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14
Needed: An Empirical Science Of Algorithms
 Operations Research
, 1994
"... this article goes to press. Journal editors can be encouraged to seek out referees who have done rigorous empirical studies. Refereeing standards will evolve, particularly as the empirical science develops. ..."
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Cited by 73 (3 self)
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this article goes to press. Journal editors can be encouraged to seek out referees who have done rigorous empirical studies. Refereeing standards will evolve, particularly as the empirical science develops.
The Theory of LEGO  A Proof Checker for the Extended Calculus of Constructions
, 1994
"... LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO ..."
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Cited by 68 (10 self)
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LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO is intended to be used for interactively constructing proofs in mathematical theories presented in these logics. I have developed LEGO over six years, starting from an implementation of the Calculus of Constructions by G erard Huet. LEGO has been used for problems at the limits of our abilities to do formal mathematics. In this thesis I explain some aspects of the metatheory of LEGO's type systems leading to a machinechecked proof that typechecking is decidable for all three type theories supported by LEGO, and to a verified algorithm for deciding their typing judgements, assuming only that they are normalizing. In order to do this, the theory of Pure Type Systems (PTS) is extended and f...
Inability to reason about an object's orientation using an axis and angle of rotation
, 1995
"... The kinematic bases by which humans imagine an object turn from one orientation to another are unknown. The studies reported here show that individuals of high spatial ability are, in most cases, unable to imagine a ShepardMetzler object rotate about an axis and angle so as to accurately envision i ..."
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Cited by 15 (0 self)
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The kinematic bases by which humans imagine an object turn from one orientation to another are unknown. The studies reported here show that individuals of high spatial ability are, in most cases, unable to imagine a ShepardMetzler object rotate about an axis and angle so as to accurately envision its appearance. Nor can they conceive of the axis and angle by which it would rotate in a shortest path between two orientations. Accuracy progressively improves across cases in which neither angle, one of the angles, or both angles between the rotation axis and viewerenvironment frame and between the axis and object limb are canonical. When canonical, the angles are more accurately observed from one viewpoint to hold constant in rotation. Such inability, with rare exceptions, is probably true for other kinematic operations requiring fine control of multiple spatial relations. Objects ' orientations are not readily represented in terms of shortest path axis and angle. Many people may have the intuition that they can imagine an object seen at one orientation to be at another. Without deliberation, the object seems to move between orientations by a systematically executed and efficient path. The ability to represent one object at another object's orientation is thought to underlie the judgment of whether the shapes of disoriented objects are the same or different (e.g., Corballis,
The journey of the four colour theorem through time
 The NZ Math. Magazine
"... This is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its proof. The problem, first stated as far back as 1850s, still causes controversy today. Its computeraided proof has forced mathematicians to question the notions of proofs and mathematical ..."
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Cited by 7 (0 self)
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This is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its proof. The problem, first stated as far back as 1850s, still causes controversy today. Its computeraided proof has forced mathematicians to question the notions of proofs and mathematical truth.
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of ..."
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Cited by 6 (2 self)
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
An ObjectOriented Architecture for Possibilistic Models
 in: Proc. 1994 Conf. ComputerAided Systems Technology
, 1994
"... . An architecture for the implementation of possibilistic models in an objectoriented programming environment (C++ in particular) is described. Fundamental classes for special and general random sets, their associated fuzzy measures, special and general distributions and fuzzy sets, and possibilist ..."
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Cited by 4 (4 self)
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. An architecture for the implementation of possibilistic models in an objectoriented programming environment (C++ in particular) is described. Fundamental classes for special and general random sets, their associated fuzzy measures, special and general distributions and fuzzy sets, and possibilistic processes are specified. Supplementary methodsincluding the fast Mobius transform, the maximum entropy and Bayesian approximations of random sets, distribution operators, compatibility measures, consonant approximations, frequency conversions, and possibilistic normalization and measurement methodsare also introduced. Empirical results to be investigated are also described. 1 Introduction Possibility theory [4] is an alternative information theory to that based on probability. Although possibility theory is logically independent of probability theory, they are related: both arise in DempsterShafer evidence theory as fuzzy measures defined on random sets; and their distributions a...
Exhaustive search, combinatorial optimization and enumeration: Exploring the potential of raw computing power
 In SOFSEM 2000, number 1963 in LNCS
, 2000
"... Abstract. For half a century since computers came into existence, the goal of finding elegant and efficient algorithms to solve “simple ” (welldefined and wellstructured) problems has dominated algorithm design. Over the same time period, both processingand storage capacity of computers have increa ..."
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Cited by 4 (1 self)
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Abstract. For half a century since computers came into existence, the goal of finding elegant and efficient algorithms to solve “simple ” (welldefined and wellstructured) problems has dominated algorithm design. Over the same time period, both processingand storage capacity of computers have increased by roughly a factor of a million. The next few decades may well give us a similar rate of growth in raw computing power, due to various factors such as continuingminiaturization, parallel and distributed computing. If a quantitative change of orders of magnitude leads to qualitative advances, where will the latter take place? Only empirical research can answer this question. Asymptotic complexity theory has emerged as a surprisingly effective tool for predictingrun times of polynomialtime algorithms. For NPhard problems, on the other hand, it yields overly pessimistic bounds. It asserts the nonexistence of algorithms that are efficient across an entire problem class, but ignores the fact that many instances, perhaps
Developing Understanding for Different Roles of Proof in Dynamic Geometry. Paper presented at ProfMat 2002
, 2002
"... ..."
What’s experimental about experimental mathematics? ∗
, 2008
"... From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, dur ..."
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From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, during the last three decades, high speed computers and sophisticated software packages such as Maple and Mathematica have entered into the domain of pure mathematics, bringing with them a new experimental flavor. They have opened up a new approach in which computerbased tools are used to experiment with the mathematical objects in a dialogue with more traditional methods of formal rigorous proof. At present, a subdiscipline of experimental mathematics is forming with its own research problems, methodology, conferences, and journals. In this paper, I first outline the role of the computer in the mathematical experiment and briefly describe the impact of high speed computing on mathematical research within the emerging subdiscipline of experimental mathematics. I then consider in more detail the epistemological claims put forward within experimental mathematics and comment on some of the discussions that experimental mathematics has provoked within the mathematical community in recent years. In the second part of the paper, I suggest the notion of exploratory experimentation as a possible framework for understanding experimental mathematics. This is illustrated by discussing the socalled PSLQ algorithm.