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An upper bound for selfdual codes
 Inform. Contr
"... Gleason has described the general form that the weight distribution of a selfdual code over GF(2) and GF(3) can have. We give an explicit formula for this weight distribution when the minimum distance d between codewords is made as large as possible. It follows that for selfdual codes of length n ..."
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Cited by 43 (12 self)
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Gleason has described the general form that the weight distribution of a selfdual code over GF(2) and GF(3) can have. We give an explicit formula for this weight distribution when the minimum distance d between codewords is made as large as possible. It follows that for selfdual codes of length n over GF(2) with all weights divisible by 4, d ~ 4[n/24] + 4; and for selfdual codes over GF(3), d < 3[n/12] + 3; where the square brackets denote the integer part. These results improve on the Elias bound. A table of this extremal weight distribution is given in the binary case for n < 200 and n = 256. I. PRELIMINARIES Let C be a linear code over GF(q) of block length n, containing qk codewords at a minimum distance of d apart. We call C an [n, k, d] code. The dual code C = consists of all vectors x such that n1 x'y = ~ x~y ~:0 r=0 for all y ~ C. Then C is selfdual if C: C:. The weight wt(u) of a vector u is the number of its nonzero components. The weight enumerator of a code C is W(X, Y) = ~ Xnwt(u)Y w~(u). u~C We consider selfdual codes in 3 cases:
Incompleteness Theorems for Random Reals
, 1987
"... We obtain some dramatic results using statistical mechanicsthermodynamics kinds of arguments concerning randomness, chaos, unpredictability, and uncertainty in mathematics. We construct an equation involving only whole numbers and addition, multiplication, and exponentiation, with the property tha ..."
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Cited by 32 (1 self)
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We obtain some dramatic results using statistical mechanicsthermodynamics kinds of arguments concerning randomness, chaos, unpredictability, and uncertainty in mathematics. We construct an equation involving only whole numbers and addition, multiplication, and exponentiation, with the property that if one varies a parameter and asks whether the number of solutions is finite or infinite, the answer to this question is indistinguishable from the result of independent tosses of a fair coin. This yields a number of powerful Godel incompletenesstype results concerning the limitations of the axiomatic method, in which entropyinformation measures are used. c fl 1987 Academic Press, Inc. 2 G. J. Chaitin 1. Introduction It is now half a century since Turing published his remarkable paper On Computable Numbers, with an Application to the Entscheidungsproblem (Turing [15]). In that paper Turing constructs a universal Turing machine that can simulate any other Turing machine. He also use...
Intelligent Camera Planning for Computer Graphics
, 2003
"... The virtual cameras used in computer graphics are normally controlled by user input or they are generated by a simple rule, which might require them to remain in position relative to a moving object. In this thesis an alternative method of control is proposed in which camera setups are derived from ..."
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Cited by 9 (1 self)
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The virtual cameras used in computer graphics are normally controlled by user input or they are generated by a simple rule, which might require them to remain in position relative to a moving object. In this thesis an alternative method of control is proposed in which camera setups are derived from a description of the properties of the picture the camera is intended to produce. An analysis of the requirements of such a declarative system and possible methods of achieving them has been carried out, and is presented.
Partial Formalizations And The Lemmings Game
, 1998
"... The computer game Lemmings can serve as a new Drosophila for AI research connecting logical formalizations with information that is incompletely formalizable in practice. In this article we discuss the features of the Lemmings world that make it a challenge to both experimental and theoretical AI an ..."
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Cited by 6 (0 self)
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The computer game Lemmings can serve as a new Drosophila for AI research connecting logical formalizations with information that is incompletely formalizable in practice. In this article we discuss the features of the Lemmings world that make it a challenge to both experimental and theoretical AI and present some steps toward formalizing the game using situation calculus. Preliminary versions of this paper lack important formulas and references to the literature on reactive AI and to computer vision. The formulas present are not yet integrated into a coherent whole. 1 Work partly supported by ARPA (ONR) grant N000149410775 and partly done while the author was Meyerhoff Visiting Professor at the Weizmann Institute of Science, Rehovot, Israel 1 This document is reachable from the WWW page http://wwwformal.stanford.edu/jmc/home.html. 1 Contents 1 Introduction 3 2 Description of the Lemmings Games 3 3 What Needs to be Formalized? 5 4 An Example of Lemming Play 6 5 Physics of t...
