society,
161:145–169, November 1971.
[ bib 
http ]
"One of the key papers in nonseparable theory. [To show that f IN measurable E U implies countable range] One may use Section 2, Corollary 7, with the observation that [0,1] satisfies the assumptions and, in your situation, the collection f^(1)(x), x∈X is even completely additiveBorel"

[162]  G. H. Hardy. A Mathematician's Apology, reprinted with a foreword by C. P. Snow. Cambridge University Press, 1993. [ bib ] 
[163] 
John Harrison.
Metatheory and reflection in theorem proving: A survey and critique.
Technical Report CRC053, SRI Cambridge, Millers Yard, Cambridge, UK,
1995.
[ bib 
.html ]
One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fullyexpansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof assistant.

[164] 
John Harrison.
Binary decision diagrams as a HOL derived rule.
The Computer Journal, 38:162–170, 1995.
[ bib 
.html ]
Binary Decision Diagrams (BDDs) are a representation for Boolean formulas which makes many operations, in particular tautologychecking, surprisingly efficient in important practical cases. In contrast to such custom decision procedures, the HOL theorem prover expands all proofs out to a sequence of extremely simple primitive inferences. In this paper we describe how the BDD algorithm may be adapted to comply with such strictures, helping us to understand the strengths and limitations of the HOL approach.

[165] 
John Harrison.
Optimizing proof search in model elimination.
In McRobbie and Slaney [511], pages 313–327.
[ bib 
.html ]
Many implementations of model elimination perform proof search by iteratively increasing a bound on the total size of the proof. We propose an optimized version of this search mode using a simple divideandconquer refinement. Optimized and unoptimized modes are compared, together with depthbounded and bestfirst search, over the entire TPTP problem library. The optimized sizebounded mode seems to be the overall winner, but for each strategy there are problems on which it performs best. Some attempt is made to analyze why. We emphasize that our optimization, and other implementation techniques like caching, are rather general: they are not dependent on the details of model elimination, or even that the search is concerned with theorem proving. As such, we believe that this study is a useful complement to research on extending the model elimination calculus.

[166] 
John Harrison.
Formalized mathematics.
Technical Report 36, Turku Centre for Computer Science (TUCS),
Lemminkäisenkatu 14 A, FIN20520 Turku, Finland, 1996.
[ bib 
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It is generally accepted that in principle it's possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.

[167] 
John Harrison.
A Mizar mode for HOL.
In von Wright et al. [581], pages 203–220.
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The HOL theorem prover is implemented in the LCF style, meaning that all inference is ultimately reduced to a collection of very simple (forward) primitive inference rules. By programming it is possible to build alternative means of proving theorems on top, while preserving security. Existing HOL proofs styles are, however, very different from those used in textbooks. Here we describe the addition of another proof style, inspired by Mizar. We believe the resulting system combines some of the best features of both HOL and Mizar.

[168] 
John Harrison.
Proof style.
In Giménex and PaulinMohring [601], pages 154–172.
[ bib 
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We are concerned with how computer theorem provers should expect users to communicate proofs to them. There are many stylistic choices that still allow the machine to generate a completely formal proof object. The most obvious choice is the amount of guidance required from the user, or from the machine perspective, the degree of automation provided. But another important consideration, which we consider particularly significant, is the bias towards a `procedural' or `declarative' proof style. We will explore this choice in depth, and discuss the strengths and weaknesses of declarative and procedural styles for proofs in pure mathematics and for verification applications. We conclude with a brief summary of our own experiments in trying to combine both approaches.

[169] 
John Harrison.
HOL light: A tutorial introduction.
In Srivas and Camilleri [528], pages 265–269.
[ bib 
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HOL Light is a new version of the HOL theorem prover. While retaining the reliability and programmability of earlier versions, it is more elegant, lightweight, powerful and automatic; it will be the basis for the Cambridge component of the HOL2000 initiative to develop the next generation of HOL theorem provers. HOL Light is written in CAML Light, and so will run well even on small machines, e.g. PCs and Macintoshes with a few megabytes of RAM. This is in stark contrast to the resourcehungry systems which are the norm in this field, other versions of HOL included. Among the new features of this version are a powerful simplifier, effective first order automation, simple higherorder matching and very general support for inductive and recursive definitions.

[170]  John Harrison. Floating point verification in HOL light: the exponential function. Technical Report 428, University of Cambridge Computer Laboratory, 1997. [ bib  .html ] 
[171]  John Harrison. Theorem Proving with the Real Numbers (Distinguished dissertations). Springer, 1998. [ bib  .html ] 
[172]  John Harrison. The HOL Light Manual (1.0), May 1998. [ bib  http ] 
[173]  John Harrison. Formalizing Dijkstra. In Grundy and Newey [584], pages 171–188. [ bib  .html ] 
[174]  John Harrison. A HOL theory of Euclidean space. In Hurd and Melham [594], pages 114–129. [ bib  .html ] 
[175]  John Harrison. Floatingpoint verification using theorem proving. In Bernardo and Cimatti [573], pages 211–242. [ bib  .html ] 
[176]  John Harrison. Verifying nonlinear real formulas via sums of squares. In Schneider and Brandt [596], pages 102–118. [ bib  .html ] 
[177]  John Harrison. Formalizing basic complex analysis. In Matuszewski and Zalewska [600], pages 151–165. [ bib  .html ] 
[178] 
John Harrison.
Formalizing an analytic proof of the prime number theorem.
Special Issue of the Journal of Automated Reasoning,
43(3):243–261, October 2009.
[ bib 
http ]
We describe the computer formalization of a complexanalytic proof of the Prime Number Theorem (PNT), a classic result from number theory characterizing the asymptotic density of the primes. The formalization, conducted using the HOL Light theorem prover, proceeds from the most basic axioms for mathematics yet builds from that foundation to develop the necessary analytic machinery including Cauchy’s integral formula, so that we are able to formalize a direct, modern and elegant proof instead of the more involved ‘elementary’ ErdösSelberg argument. As well as setting the work in context and describing the highlights of the formalization, we analyze the relationship between the formal proof and its informal counterpart and so attempt to derive some general lessons about the formalization of mathematics.

[179]  Sergiu Hart, Micha Sharir, and Amir Pnueli. Termination of probabilistic concurrent programs. ACM Transactions on Programming Languages and Systems (TOPLAS), 5(3):356–380, July 1983. [ bib ] 
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E. A. Heinz.
Endgame databases and efficient index schemes.
ICCA Journal, 22(1):22–32, March 1999.
[ bib 
.ps.gz ]
Endgame databases have become an integral part of modern chess programs during the past few years. Since the early 1990s two different kinds of endgame databases are publicly available, namely Edwards' socalled "tablebases" and Thompson's collection of 5piece databases. Although Thompson's databases enjoy much wider international fame, most current chess programs use tablebases because they integrate far better with the rest of the search. For the benefit of all those enthusi asts who intend to incorporate endgame databases into their own chess programs, this article describes the index schemes of Edwards' tablebases and Thompson's databases in detail, explains their differences, and provides a comparative evaluation of both.

[185]  Jörg Heitkötter and David Beasley. The hitchhiker's guide to evolutionary computation: A list of frequently asked questions (faq). USENET: comp.ai.genetic, 1998. [ bib  ftp ] 
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[200] 
Joe Hurd.
The Real Number Theories in hol98, November 1998.
Part of the documentation for the hol98 theorem prover.
[ bib ]
A description of the hol98 reals library, ported partly from the reals library in HOL90 and partly from the real library in hollight.

[201] 
Joe Hurd.
Integrating Gandalf and HOL.
In Bertot et al. [585], pages 311–321.
[ bib 
http ]
Gandalf is a firstorder resolution theoremprover, optimized for speed and specializing in manipulations of large clauses. In this paper I describe GANDALF_TAC, a HOL tactic that proves goals by calling Gandalf and mirroring the resulting proofs in HOL. This call can occur over a network, and a Gandalf server may be set up servicing multiple HOL clients. In addition, the translation of the Gandalf proof into HOL fits in with the LCF model and guarantees logical consistency.

[202]  Joe Hurd. Integrating Gandalf and HOL. Technical Report 461, University of Cambridge Computer Laboratory, May 1999. Second edition. [ bib  .html ] 
[203]  Joe Hurd. Congruence classes with logic variables. In Aagaard et al. [587], pages 87–101. [ bib ] 
[204] 
Joe Hurd.
Lightweight probability theory for verification.
In Aagaard et al. [587], pages 103–113.
[ bib ]
There are many algorithms that make use of probabilistic choice, but a lack of tools available to specify and verify their operation. The primary contribution of this paper is a lightweight modelling of such algorithms in higherorder logic, together with some key properties that enable verification. The theory is applied to a uniform random number generator and some basic properties are established. As a secondary contribution, all the theory developed has been mechanized in the hol98 theoremprover.

[205] 
Joe Hurd.
The Probability Theories in hol98, June 2000.
Part of the documentation for the hol98 theorem prover.
[ bib ]
A description of the hol98 probability library.