Informationtheoretic Incompleteness
 APPLIED MATHEMATICS AND COMPUTATION
, 1992
"... We propose an improved definition of the complexity of a formal axiomatic system: this is now taken to be the minimum size of a selfdelimiting program for enumerating the set of theorems of the formal system. Using this new definition, we show (a) that no formal system of complexity n can exhibit a ..."
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Cited by 6 (1 self)
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We propose an improved definition of the complexity of a formal axiomatic system: this is now taken to be the minimum size of a selfdelimiting program for enumerating the set of theorems of the formal system. Using this new definition, we show (a) that no formal system of complexity n can exhibit a specific object with complexity greater than n + c, and (b) that a formal system of complexity n can determine at most n + c scattered bits of the halting probability\Omega . We also present a short, selfcontained proof of (b).
From Unicode to Typography, a Case Study: the Greek Script
 Proceedings of 14th International Unicode Conference, available from http://omega.enstb.org/yannis/pdf/boston99.pdf
, 1999
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"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how ..."
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Cited by 5 (2 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
A lambda calculus for real analysis
, 2005
"... Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoni ..."
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Cited by 4 (0 self)
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Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoning looks remarkably like a sanitised form of that in classical topology. This paper is an introduction to ASD for the general mathematician, and applies it to elementary real analysis. It culminates in the Intermediate Value Theorem, i.e. the solution of equations fx = 0 for continuous f: R → R. As is well known from both numerical and constructive considerations, the equation cannot be solved if f “hovers ” near 0, whilst tangential solutions will never be found. In ASD, both of these failures and the general method of finding solutions of the equation when they exist are explained by the new concept of “overtness”. The zeroes are captured, not as a set, but by highertype operators � and ♦ that remain (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than sets of points leads to
On The Number of Domino Tilings of the Rectangular Grid, The Holey Square and Related Problems
, 1997
"... The study of domino tiling is novel, with many intriguing questions. The majority of which are still unanswered. Recently, in separate investigations, Noam Elkies and others proved that the total number of tilings of both the Square Grid and the Aztec diamond contain a nice power of two. In this pap ..."
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Cited by 3 (0 self)
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The study of domino tiling is novel, with many intriguing questions. The majority of which are still unanswered. Recently, in separate investigations, Noam Elkies and others proved that the total number of tilings of both the Square Grid and the Aztec diamond contain a nice power of two. In this paper, we will generalize these results to a wide variety of grids by combining powerful and new ideas from graph theory, linear algebra, and geometry. 1
A Solution Based H Norm Triangular Mesh Quality Indicator. Grid Generation and Adaptive
 Mathematics: Proceedings of the IMA Workshop on Parallel and Adaptive Methods
, 1999
"... Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider manyvalued functions, s ..."
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Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider manyvalued functions, since they too bring in plural terms—terms such as ‘�4 ’ or the descriptive ‘the inhabitants of London ’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference seriously if only because mathematicians take manyvalued functions seriously. We assess the objection (by Russell, Frege and others) that manyvalued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection illfounded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more. 1. Russell’s theory of plural descriptions Everybody knows that Russell had a theory of definite descriptions. Not everybody realizes that he had two: one for singular descriptions, another for plural descriptions. The contents of ‘On Denoting ’ have blinkered the popular conception of his agenda. 1.1 The Principles of Mathematics In the Principles class talk is plural talk: ‘soandso’s children, or the children of Londoners, afford illustrations ’ of classes; ‘the children of Israel are a class ’ (1903c, pp. 24, 83). Readers brought up on modern set theory must beware. Russell’s plural descriptions each stand for many things, and accordingly his classes are ‘classes as many’: they are many things—the children of Israel are a class—not one. (Unless, of course, they only have a single member. Throughout this paper we use the plural idiom inclusively, to cover the singular as a limiting case. Purists should read ‘the Fs ’ as ‘the F or Fs ’ and adjust the context to suit.) Russell first investigates plural idioms in the chapter on ‘Denoting’. His exposition is complicated, however, by his insistence that distribu