[206] 
Joe Hurd.
Congruence classes with logic variables.
Logic Journal of the IGPL, 9(1):59–75, January 2001.
[ bib 
http ]
We are improving equality reasoning in automatic theoremprovers, and congruence classes provide an efficient storage mechanism for terms, as well as the congruence closure decision procedure. We describe the technical steps involved in integrating logic variables with congruence classes, and present an algorithm that can be proved to find all matches between classes (modulo certain equalities). An application of this algorithm makes possible a percolation algorithm for undirected rewriting in minimal space; this is described and an implementation in hol98 is examined in some detail.

[207] 
Joe Hurd.
Predicate subtyping with predicate sets.
In Boulton and Jackson [588], pages 265–280.
[ bib 
http ]
We show how PVSstyle predicate subtyping can be simulated in HOL using predicate sets, and explain how to perform subtype checking using this model. We illustrate some applications of this to specification and verification in HOL, and also demonstrate some limits of the approach. Finally we show preliminary results of a subtype checker that has been integrated into a contextual rewriter.

[208]  Joe Hurd. Verification of the MillerRabin probabilistic primality test. In Boulton and Jackson [589], pages 223–238. [ bib ] 
[209] 
Joe Hurd.
Formal Verification of Probabilistic Algorithms.
PhD thesis, University of Cambridge, 2002.
[ bib 
http ]
This thesis shows how probabilistic algorithms can be formally verified using a mechanical theorem prover.

[210] 
Joe Hurd.
A formal approach to probabilistic termination.
In Carreño et al. [590], pages 230–245.
[ bib 
http ]
We present a probabilistic version of the while loop, in the context of our mechanized framework for verifying probabilistic programs. The while loop preserves useful program properties of measurability and independence, provided a certain condition is met. This condition is naturally interpreted as “from every starting state, the while loop will terminate with probability 1”, and we compare it to other probabilistic termination conditions in the literature. For illustration, we verify in HOL two example probabilistic algorithms that necessarily rely on probabilistic termination: an algorithm to sample the Bernoulli(p) distribution using coinflips; and the symmetric simple random walk.

[211] 
Joe Hurd.
HOL theorem prover case study: Verifying probabilistic programs.
In Norman et al. [507], pages 83–92.
[ bib ]
The focus of this paper is the question: “How suited is the HOL theorem prover to the verification of probabilistic programs?” To answer this, we give a brief introduction to our model of probabilistic programs in HOL, and then compare this approach to other formal tools that have been used to verify probabilistic programs: the Prism model checker, the Coq theorem prover, and the B method.

[212]  Joe Hurd. Fast normalization in the HOL theorem prover. In Walsh [505]. An extended abstract. [ bib  http ] 
[213]  Joe Hurd. An LCFstyle interface between HOL and firstorder logic. In Voronkov [515], pages 134–138. [ bib  http ] 
[214] 
Joe Hurd.
Verification of the MillerRabin probabilistic primality test.
Journal of Logic and Algebraic Programming, 50(1–2):3–21,
May–August 2003.
Special issue on Probabilistic Techniques for the Design and Analysis
of Systems.
[ bib 
http ]
Using the HOL theorem prover, we apply our formalization of probability theory to specify and verify the MillerRabin probabilistic primality test. The version of the test commonly found in algorithm textbooks implicitly accepts probabilistic termination, but our own verified implementation satisfies the stronger property of guaranteed termination. Completing the proof of correctness requires a significant body of group theory and computational number theory to be formalized in the theorem prover. Once verified, the primality test can either be executed in the logic (using rewriting) and used to prove the compositeness of numbers, or manually extracted to Standard ML and used to find highly probable primes.

[215] 
Joe Hurd.
Using inequalities as term ordering constraints.
Technical Report 567, University of Cambridge Computer Laboratory,
June 2003.
[ bib 
http ]
In this paper we show how linear inequalities can be used to approximate KnuthBendix term ordering constraints, and how term operations such as substitution can be carried out on systems of inequalities. Using this representation allows an offtheshelf linear arithmetic decision procedure to check the satisfiability of a set of ordering constraints. We present a formal description of a resolution calculus where systems of inequalities are used to constrain clauses, and implement this using the Omega test as a satisfiability checker. We give the results of an experiment over problems in the TPTP archive, comparing the practical performance of the resolution calculus with and without inherited inequality constraints.

[216]  Joe Hurd. Formal verification of probabilistic algorithms. Technical Report 566, University of Cambridge Computer Laboratory, May 2003. [ bib  .html ] 
[217] 
Joe Hurd.
Firstorder proof tactics in higherorder logic theorem provers.
In Archer et al. [579], pages 56–68.
[ bib 
http ]
In this paper we evaluate the effectiveness of firstorder proof procedures when used as tactics for proving subgoals in a higherorder logic interactive theorem prover. We first motivate why such firstorder proof tactics are useful, and then describe the core integrating technology: an `LCFstyle' logical kernel for clausal firstorder logic. This allows the choice of different logical mappings between higherorder logic and firstorder logic to be used depending on the subgoal, and also enables several different firstorder proof procedures to cooperate on constructing the proof. This work was carried out using the HOL4 theorem prover; we comment on the ease of transferring the technology to other higherorder logic theorem provers.

[218]  Joe Hurd, Annabelle McIver, and Carroll Morgan. Probabilistic guarded commands mechanized in HOL. In Cerone and Pierro [569], pages 95–111. [ bib ] 
[219] 
Joe Hurd.
Compiling HOL4 to native code.
In Slind [593].
[ bib 
http ]
We present a framework for extracting and compiling proof tools and theories from a higher order logic theorem prover, so that the theorem prover can be used as a platform for supporting reasoning in other applications. The framework is demonstrated on a small application that uses HOL4 to find proofs of arbitrary first order logic formulas.

[220] 
Joe Hurd.
Formal verification of chess endgame databases.
In Hurd et al. [595], pages 85–100.
[ bib 
http ]
Chess endgame databases store the number of moves required to force checkmate for all winning positions: with such a database it is possible to play perfect chess. This paper describes a method to construct endgame databases that are formally verified to logically follow from the laws of chess. The method employs a theorem prover to model the laws of chess and ensure that the construction is correct, and also a BDD engine to compactly represent and calculate with large sets of chess positions. An implementation using the HOL4 theorem prover and the BuDDY BDD engine is able to solve all four piece pawnless endgames.

[221] 
Joe Hurd.
Formalizing elliptic curve cryptography in higher order logic.
Available from the author's website, October 2005.
[ bib 
http ]
Formalizing a mathematical theory using a theorem prover is a necessary first step to proving the correctness of programs that refer to that theory in their specification. This report demonstrates how the mathematical theory of elliptic curves and their application to cryptography can be formalized in higher order logic. This formal development is mechanized using the HOL4 theorem prover, resulting in a collection of formally verified functional programs (expressed as higher order logic functions) that correctly implement the primitive operations of elliptic curve cryptography.

[222] 
Joe Hurd, Annabelle McIver, and Carroll Morgan.
Probabilistic guarded commands mechanized in HOL.
Theoretical Computer Science, 346:96–112, November 2005.
[ bib 
http ]
The probabilistic guardedcommand language pGCL contains both demonic and probabilistic nondeterminism, which makes it suitable for reasoning about distributed random algorithms Proofs are based on weakest precondition semantics, using an underlying logic of real (rather than Boolean) valued functions.

[223]  Joe Hurd. First order proof for higher order theorem provers (abstract). In Benzmueller et al. [524], pages 1–3. [ bib ] 
[224]  Joe Hurd. System description: The Metis proof tactic. In Benzmueller et al. [524], pages 103–104. [ bib ] 
[225] 
Joe Hurd, Mike Gordon, and Anthony Fox.
Formalized elliptic curve cryptography.
In HCSS2006 [538].
[ bib 
http ]
Formalizing a mathematical theory is a necessary first step to proving the correctness of programs that refer to that theory in their specification. This paper demonstrates how the mathematical theory of elliptic curves and their application to cryptography can be formalized in higher order logic. This formal development is mechanized using the HOL4 theorem prover, resulting in a collection of higher order logic functions that correctly implement the primitive operations of elliptic curve cryptography.

[226]  Joe Hurd. Book review: Rippling: Metalevel guidance for mathematical reasoning by A. Bundy, D. Basin, D. Hutter and A. Ireland. Bulletin of Symbolic Logic, 12(3):498–499, 2006. [ bib  http ] 
[227] 
Joe Hurd.
Embedding Cryptol in higher order logic.
Available from the author's website, March 2007.
[ bib 
http ]
This report surveys existing approaches to embedding Cryptol programs in higher order logic, and presents a new approach that aims to simplify as much as possible reasoning about the embedded programs.

[228] 
Joe Hurd.
Proof pearl: The termination analysis of TERMINATOR.
In Schneider and Brandt [596], pages 151–156.
[ bib 
http ]
TERMINATOR is a static analysis tool developed by Microsoft Research for proving termination of Windows device drivers written in C. This proof pearl describes a formalization in higher order logic of the program analysis employed by TERMINATOR, and verifies that if the analysis succeeds then program termination logically follows.

[229]  Joe Hurd, Anthony Fox, Mike Gordon, and Konrad Slind. ARM verification (abstract). In HCSS2007 [539]. [ bib  http ] 
[230] 
Joe Hurd.
OpenTheory: Package management for higher order logic theories.
In Reis and Théry [564], pages 31–37.
[ bib 
http ]
Interactive theorem proving has grown from toy examples to major projects formalizing mathematics and verifying software, and there is now a critical need for theory engineering techniques to support these efforts. This paper introduces the OpenTheory project, which aims to provide an effective package management system for logical theories. The OpenTheory article format allows higher order logic theories to be exported from one theorem prover, compressed by a standalone tool, and imported into a different theorem prover. Articles naturally support theory interpretations, which is the mechanism by which theories can be cleanly transferred from one theorem prover context to another, and which also leads to more efficient developments of standard theories.

[231] 
Joe Hurd, Magnus Carlsson, Sigbjorn Finne, Brett Letner, Joel Stanley, and
Peter White.
Policy DSL: Highlevel specifications of information flows for
security policies.
In HCSS2009 [540].
[ bib 
http ]
SELinux security policies are powerful tools to implement properties such as process confinement and least privilege. They can also be used to support MLS policies on SELinux. However, the policies are very complex, and creating them is a difficult and errorprone process. Furthermore, it is not possible to state explicit constraints on an SELinux policy such as “information flowing to the network must be encrypted”.

[232] 
Joe Hurd and Guy Haworth.
Data assurance in opaque computations.
In Van den Herik and Spronck [500], pages 221–231.
[ bib 
http ]
The chess endgame is increasingly being seen through the lens of, and therefore effectively defined by, a data `model' of itself. It is vital that such models are clearly faithful to the reality they purport to represent. This paper examines that issue and systems engineering responses to it, using the chess endgame as the exemplar scenario. A structured survey has been carried out of the intrinsic challenges and complexity of creating endgame data by reviewing the past pattern of errors during work in progress, surfacing in publications and occurring after the data was generated. Specific measures are proposed to counter observed classes of errorrisk, including a preliminary survey of techniques for using stateoftheart verification tools to generate EGTs that are correct by construction. The approach may be applied generically beyond the game domain.

[233] 
Joe Hurd.
Composable packages for higher order logic theories.
In Aderhold et al. [603].
[ bib 
http ]
Interactive theorem proving is tackling ever larger formalization and verification projects, and there is a critical need for theory engineering techniques to support these efforts. One such technique is effective package management, which has the potential to simplify the development of logical theories by precisely checking dependencies and promoting reuse. This paper introduces a domainspecific language for defining composable packages of higher order logic theories, which is designed to naturally handle the complex dependency structures that often arise in theory development. The package composition language functions as a module system for theories, and the paper presents a welldefined semantics for the supported operations. Preliminary tests of the package language and its toolset have been made by packaging the theories distributed with the HOL Light theorem prover. This experience is described, leading to some initial theory engineering discussion on the ideal properties of a reusable theory.

[234]  Joe Hurd. Evaluation opportunities in mechanized theories (invited talk abstract). In McGuinness et al. [523]. [ bib  http ] 
[235]  Joe Hurd. OpenTheory Article Format, August 2010. Available for download at http://www.gilith.com/opentheory/article.html. [ bib  .html ] 
[236] 
Joe Hurd.
The OpenTheory standard theory library.
In Bobaru et al. [562], pages 177–191.
[ bib 
http ]
Interactive theorem proving is tackling ever larger formalization and verification projects, and there is a critical need for theory engineering techniques to support these efforts. One such technique is crossprover package management, which has the potential to simplify the development of logical theories and effectively share theories between different theorem prover implementations. The OpenTheory project has developed standards for packaging theories of the higher order logic implemented by the HOL family of theorem provers. What is currently missing is a standard theory library that can serve as a published contract of interoperability and contain proofs of basic properties that would otherwise appear in many theory packages. The core contribution of this paper is the presentation of a standard theory library for higher order logic represented as an OpenTheory package. We identify the core theory set of the HOL family of theorem provers, and describe the process of instrumenting the HOL Light theorem prover to extract a standardized version of its core theory development. We profile the axioms and theorems of our standard theory library and investigate the performance cost of separating the standard theory library into coherent hierarchical theory packages.

[237]  Joe Hurd. FUSE: Interapplication security for android (abstract). In Launchbury and Richards [541], pages 53–54. [ bib  http ] 
[238]  Michael Huth. The interval domain: A matchmaker for aCTL and aPCTL. In Rance Cleaveland, Michael Mislove, and Philip Mulry, editors, US  Brazil Joint Workshops on the Formal Foundations of Software Systems, volume 14 of Electronic Notes in Theoretical Computer Science. Elsevier, 2000. [ bib  .html ] 
[239] 
Graham Hutton and Erik Meijer.
Monadic parser combinators.
Technical Report NOTTCSTR964, Department of Computer Science,
University of Nottingham, 1996.
[ bib 
http ]
In functional programming, a popular approach to building recursive descent parsers is to model parsers as functions, and to define higherorder functions (or combinators) that implement grammar constructions such as sequencing, choice, and repetition. Such parsers form an instance of a monad, an algebraic structure from mathematics that has proved useful for addressing a number of computational problems. The purpose of this report is to provide a stepbystep tutorial on the monadic approach to building functional parsers, and to explain some of the benefits that result from exploiting monads. No prior knowledge of parser combinators or of monads is assumed. Indeed, this report can also be viewed as a first introduction to the use of monads in programming.

[240] 
Paul B. Jackson.
Expressive typing and abstract theories in Nuprl and PVS
(tutorial).
In von Wright et al. [581].
[ bib 
.ps.gz ]
This 90min tutorial covered two of the more novel aspects of the Nuprl and PVS theorem provers:

[241]  Mateja Jamnik, Alan Bundy, and Ian Green. On automating diagrammatic proofs of arithmetic arguments. Journal of Logic, Language and Information, 8(3):297–321, 1999. Also available as Department of Artificial Intelligence Research Paper No. 910. [ bib  .ps ] 
[242]  Claus Skaanning Jensen, Augustine Kong, and Uffe Kjaerulff. Blocking Gibbs sampling in very large probabilistic expert systems. Technical Report R932031, Aalborg University, October 1993. [ bib ] 
[243] 
D. S. Johnson.
A catalog of complexity classes.
In J. van Leeuwen, editor, Handbook of Theoretical Computer
Science, Volume A: Algorithms and Complexity, chapter 9, pages 67–161.
Elsevier and The MIT Press (copublishers), 1990.
[ bib ]
"Johnson presents an extensive survey of computational complexity classes. Of particular interest here is his discussion of randomized, probabilistic, and stochastic complexity classes."

[244]  Peter T. Johnstone. Notes on logic and set theory. Cambridge University Press, 1987. [ bib ] 
[245]  Claire Jones. Probabilistic NonDeterminism. PhD thesis, University of Edinburgh, 1990. [ bib  http ] 
[246]  Michael D. Jones. Restricted types for HOL. In Gunter [583]. [ bib  .html ] 
[247]  F. Kammüller and L. C. Paulson. A formal proof of Sylow's first theorem – an experiment in abstract algebra with Isabelle HOL. Journal of Automated Reasoning, 23(34):235–264, 1999. [ bib ] 
[248]  F. Kammüller. Modular reasoning in Isabelle. In McAllester [514]. [ bib ] 
[249]  David R. Karger. Global mincuts in RNC, and other ramifications of a simple mincut algorithm. In Proceedings of the 4th Annual ACMSIAM Symposium on Discrete Algorithms (SODA '93), pages 21–30, Austin, TX, USA, January 1993. SIAM. [ bib ] 
[250]  R. M. Karp. The probabilistic analysis of some combinatorial search algorithms. In Traub [452], pages 1–20. [ bib ] 
[251]  Matt Kaufmann and J Strother Moore. An industrial strength theorem prover for a logic based on Common Lisp. IEEE Transactions on Software Engineering, 23(4):203–213, April 1997. [ bib  http ] 
[252]  Matt Kaufmann and J Strother Moore. A precise description of the ACL2 logic, 1998. [ bib  .ps.Z ] 
[253]  Matt Kaufmann, Panagiotis Manolios, and J Strother Moore. ComputerAided Reasoning: An Approach. Kluwer Academic Publishers, June 2000. [ bib ] 
[254]  Matt Kaufmann, Panagiotis Manolios, and J Strother Moore, editors. ComputerAided Reasoning: ACL2 Case Studies. Kluwer Academic Publishers, June 2000. [ bib ] 
[255]  F. P. Kelly. Notes on effective bandwidths. In F.P. Kelly, S. Zachary, and I.B. Ziedins, editors, Stochastic Networks: Theory and Applications, number 4 in Royal Statistical Society Lecture Notes Series, pages 141–168. Oxford University Press, 1996. [ bib ] 
[256]  H. J. Keisler. Probability quantifiers. In J. Barwise and S. Feferman, editors, ModelTheoretic Logics, pages 509–556. Springer, New York, 1985. [ bib ] 
[257] 
D. J. King and R. D. Arthan.
Development of practical verification tools.
ICL Systems Journal, 11(1), May 1996.
[ bib 
.html ]
Current best practice for highassurance security and safetycritical software often requires the use of machinechecked rigorous mathematical techniques. Unfortunately, while there have been some notable successes, the provision of software tools that adequately support such techniques is a hard research problem, albeit one that is slowly being solved. This paper describes some contributions to this area of research and development in ICL in recent years. This research builds both on ICL’s own experiences in building and using tools to support security applications and on work carried out by the Defence Research Agency into notations and methods for program verification.

[258] 
Steve King, Jonathan Hammond, Rod Chapman, and Andy Pryor.
Is proof more costeffective than testing?
IEEE Transactions on Software Engineering, 26(8):675–686,
August 2000.
[ bib ]
This paper describes the use of formal development methods on an industrial safetycritical application. The Z notation was used for documenting the system specification and part of the design, and the SPARK1 subset of Ada was used for coding. However, perhaps the most distinctive nature of the project lies in the amount of proof that was carried out: proofs were carried out both at the Z levelapproximately 150 proofs in 500 pagesand at the SPARK code levelapproximately 9,000 verification conditions generated and discharged. The project was carried out under UK Interim Defence Standards 0055 and 0056, which require the use of formal methods on safetycritical applications. It is believed to be the first to be completed against the rigorous demands of the 1991 version of these standards. The paper includes comparisons of proof with the various types of testing employed, in terms of their efficiency at finding faults. The most striking result is that the Z proof appears to be substantially more efficient at finding faults than the most efficient testing phase. Given the importance of early fault detection, we believe this helps to show the significant benefit and practicality of largescale proof on projects of this kind.

[259] 
Gerwin Klein, Kevin Elphinstone, Gernot Heiser, June Andronick, David Cock,
Philip Derrin, Dhammika Elkaduwe, Kai Engelhardt, Rafal Kolanski, Michael
Norrish, Thomas Sewell, Harvey Tuch, and Simon Winwood.
seL4: Formal verification of an OS kernel.
In Matthews and Anderson [576], pages 207–220.
[ bib 
http ]
Complete formal verification is the only way to guarantee that a system is free of programming errors.

[260]  D. E. Knuth and P. B. Bendix. Simple word problems in universal algebra. In J. Leech, editor, Computational problems in abstract algebra, pages 263–297. Pergamon Press, Elmsford, N.Y., 1970. [ bib ] 
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[270]  Jesper Torp Kristensen. Generation and compression of endgame tables in chess with fast random access using OBDDs. Master's thesis, University of Aarhus, Department of Computer Science, February 2005. [ bib  http ] 
[271]  R. Kumar, T. Kropf, and K. Schneider. Integrating a firstorder automatic prover in the HOL environment. In Archer et al. [543], pages 170–176. [ bib ] 
[272] 
Ramana Kumar and Joe Hurd.
Standalone tactics using OpenTheory.
In Beringer and Felty [554], pages 405–411.
[ bib 
http ]
Proof tools in interactive theorem provers are usually developed within and tied to a specific system, which leads to a duplication of effort to make the functionality available in different systems. Many verification projects would benefit from access to proof tools developed in other systems. Using OpenTheory as a language for communicating between systems, we show how to turn a proof tool implemented for one system into a standalone tactic available to many systems via the internet. This enables, for example, LCFstyle proof reconstruction efforts to be shared by users of different interactive theorem provers and removes the need for each user to install the external tool being integrated.

[273] 
Ramana Kumar.
Challenges in using OpenTheory to transport Harrison's HOL
model from HOL Light to HOL4.
In Blanchette and Urban [568], pages 110–116.
[ bib 
http ]
OpenTheory is being used for the first time (in work to be described at ITP 2013) as a tool in a larger project, as opposed to in an example demonstrating OpenTheory's capability. The tool works, demonstrating its viability. But it does not work completely smoothly, because the use case is somewhat at odds with OpenTheory's primary design goals. In this extended abstract, we explore the tensions between the goals that OpenTheorylike systems might have, and question the relative importance of various kinds of use. My hope is that describing issues arising from work in progress will stimulate fruitful discussion relevant to the development of proof exchange systems.

[274]  Eyal Kushilevitz and Michael O. Rabin. Randomized mutual exclusion algorithms revisited. In Maurice Herlihy, editor, Proceedings of the 11th Annual Symposium on Principles of Distributed Computing, pages 275–283, Vancouver, BC, Canada, August 1992. ACM Press. [ bib ] 
[275] 
Marta Kwiatkowska, Gethin Norman, Roberto Segala, and Jeremy Sproston.
Automatic verification of realtime systems with discrete probability
distributions.
In Katoen [506], pages 75–95.
[ bib 
.ps.gz ]
We consider the timed automata model of [3], which allows the analysis of realtime systems expressed in terms of quantitative timing constraints. Traditional approaches to realtime system description express the model purely in terms of nondeterminism; however, we may wish to express the likelihood of the system making certain transitions. In this paper, we present a model for realtime systems augmented with discrete probability distributions. Furthermore, using the algorithm of [5] with fairness, we develop a model checking method for such models against temporal logic properties which can refer both to timing properties and probabilities, such as, “with probability 0.6 or greater, the clock x remains below 5 until clock y exceeds 2”.

[276]  M. Kwiatkowska, G. Norman, and D. Parker. Prism: Probabilistic symbolic model checker. In Proceedings of PAPM/PROBMIV 2001 Tools Session, September 2001. [ bib  .ps.gz ] 
[277] 
B. A. LaMacchia and A. M. Odlyzko.
Computation of discrete logarithms in prime fields.
Designs, Codes, and Cryptography, 1(1):46–62, May 1991.
[ bib 
.html ]
The presumed difficulty of computing discrete logarithms in finite fields is the basis of several popular public key cryptosystems. The secure identification option of the Sun Network File System, for example, uses discrete logarithms in a field GF(p) with p a prime of 192 bits. This paper describes an implementation of a discrete logarithm algorithm which shows that primes of under 200 bits, such as that in the Sun system, are very insecure. Some enhancements to this system are suggested.

[278] 
Leslie Lamport.
How to write a proof, February 1993.
[ bib 
http ]
A method of writing proofs is proposed that makes it much harder to prove things that are not true. The method, based on hierarchical structuring, is simple and practical.

[279]  Leslie Lamport. Latex: A document preparation system. AddisonWesley, 2nd edition, 1994. [ bib ] 
[280] 
Leslie Lamport and Lawrence C. Paulson.
Should your specification language be typed?
ACM Transactions on Programming Languages and Systems,
21(3):502–526, May 1999.
[ bib 
http ]
Most specification languages have a type system. Type systems are hard to get right, and getting them wrong can lead to inconsistencies. Set theory can serve as the basis for a specification languuage without types. This possibility, which has been widely overlooked, offers many advantages. Untyped set theory is simple and is more flexible than any simple typed formalism. Polymorphism, overloading, and subtyping can make a type system more powerful, but at the cost of increased complexity,and such refinements can never attain the flexibility of having no types at all. Typed formalisms have advantages too, stemming from the power of mechanical type checking. While types serve little purpose in hand proofs, they do help with mechanized proofs. In the absence of verification, type checking can catch errors in specifications. It may be possible to have the best of both worlds by adding typing annotations to an untyped specification language. We consider only specification languages, not programming languages.

[281]  Bruce M. Landman and Aaron Robertson. Ramsey Theory on the Integers. American Mathematical Society, February 2004. [ bib ] 
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[287]  Xavier Leroy. Formal certification of a compiler backend or: programming a compiler with a proof assistant. In Morrisett and Jones [567], pages 42–54. [ bib  .pdf ] 
[288] 
Joe LeslieHurd and Guy Haworth.
Computer theorem proving and HoTT.
ICGA Journal, 36(2):100–103, June 2013.
[ bib 
http ]
Theoremproving is a oneplayer game. The history of computer programs being the players goes back to 1956 and the `LT' Logic Theory Machine of Newell, Shaw and Simon. In gameplaying terms, the `initial position' is the core set of axioms chosen for the particular logic and the `moves' are the rules of inference. Now, the Univalent Foundations Program at IAS Princeton and the resulting `HoTT' book on Homotopy Type Theory have demonstrated the success of a new kind of experimental mathematics using computer theorem proving.

[289] 
Joe LeslieHurd.
Maintaining verified software.
In Shan [537], pages 71–80.
[ bib 
http ]
Maintaining software in the face of evolving dependencies is a challenging problem, and in addition to good release practices there is a need for automatic dependency analysis tools to avoid errors creeping in. Verified software reveals more semantic information in the form of mechanized proofs of functional specifications, and this can be used for dependency analysis. In this paper we present a scheme for automatic dependency analysis of verified software, which for each program checks that the collection of installed libraries is sufficient to guarantee its functional correctness. We illustrate the scheme with a case study of Haskell packages verified in higher order logic. The dependency analysis reduces the burden of maintaining verified Haskell packages by automatically computing version ranges for the packages they depend on, such that any combination provides the functionality required for correct operation.

[290]  Clarence Irving Lewis. A Survey of Symbolic Logic. Univ. of California Press, Berkeley, Berkeley, 1918. Reprint of Chapters I–IV by Dover Publications, 1960, New York. [ bib ] 
[291] 
J. R. Lewis and B. Martin.
Cryptol: High assurance, retargetable crypto development and
validation.
In MILCOM2003 [559], pages 820–825.
[ bib 
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As cryptography becomes more vital to the infrastructure of computing systems, it becomes increasingly vital to be able to rapidly and correctly produce new implementations of cryptographic algorithms. To address these challenges, we introduce a new, formal methodsbased approach to the specification and implementation of cryptography, present a number of scenarios of use, an overview of the language, and present part of a specification of the advanced encryption standard.

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Christopher Lynch and Barbara Morawska.
Goaldirected Eunification.
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We give a general goal directed method for solving the Eunification problem. Our inference system is a generalization of the inference rules for Syntactic Theories, except that our inference system is proved complete for any equational theory. We also show how to easily modify our inference system into a more restricted inference system for Syntactic Theories, and show that our completeness techniques prove completeness there also.

[295] 
Christopher Lynch and Barbara Morawska.
Goaldirected Eunification.
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We give a general goal directed method for solving the unification problem. Our inference system is a generalization of the inference rules for Syntactic Theories, except that our inference system is proved complete for any equational theory. We also show how to easily modify our inference system into a more restricted inference system for Syntactic Theories, and show that our completeness techniques prove completeness there also.

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David MacQueen.
Modules for Standard ML.
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The functional programming language ML has been undergoing a thorough redesign during the past year, and the module facility described here has been proposed as part of the revised language, now called Standard ML. The design has three main goals: (1) to facilitate the structuring of large ML programs; (2) to support separate compilation and generic library units; and (3) to employ new ideas in the semantics of data types to extend the power of ML's polymorphic type system. It is based on concepts inherent in the structure of ML, primarily the notions of a declaration, its type signature, and the environment that it denotes.

[298]  Jeffrey M. Maddalon, Ricky W. Butler, and César Muñoz. A mathematical basis for the safety analysis of conflict prevention algorithms. Technical Report NASA/TM2009215768, National Aeronautics and Space Administration, Langley Research Center, Hampton VA 236812199, USA, June 2009. [ bib  .pdf ] 
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Annabelle McIver and Carroll Morgan.
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Probabilistic predicate transformers guarantee standard (ordinary) predicate transformers to incorporate a notion of probabilistic choice in imperative programs. The basic theory of that, for finite state spaces, is set out in [5], together with a statements of their `healthiness conditions'. Here the earlier results are extended to infinite state spaces, and several more specialised topics are explored: the characterisation of standard and deterministic programs; and the structure of the extended space generated when `angelic choice' is added to the system.

[308] 
Annabelle McIver, Carroll Morgan, and Elena Troubitsyna.
The probabilistic steam boiler: a case study in probabilistic data
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Probabilistic choice and demonic nondeterminism have been combined in a model for sequential programs in which program refinement is defined by removing demonic nondeterminism. Here we study the more general topic of data refinement in the probabilistic setting, extending standard techniques to probabilistic programs. We use the method to obtain a quantitative assessment of safety of a (probabilistic) version of the steam boiler.

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Following earlier models, we lift standard deterministic and nondeterministic semantics of imperative programs to probabilistic semantics. This semantics allows for random external inputs of known or unknown probability and random number generators. We then propose a method of analysis of programs according to this semantics, in the general framework of abstract interpretation. This method lifts an ordinary abstract lattice, for nonprobabilistic programs, to one suitable for probabilistic programs. Our construction is highly generic. We discuss the influence of certain parameters on the precision of the analysis, basing ourselves on experimental results.

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David Monniaux.
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We introduce a new method, combination of random testing and abstract interpretation, for the analysis of programs featuring both probabilistic and nonprobabilistic nondeterminism. After introducing ordinary testing, we show how to combine testing and abstract interpretation and give formulas linking the precision of the results to the number of iterations. We then discuss complexity and optimization issues and end with some experimental results.

[324] 
Carroll Morgan, Annabelle McIver, Karen Seidel, and J. W. Sanders.
Probabilistic predicate transformers.
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Predicate transformers facilitate reasoning about imperative programs, including those exhibiting demonic nondeterministic choice. Probabilistic predicate transformers extend that facility to programs containing probabilistic choice, so that one can in principle determine not only whether a program is guaranteed to establish a certain result, but also its probability of doing so. We bring together independent work of Claire Jones and Jifeng He, showing how their constructions can be made to correspond; from that link between a predicatebased and a relationbased view of probabilistic execution we are able to propose "probabilistic healthiness conditions", generalising those of Dijkstra for ordinary predicate transformers. The associated calculus seems suitable for exploring further the rigorous derivation of imperative probabilistic programs.

[325]  Carroll Morgan. Proof rules for probabilistic loops. In He Jifeng, John Cooke, and Peter Wallis, editors, Proceedings of the BCSFACS 7th Refinement Workshop, Workshops in Computing. Springer, 1996. [ bib  .html ] 
[326]  Carroll Morgan, Annabelle McIver, and Karen Seidel. Probabilistic predicate transformers. ACM Transactions on Programming Languages and Systems, 18(3):325–353, May 1996. [ bib ] 
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We present a new approach for goaldirected theorem proving with equality which integrates Basic Ordered Paramodulation into a Model Elimination framework. In order to be able to use orderings and to restrict the applications of equations to nonvariable positions, the goaldirected tableau construction is combined with bottomup completion where only positive literals are overlapped. The resulting calculus thus keeps the best properties of completion while giving up only part of the goaldirectedness.

[329] 
Max Moser, Ortrun Ibens, Reinhold Letz, Joachim Steinbach, Christoph Goller,
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SETHEO and ESETHEO – the CADE13 systems.
Journal of Automated Reasoning, 18:237–246, 1997.
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The model elimination theorem prover SETHEO (version V3.3) and its equational extension ESETHEO are presented. SETHEO employs sophisticated mechanisms of subgoal selection, elaborate iterative deepening techniques, and local failure caching methods. Its equational counterpart ESETHEO transforms formulae containing equality (using a variant of Brand's modification method) and processes the output with the standard SETHEO system. The paper gives an overview of the theoretical background, the system architecture, and the performance of both systems.

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Chess endgame tables should provide efficiently the value and depth of any required position during play. The indexing of an endgame's positions is crucial to meeting this objective. This paper updates Heinz' previous review of approaches to indexing and describes the latest approach by the first and third authors.

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Tobias Nipkow.
Functional unification of higherorder patterns.
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Higherorder patterns (HOPs) are a class of lambdaterms which behave almost like firstorder terms w.r.t. unification: unification is decidable and unifiable terms have most general unifiers which are easy to compute. HOPs were first discovered by Dale Miller and subsequently developed and applied by Pfenning and Nipkow. This paper presents a stepwise development of a functional unification algorithm for HOPs. Both the usual representation of lambdaterms with alphabetic bound variables and de Bruijn's notation are treated. The appendix contains a complete listing of an implementation in Standard ML.

[346]  Tobias Nipkow. Hoare logics in Isabelle/HOL. In H. Schwichtenberg and R. Steinbrüggen, editors, Proof and SystemReliability, pages 341–367. Kluwer, 2002. [ bib  .html ] 
[347]  Tobias Nipkow, Lawrence C. Paulson, and Markus Wenzel. Isabelle/HOL—A Proof Assistant for HigherOrder Logic, volume 2283 of LNCS. Springer, 2002. [ bib  http ] 
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Tobias Nipkow.
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This article presents formalizations in higherorder logic of two proofs of Arrow’s impossibility theorem due to Geanakoplos. The GibbardSatterthwaite theorem is derived as a corollary. Lacunae found in the literature are discussed.

[349]  Tobias Nipkow, Lawrence C. Paulson, and Markus Wenzel. Isabelle/HOL—A Proof Assistant for HigherOrder Logic, October 2011. Available for download at http://isabelle.in.tum.de/dist/Isabelle20111/doc/tutorial.pdf. [ bib  .pdf ] 
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Michael Norrish and Konrad Slind.
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The HOL system is a mechanized proof assistant for higherorder logic that has been under continuous development since the mid1980s, by an everchanging group of developers and external contributors. We give a brief overview of various implementations of the HOL logic before focusing on the evolution of certain important features available in a recent implementation. We also illustrate how the module system of Standard ML provided security and modularity in the construction of the HOL kernel, as well as serving in a separate capacity as a useful representation medium for persistent, hierarchical logical theories.

[353] 
Michael Norrish.
Rewriting conversions implemented with continuations.
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We give a continuationbased implementation of rewriting for systems in the LCF tradition. These systems must construct explicit proofs of equations when rewriting, and currently do so in a way that can be very spaceinefficient. An explicit representation of continuations improves performance on large terms, and on longrunning computations.

[354]  L. Northrup, P. Feiler, R. P. Gabriel, J. Goodenough, R. Linger, T. Longstaff, R. Kazman, M. Klein, D. Schmidt, K. Sullivan, and K. Wallnau. UltraLargeScale Systems: The Software Challenge of the Future. Software Engineering Institute, Carnegie Mellon, 2006. The 2006 report for a 12month study of ultralargescale systems software, sponsored by the United States Department of Defense. [ bib  http ] 
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Steven Obua and Sebastian Skalberg.
Importing HOL into Isabelle/HOL.
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We developed an importer from both HOL 4 and HOLlight into Isabelle/HOL. The importer works by replaying proofs within Isabelle/HOL that have been recorded in HOL 4 or HOLlight and is therefore completely safe. Concepts in the source HOL system, that is types and constants, can be mapped to concepts in Isabelle/HOL; this facilitates a true integration of imported theorems and theorems that are already available in Isabelle/HOL. The importer is part of the standard Isabelle distribution.

[358] 
Chris Okasaki.
Purely Functional Data Structures.
Cambridge University Press, Cambridge, UK, 1998.
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Data structures and efficiency analysis for functional programming. Code in ML and Haskell. Many references.

[359]  John O'Leary, Roope Kaivola, and Tom Melham. Relational STE and theorem proving for formal verification of industrial circuit designs. In Jobstmann and Ray [530], pages 97–104. [ bib  .pdf ] 
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Michael Parks.
Numbertheoretic test generation for directed rounding.
IEEE Transactions on Computers, 49(7):651–658, 2000.
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We present methods to generate systematically the hardest test cases for multiplication, division, and square root subject to directed rounding, essentially extending previous work on numbertheoretic floatingpoint testing to rounding modes other than tonearest. The algorithms focus upon the rounding boundaries of the modes truncate, tominusinfinity, and toinfinity, and programs based on them require little beyond exact arithmetic in the working precision to create billions of edge cases. We will show that the amount of work required to calculate trial multiplicands pays off in the form of free extra tests due to an interconnection among the operations considered herein. Although these tests do not replace proofs of correctness, they can be used to gain a high degree of confidence that the accuracy requirements as mandated by IEEE Standard 754 have been satisfied.

[365]  Christine Paulin, Philippe Audebaud, and Richard Lassaigne. Randomized algorithms in type theory. Slides from a talk delivered at Dagstuhl seminar 01341: Dependent Type Theory meets Practical Programming, August 2001. [ bib  .ps.gz ] 
[366] 
Lawrence C. Paulson.
A higherorder implementation of rewriting.
Science of Computer Programming, 3:119–149, 1983.
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Many automatic theoremprovers rely on rewriting. Using theorems as rewrite rules helps to simplify the subgoals that arise during a proof. LCF is an interactive theoremprover intended for reasoning about computation. Its implementation of rewriting is presented in detail. LCF provides a family of rewriting functions, and operators to combine them. A succession of functions is described, from pattern matching primitives to the rewriting tool that performs most inferences in LCF proofs. The design is highly modular. Each function performs a basic, specific task, such as recognizing a certain form of tautology. Each operator implements one method of building a rewriting function from sim pler ones. These pieces can be put together in numerous ways, yielding a variety of rewriting strategies. The approach involves programming with higherorder functions. Rewriting functions are data val ues, produced by computation on other rewriting functions. The code is in daily use at Cambridge, demon strating the practical use of functional programming.

[367] 
Lawrence C. Paulson.
Set theory for verification: I. From foundations to functions.
Journal of Automated Reasoning, 11(3):353–389, 1993.
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor's Theorem, the Composition of Homomorphisms challenge [9], and Ramsey's Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics.

[368]  Lawrence C. Paulson. Isabelle: A generic theorem prover. Lecture Notes in Computer Science, 828:xvii + 321, 1994. [ bib ] 
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Lawrence C. Paulson.
Set theory for verification: II. Induction and recursion.
Journal of Automated Reasoning, 15(2):167–215, 1995.
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A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. Inductively defined sets are expressed as least fixedpoints, applying the KnasterTarski Theorem over a suitable set. Recursive functions are defined by wellfounded recursion and its derivatives, such as transfinite recursion. Recursive data structures are expressed by applying the KnasterTarski Theorem to a set, such as V_{ω}, that is closed under Cartesian product and disjoint sum. Worked examples include the transitive closure of a relation, lists, variablebranching trees and mutually recursive trees and forests. The SchröderBernstein Theorem and the soundness of propositional logic are proved in Isabelle sessions.

[370]  Lawrence C. Paulson. Proving properties of security protocols by induction. In CSFW1997 [508], pages 70–83. [ bib  .ps.gz ] 
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[372]  Lawrence C. Paulson. The Isabelle Reference Manual, October 1998. With contributions by Tobias Nipkow and Markus Wenzel. [ bib ] 
[373]  Lawrence C. Paulson. Inductive analysis of the Internet protocol TLS. TISSEC, 2(3):332–351, August 1999. [ bib ] 
[374]  L. C. Paulson. A generic tableau prover and its integration with Isabelle. Journal of Universal Computer Science, 5(3), March 1999. [ bib  http ] 
[375]  Lawrence C. Paulson. Mechanizing UNITY in Isabelle. ACM Transactions on Computational Logic, 2000. In press. [ bib ] 
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Lee Pike, Mark Shields, and John Matthews.
A verifying core for a cryptographic language compiler.
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A verifying compiler is one that emits both object code and a proof of correspondence between object and source code. We report the use of ACL2 in building a verifying compiler for Cryptol, a streambased language for encryption algorithm specification that targets Rockwell Collins' AAMP7 microprocessor (and is designed to compile efficiently to hardware, too). This paper reports on our success in verifying the "core" transformations of the compiler – those transformations over the sublanguage of Cryptol that begin after "higherorder" aspects of the language are compiled away, and finish just before hardware or software specific transformations are exercised. The core transformations are responsible for aggressive optimizations. We have written an ACL2 macro that automatically generates both the correspondence theorems and their proofs. The compiler also supplies measure functions that ACL2 uses to automatically prove termination of Cryptol programs, including programs with mutuallyrecursive cliques of streams. Our verifying compiler has proved the correctness of its core transformations for multiple algorithms, including TEA, RC6, and AES. Finally, we describe an ACL2 book of primitive operations for the general specification and verification of encryption algorithms.

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William Pugh.
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The Omega test is an integer programming algorithm that can determine whether a dependence exists between two array references, and if so, under what conditions. Conventional wisdom holds that integer programming techniques are far too expensive to be used for dependence analysis, except as a method of last resort for situations that cannot be decided by simpler methods. We present evidence that suggests this wisdom is wrong, and that the Omega test is competitive with approximate algorithms used in practice and suitable for use in production compilers. The Omega test is based on an extension of FourierMotzkin variable elimination to integer programming, and has worstcase exponential time complexity. However, we show that for many situations in which other (polynomial) methods are accurate, the Omega test has low order polynomial time complexity. The Omega test can be used to simplify integer programming problems, rather than just deciding them. This has many applications, including accurately and efficiently computing dependence direction and distance vectors.

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Probability distributions are useful for expressing the meanings of probabilistic languages, which support formal modeling of and reasoning about uncertainty. Probability distributions form a monad, and the monadic definition leads to a simple, natural semantics for a stochastic lambda calculus, as well as simple, clean implementations of common queries. But the monadic implementation of the expectation query can be much less efficient than current best practices in probabilistic modeling. We therefore present a language of measure terms, which can not only denote discrete probability distributions but can also support the best known modeling techniques. We give a translation of stochastic lambda calculus into measure terms. Whether one translates into the probability monad or into measure terms, the results of the translations denote the same probability distribution.

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The authors describe how the Crowds system works – essentially, a group of users act as web forwarders for each other in a way that appears random to outsiders. They analyse the anonymity properties of the system and compare it with other systems. Crowds enables the retrieval of information over the web with only a small amount of private information leakage to other parties.

[396]  Stefan Richter. Formalizing integration theory with an application to probabilistic algorithms. In Slind et al. [592], pages 271–286. [ bib ] 
[397]  J. A. Robinson. A machineoriented logic based on the resolution principle. Journal of the ACM, 12(1):23–49, January 1965. [ bib ] 
[398]  J. A. Robinson. Automatic deduction with hyperresolution. International Journal of Computer Mathematics, 1:227–234, 1965. [ bib ] 
[399]  J. A. Robinson. A note on mechanizing higher order logic. Machine Intelligence, 5:121–135, 1970. [ bib ] 
[400]  Abraham Robinson. Nonstandard Analysis. Princeton University Press, 1996. [ bib ] 
[401]  A. Robinson and A. Voronkov, editors. Handbook of Automated Reasoning. Elsevier Science, 2001. [ bib ] 
[402]  A. J. Roycroft. *C*. EG, 7(119):771, 1996. [ bib ] 
[403]  A. J. Roycroft. The computer section. EG, 8(123):47–48, 1997. [ bib ] 
[404]  A. J. Roycroft. *C*. EG, 8(130 Supplement):428, 1998. [ bib ] 
[405]  A. J. Roycroft. AJRs snippets. EG, 8(131):476, 1999. [ bib ] 
[406]  Piotr Rudnicki. An overview of the Mizar project. Notes to a talk at the workshop on Types for Proofs and Programs, June 1992. [ bib  .html ] 
[407]  John Rushby. Formalism in safety cases. In Dale and Anderson [577], pages 3–17. [ bib  http ] 
[408]  Bertrand Russell. The Autobiography of Bertrand Russell. George Allen & Unwin, London, 1968. 3 volumes. [ bib ] 
[409]  David M. Russinoff. An experiment with the BoyerMoore theorem prover: A proof of Wilson's theorem. Journal of Automated Reasoning, 1:121–139, 1985. [ bib ] 
[410]  Mark Saaltink. Domain checking Z specifications. In C. Michael Holloway and Kelly J. Hayhurst, editors, LFM' 97: Fourth NASA Langley Formal Methods Workshop, pages 185–192, Hampton, VA, September 1997. [ bib  http ] 
[411]  R. Sattler. Further to the KRP(a2)KbBP(a3) database. ICCA Journal, 11(2/3):82–87, 1988. [ bib ] 
[412]  J. Schaeffer. One Jump Ahead: Challenging Human Supremacy in Checkers. Springer, New York, 1997. [ bib ] 
[413]  J. Schaeffer, Y. Bjornsson, N. Burch, R. Lake, P. Lu, and S Sutphen. Building the checkers 10piece endgame databases. In Herik et al. [519]. [ bib ] 
[414]  Fred Schneider. Trust in Cyberspace. National Academy Press, 1999. [ bib ] 
[415]  Bruce Schneier. Applied Cryptography. Wiley, second edition, 1996. [ bib ] 
[416]  Johann Ph. Schumann. DELTA — A bottomup processor for topdown theorem provers (system abstract). In Bundy [510]. [ bib  .html ] 
[417] 
Johann Schumann.
Tableauxbased theorem provers: Systems and implementations.
Journal of Automated Reasoning, 13:409–421, 1994.
[ bib 
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The following list of tableauxbased theorem provers was assembled in the Spring and Summer 1993 as the result of a widespread enquiry via email. It is intended to provide a short overview of the field and existing implementations. For each system, a short description is given. Additionally, useful information about the system is presented in tabular form. This includes the type of logic which can be handled by the system (input), the implementation language, hardware and operating systems requirements (implementation). Most of the systems are available as binaries or as sources with documentation and can be obtained via anonymous ftp or upon request. The descriptions and further information have been submitted by the individuals whose names are given as contact address. The provers are ordered alphabetically by their name (or the author's name).

[418]  J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM, 27(4):701–717, October 1980. [ bib  .pdf ] 
[419] 
CarlJohan Seger.
Introduction to formal hardware verification.
Technical Report TR9213, Department of Computer Science, University
of British Columbia, June 1992.
[ bib 
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Formal hardware verification has recently attracted considerable interest. The need for “correct” designs in safetycritical applications, coupled with the major cost associated with products delivered late, are two of the main factors behind this. In addition, as the complexity of the designs increase, an ever smaller percentage of the possible behaviors of the designs will be simulated. Hence, the confidence in the designs obtained by simulation is rapidly diminishing. This paper provides an introduction to the topic by describing three of the main approaches to formal hardware verification: theoremproving, model checking and symbolic simulation. We outline the underlying theory behind each approach, we illustrate the approaches by applying them to simple examples and we discuss their strengths and weaknesses. We conclude the paper by describing current ongoing work on combining the approaches to achieve multilevel verification approaches.

[420]  Karen Seidel, Carroll Morgan, and Annabelle McIver. An introduction to probabilistic predicate transformers. Technical Report TR696, Oxford Programming Research Group Technical Report, 1996. [ bib ] 
[421]  Karen Seidel, Carroll Morgan, and Annabelle McIver. Probabilistic imperative programming: a rigorous approach. In Groves and Reeves [532]. [ bib ] 
[422]  Natarajan Shankar and Sam Owre. Principles and pragmatics of subtyping in PVS. In D. Bert, C. Choppy, and P. D. Mosses, editors, Recent Trends in Algebraic Development Techniques, WADT '99, volume 1827 of Lecture Notes in Computer Science, pages 37–52, Toulouse, France, September 1999. Springer. [ bib  .html ] 
[423] 
Mark Shields.
A language for symmetrickey cryptographic algorithms and its
efficient implementation.
Available from the author's website, March 2006.
[ bib 
http ]
The development of cryptographic hardware for classified data is expensive and time consuming. We present a domainspecific language, microCryptol, and a corresponding compiler, mcc, to address these costs. microCryptol supports the capture of mathematically precise specifications of algorithms, while also allowing those specifications to be compiled to efficient imperative code able to execute on embedded microprocessors.

[424]  Jamie Shield, Ian J. Hayes, and David A. Carrington. Using theory interpretation to mechanise the reals in a theorem prover. In Colin Fidge, editor, Electronic Notes in Theoretical Computer Science, volume 42. Elsevier, 2001. [ bib  .html ] 
[425]  J. H. Silverman. The Arithmetic of Elliptic Curves. Number 106 in Graduate Texts in Mathematics. Springer, 1986. [ bib ] 
[426] 
John Slaney, Arnold Binas, and David Price.
Guiding a theorem prover with soft constraints.
In de Mántaras and Saitta [522], pages 221–225.
[ bib 
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Attempts to use finite models to guide the search for proofs by resolution and the like in forst order logic all suffer from the need to trade off the expense of generating and maintaining models against the improvement in quality of guidance as investment in the semantic aspect of reasoning is increased. Previous attempts to resolve this tradeoff have resulted either in poor selection of models, or in fragility as the search becomes oversensitive to the order of clauses, or in extreme slowness. Here we present a fresh approach, whereby most of the clauses for which a model is sought are treated as soft constraints. The result is a partial model of all of the clauses rather than an exact model of only a subset of them. This allows our system to combine the speed of maintaining just a single model with the robustness previously requiring multiple models. We present experimental evidence of benefits over a range of first order problem domains.

[427]  Konrad Slind. HOL98 Draft User's Manual, Athabasca Release, Version 2, January 1999. Part of the documentation included with the hol98 theoremprover. [ bib ] 
[428]  Konrad Slind. Reasoning about Terminating Functional Programs. PhD thesis, Technical University of Munich, 1999. [ bib ] 
[429]  Konrad Slind and Michael Norrish. The HOL System Tutorial, February 2001. Part of the documentation included with the HOL4 theoremprover. [ bib  http ] 
[430] 
Konrad Slind and Joe Hurd.
Applications of polytypism in theorem proving.
In Basin and Wolff [591], pages 103–119.
[ bib 
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Polytypic functions have mainly been studied in the context of functional programming languages. In that setting, applications of polytypism include elegant treatments of polymorphic equality, prettyprinting, and the encoding and decoding of highlevel datatypes to and from lowlevel binary formats. In this paper, we discuss how polytypism supports some aspects of theorem proving: automated termination proofs of recursive functions, incorporation of the results of metalanguage evaluation, and equivalencepreserving translation to a lowlevel format suitable for propositional methods. The approach is based on an interpretation of higher order logic types as terms, and easily deals with mutual and nested recursive types.

[431]  R. Solovay and V. Strassen. A fast MonteCarlo test for primality. SIAM Journal on Computing, 6(1):84–85, March 1977. [ bib ] 
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[433]  Daryl Stewart. Formal for everyone  Challenges in achievable multicore design and verification. In Cabodi and Singh [529], page 186. [ bib  .pdf ] 
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[435]  Colin Stirling. Bisimulation, model checking and other games, June 1997. Notes for a Mathfit instructional meeting on games and computation, held in Edinburgh, Scotland. [ bib  .ps ] 
[436]  David Stirzaker. Elementary Probability. Cambridge University Press, 1994. [ bib ] 
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[438]  Bryan O'Sullivan, John Goerzen, and Don Stewart. Real World Haskell. O'Reilly Media, Inc., 1st edition, 2008. [ bib ] 
[439]  Christian B. Suttner and Geoff Sutcliffe. The TPTP problem library — v2.1.0. Technical Report JCUCS97/8, Department of Computer Science, James Cook University, December 1997. [ bib  .ps.gz ] 
[440]  Donald Syme. Declarative Theorem Proving for Operational Semantics. PhD thesis, University of Cambridge, 1998. [ bib ] 
[441]  Donald Syme. A Simplifier/Programmable Grinder for hol98, January 1998. Part of the documentation included with the hol98 theoremprover. [ bib ] 
[442]  Tanel Tammet. A resolution theorem prover for intuitionistic logic. In McRobbie and Slaney [511]. [ bib ] 
[443]  Tanel Tammet. Gandalf version c1.0c Reference Manual, October 1997. [ bib  http ] 
[444]  Tanel Tammet. Gandalf. Journal of Automated Reasoning, 18(2):199–204, April 1997. [ bib ] 
[445]  Tanel Tammet. Towards efficient subsumption. In Kirchner and Kirchner [512]. [ bib ] 
[446]  J. Tamplin. EGTquery service extending to 6man pawnless endgame EGTs in DTC, DTM, DTZ and DTZ_{50} metrics. Available from the web page http://chess.jaet.org/endings/, 2006. [ bib ] 
[447]  Aaron Tay. A guide to endgames tablebase. Available from the web page http://www.aarontay.per.sg/Winboard/egtb.html, 2006. [ bib ] 
[448]  Laurent Théry. A quick overview of HOL and PVS, August 1999. Lecture Notes from the Types Summer School '99: Theory and Practice of Formal Proofs, held in Giens, France. [ bib  .html ] 
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J. F. Traub, editor.
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Academic Press, New York, 1976.
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"This book provides a record of a conference, and contains many novel ideas for probabilistic algorithms."

[453]  A. Trybulec and H. A. Blair. Computer aided reasoning. In Rohit Parikh, editor, Proceedings of the Conference on Logic of Programs, volume 193 of Lecture Notes in Computer Science, pages 406–412, Brooklyn, NY, June 1985. Springer. [ bib ] 
[454]  Andrzej Trybulec and Howard Blair. Computer assisted reasoning with MIZAR. In Aravind Joshi, editor, Proceedings of the 9th International Joint Conference on Artificial Intelligence, pages 26–28, Los Angeles, CA, August 1985. Morgan Kaufmann. [ bib ] 
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D. A. Turner.
A new implementation technique for applicative languages.
Software – Practice and Experience, 9(1):31–49, January 1979.
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Combinators, a "curried" version of lambda calculus that eliminates the need for symbol binding. Combinators can be reduced (evaluated) locally and in parallel, so they make an interesting model of parallel computation. Combinator hackers: this paper introduces some new combinators, besides SKI, that help keep the translation from blowing up in space.

[457]  Thomas Tymoczko. The four color problems. Journal of Philosophy, 76, 1979. [ bib ] 
[458]  Peter Van Roy and Seif Haridi. Concepts, Techniques, and Models of Computer Programming. MIT Press, March 2004. [ bib ] 
[459] 
Moshe Y. Vardi.
Why is modal logic so robustly decidable?
Technical Report TR97274, Rice University, April 1997.
[ bib ]
In the last 20 years modal logic has been applied to numerous areas of computer science, including artificial intelligence, program verification, hardware verification, database theory, and distributed computing. There are two main computational problems associated with modal logic. The first problem is checking if a given formula is true in a given state of a given structure. This problem is known as the modelchecking problem. The second problem is checking if a given formula is true in all states of all structures. This problem is known as the validity problem. Both problems are decidable. The modelchecking problem can be solved in linear time, while the validity problem is PSPACEcomplete. This is rather surprising when one considers the fact that modal logic, in spite of its apparent propositional syntax, is essentially a firstorder logic, since the necessity and possibility modalities quantify over the set of possible worlds, and model checking and validity for firstorder logic are computationally hard problems. Why, then, is modal logic so robustly decidable? To answer this question, we have to take a close look at modal logic as a fragment of firstorder logic. A careful examination reveals that propositional modal logic can in fact be viewed as a fragment of 2variable firstorder logic. It turns out that this fragment is computationally much more tractable than full firstorder logic, which provides some explanation for the tractability of modal logic. Upon a deeper examination, however, we discover that this explanation is not too satisfactory. The tractability of modal logic is quite and cannot be explained by the relationship to twovariable firstorder logic. We argue that the robust decidability of modal logic can be explained by the socalled treemodel property, and we show how the treemodel property leads to automatabased decision procedures.

[460]  Norbert Völker. HOL2P  A system of classical higher order logic with second order polymorphism. In Schneider and Brandt [596], pages 334–351. [ bib  http ] 
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[462]  Tanja E. J. Vos. UNITY in Diversity: A Stratified Approach to the Verification of Distributed Algorithms. PhD thesis, Utrecht University, 2000. [ bib ] 
[463]  Philip Wadler. The essence of functional programming. In 19th Symposium on Principles of Programming Languages. ACM Press, January 1992. [ bib ] 
[464] 
Philip Wadler.
Comprehending monads.
Mathematical Structures in Computer Science, 2:461–493, 1992.
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Category theorists invented monads in the 1960's to concisely express certain aspects of universal algebra. Functional programmers invented list comprehensions in the 1970's to concisely express certain programs involving lists. This paper shows how list comprehensions may be generalised to an arbitrary monad, and how the resulting programming feature can concisely express in a pure functional language some programs that manipulate state, handle exceptions, parse text, or invoke continuations. A new solution to the old problem of destructive array update is also presented. No knowledge of category theory is assumed.

[465]  Stan Wagon. The BanachTarski Paradox. Cambridge University Press, 1993. [ bib ] 
[466]  Keith Wansbrough, Michael Norrish, Peter Sewell, and Andrei Serjantov. Timing UDP: mechanized semantics for sockets, threads and failures. Draft, 2001. [ bib  http ] 
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[468] 
Morten Welinder.
Very efficient conversions.
In Schubert et al. [546], pages 340–352.
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Using program transformation techniques from the field of partial evaluation an automatic tool for generating very efficient conversions from equalitystating theorems has been implemented. In the situation where a Hol user would normally employ the builtin function GEN_REWRITE_CONV, a function that directly produces a conversion of the desired functionality, this article demonstrates how producing the conversion in the form of a program text instead of as a closure can lead to significant speedups. The Hol system uses a set of 31 simplifying equations on a very large number of intermediate terms derived, e.g., during backwards proofs. For this set the conversion generated by the twostep method is about twice as fast as the method currently used. When installing the new conversion, tests show that the overall running times of Hol proofs are reduced by about 10%. Apart from the speedup this is completely invisible to the user. With cooperation from the user further speedup is possible.

[469]  Markus Wenzel. Isar  A generic interpretative approach to readable formal proof documents. In Bertot et al. [585], pages 167–184. [ bib ] 
[470]  Stephen Westfold. Integrating Isabelle/HOL with Specware. In Schneider and Brandt [597]. [ bib ] 
[471]  Arthur White. Limerick. Mathematical Magazine, 64(2):91, April 1991. [ bib ] 
[472]  Alfred North Whitehead and Bertrand Russell. Principia Mathematica. Cambridge University Press, Cambridge, 1910. [ bib ] 
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[474]  Freek Wiedijk. Mizar light for HOL light. In Boulton and Jackson [588]. [ bib ] 
[475]  Freek Wiedijk. Mizar's soft type system. In Schneider and Brandt [596], pages 383–399. [ bib ] 
[476]  David Williams. Probability with Martingales. Cambridge University Press, 1991. [ bib ] 
[477] 
Carl Roger Witty.
The Ontic inference language.
Master's thesis, MIT, June 1995.
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OIL, the Ontic Inference Language, is a simple bottomup logic programming language with equality reasoning. Although intended for use in writing proof verification systems, OIL is an interesting generalpurpose programming language. In some cases, very simple OIL programs can achieve an efficiency which requires a much more complicated algorithm in a traditional programming language. This thesis gives a formal semantics for OIL and some new results related to its efficient implementation. The main new result is a method of transforming bottomup logic programs with equality to bottomup logic programs without equality. An introduction to OIL and several examples are also included.

[478]  Wai Wong. The HOL res_quan library, 1993. HOL88 documentation. [ bib  .html ] 
[479] 
W. Wong.
Recording and checking HOL proofs.
In Schubert et al. [546], pages 353–368.
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Formal proofs generated by mechanized theorem proving systems may consist of a large number of inferences. As these theorem proving systems are usually very complex, it is extremely difficult if not impossible to formally verify them. This calls for an independent means of ensuring the consistency of mechanically generated proofs. This paper describes a method of recording HOL proofs in terms of a sequence of applications of inference rules. The recorded proofs can then be checked by an independent proof checker. Also described in this paper is an efficient proof checker which is able to check a practical proof consisting of thousands of inference steps.

[480]  Wai Wong. A proof checker for HOL. Technical Report 389, University of Cambridge Computer Laboratory, March 1996. [ bib ] 
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[485]  Lotfi A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965. [ bib ] 
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[487]  Lintao Zhang and Sharad Malik. The quest for efficient boolean satisfiability solvers. In Voronkov [515], pages 295–313. [ bib  .pdf ] 
[488]  Junxing Zhang and Konrad Slind. Verification of Euclid's algorithm for finding multiplicative inverses. In Hurd et al. [595], pages 205–220. [ bib ] 
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[492] 
IEEE.
Standard for Verilog Hardware Description Language.
IEEE Standard 13642001. The Institute of Electrical and Electronic
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[ bib ]
The Verilog Hardware Description Language (HDL) is defined in this standard. Verilog HDL is a formal notation intended for use in all phases of the creation of electronic systems. Because it is both machine readable and human readable, it supports the development, verification, synthesis, and testing of hardware designs; the communication of hardware design data; and the maintenance, modification, and procurement of hardware. The primary audiences for this standard are the implementors of tools supporting the language and advanced users of the language.

[493] 
IEEE.
Standard for Information Technology—Portable Operating System
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IEEE Standard 1003.12001. The Institute of Electrical and Electronic
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This standard defines a standard operating system interface and environment, including a command interpreter (or shell), and common utility programs to support applications portability at the source code level. It is the single common revision to IEEE Std 1003.11996, IEEE Std 1003.21992, and the Base Specifications of The Open Group Single UNIX Specification, Version 2.

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Certification of critical software systems (e.g., for safety and security) is important to help ensure their dependability. Today, certification relies as much on evaluation of the software development process as it does on the system s properties. While the latter are preferable, the complexity of these systems usually makes them extremely difficult to evaluate. To explore these and related issues, the National Coordination Office for Information technology Research and Development asked the NRC to undertake a study to assess the current state of certification in dependable systems. The study is in two phases: the first to frame the problem and the second to assess it. This report presents a summary of a workshop held as part of the first phase. The report presents a summary of workshop participants presentations and subsequent discussion. It covers, among other things, the strengths and limitations of process; new challenges and opportunities; experience to date; organization context; and costeffectiveness of software engineering techniques. A consensus report will be issued upon completion of the second phase.

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