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Playing games with algorithms: Algorithmic combinatorial game theory
 IN: PROC. 26TH SYMP. ON MATH FOUND. IN COMP. SCI., LECT. NOTES IN COMP. SCI., SPRINGERVERLAG
, 2001
"... Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in combinatorial game theory, ..."
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Cited by 46 (11 self)
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Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in combinatorial game theory, which analyzes ideal play in perfectinformation games. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomialtime algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.
An Exploration of the BirdMeertens Formalism
 In STOP Summer School on Constructive Algorithmics, Abeland
, 1989
"... Two formalisms that have been used extensively in the last few years for the calculation of programs are the Eindhoven quantifier notation and the formalism developed by Bird and Meertens. Although the former has always been applied with ultimate goal the derivation of imperative programs and th ..."
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Cited by 32 (3 self)
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Two formalisms that have been used extensively in the last few years for the calculation of programs are the Eindhoven quantifier notation and the formalism developed by Bird and Meertens. Although the former has always been applied with ultimate goal the derivation of imperative programs and the latter with ultimate goal the derivation of functional programs there is a remarkable similarity in the formal games that are played. This paper explores the BirdMeertens formalism by expressing and deriving within it the basic rules applicable in the Eindhoven quantifier notation. 1 Calculation was an endless delight to Moorish scholars. They loved problems, they enjoyed finding ingenious methods to solve them, and sometimes they turned their methods into mechanical devices. (J. Bronowski, The Ascent of Man. Book Club Associates: London (1977).) 1 Introduction Our ability to calculate  whether it be sums, products, differentials, integrals, or whatever  would be woefull...
On the Complexity of Finding the Chromatic Number of a Recursive Graph I: The Bounded Case
 Annals of Pure and Applied Logic
, 1989
"... We classify functions in recursive graph theory in terms of how many queries to K (or # ## or # ### ) are required to compute them. We show that (1) binary search is optimal (in terms of the number of queries to K) for finding the chromatic number of a recursive graph and that no set of Turing d ..."
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Cited by 17 (10 self)
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We classify functions in recursive graph theory in terms of how many queries to K (or # ## or # ### ) are required to compute them. We show that (1) binary search is optimal (in terms of the number of queries to K) for finding the chromatic number of a recursive graph and that no set of Turing degree less than 0 # will su#ce, (2) determining if a recursive graph has a finite chromatic number is # 2 complete, and (3) binary search is optimal (in terms of the number of queries to # ### ) for finding the recursive chromatic number of a recursive graph and that no set of Turing degree less than 0 ### will su#ce. We also explore how much help queries to a weaker set may provide. Some of our results have analogues in terms of asking p questions at a time, but some do not. In particular, (p + 1)ary search is not always optimal for finding the chromatic number of a recursive graph. Most of our results are also true for highly recursive graphs, though there are some interesting di#erenc...
On dynamically presenting a topology course
 Annals of Mathematics and Artificial Intelligence
, 2001
"... www.cs.mdx.ac.uk/imp Authors of traditional mathematical texts often have difficulty balancing the amount of contextual information and proof detail. We propose a simple hypermedia framework that can assist in the organisation and presentation of mathematical theorems and definitions. We describe th ..."
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Cited by 10 (5 self)
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www.cs.mdx.ac.uk/imp Authors of traditional mathematical texts often have difficulty balancing the amount of contextual information and proof detail. We propose a simple hypermedia framework that can assist in the organisation and presentation of mathematical theorems and definitions. We describe the application of the framework to convert an existing course in general topology to a webbased set of materials. A pilot study of the materials indicated a high level of user satisfaction. We discuss further lines of investigation, in particular, the presentation of larger bodies of work. 1
Algorithmically Coding the Universe
 Developments in Language Theory, World Scientific
, 1994
"... All science is founded on the assumption that the physical universe is ordered. Our aim is to challenge this hypothesis using arguments from the algorithmic information theory. 1 Introduction Algorithmic information theory opens new vistas that extend far beyond the traditional boundaries of mathem ..."
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Cited by 9 (7 self)
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All science is founded on the assumption that the physical universe is ordered. Our aim is to challenge this hypothesis using arguments from the algorithmic information theory. 1 Introduction Algorithmic information theory opens new vistas that extend far beyond the traditional boundaries of mathematics and computer science. How can we describe the seemingly random processes in nature and reconcile them with the supposed order? How much can a given piece of information be compressed? These are matters of fundamental scientific importance that will be discussed below, mainly from an informal or semiformal point of view. The descriptional complexity of a sequence of bits, finite or infinite, is the length of the shortest sequence of bits defining the originally given sequence. A given sequence being random means, roughly, that its descriptional complexity equals its length. In other words the simplest way to define the sequence is to write it down. This seems to be the case for the seq...
Unbounded Searching Algorithms
 SIAM J. Comput
, 1990
"... The unbounded search problem was posed by Bentley and Yao. It is the problem of finding a key in a linearly ordered unbounded table, with the proviso that the number of comparisons is to be minimized. We show that Bentley and Yao's lower bound is essentially optimal, and we prove some new upper boun ..."
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Cited by 7 (2 self)
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The unbounded search problem was posed by Bentley and Yao. It is the problem of finding a key in a linearly ordered unbounded table, with the proviso that the number of comparisons is to be minimized. We show that Bentley and Yao's lower bound is essentially optimal, and we prove some new upper bounds for the unbounded search problem. We show how to solve this problem in parallel as well. 1. Introduction As posed in [BY76], unbounded search is the problem of searching for a key in a sorted table of unbounded size. The following twoplayer game is equivalent to the unbounded search problem: Player A chooses an arbitrary positive integer, n. Player B is allowed to ask whether an integer x is less than n. In general the number of questions that B has to ask in order to determine n is a function of n. In this paper, we present lower and upper bounds on the size of this function. The following theorem from [BY76] is instrumental in providing lower bounds on the number of questions needed i...
LCF: A lexicographic binary representation of the rationals
 J. Universal Comput. Sci
, 1995
"... Abstract: A binary representation of the rationals derived from their continued fraction expansions is described and analysed. The concepts \adjacency", \mediant " and \convergent " from the literature on Farey fractions and continued fractions are suitably extended to provide a found ..."
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Cited by 7 (0 self)
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Abstract: A binary representation of the rationals derived from their continued fraction expansions is described and analysed. The concepts \adjacency", \mediant " and \convergent " from the literature on Farey fractions and continued fractions are suitably extended to provide a foundation for this new binary representation system. Worst case representationinduced precision loss for any real number by a xed length representable number of the system is shown to be at most 19 % of bit word length, with no precision loss whatsoever induced in the representation of any reasonably sized rational number. The representation is supported by a computer arithmetic system implementing exact rational and approximate real computations in an online fashion.
Integration on Surreal Numbers
, 2003
"... The thesis concerns the (class) structure No of Conway’s surreal numbers. The main concern is the behaviour on No of some of the classical functions of real analysis, and a definition of integral for such functions. In the main texts on No, most definitions and proofs are done by transfinite recursi ..."
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Cited by 3 (3 self)
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The thesis concerns the (class) structure No of Conway’s surreal numbers. The main concern is the behaviour on No of some of the classical functions of real analysis, and a definition of integral for such functions. In the main texts on No, most definitions and proofs are done by transfinite recursion and induction on the complexity of elements. In the thesis I consider a general scheme of definition for functions on No, generalising those for sum, product and exponential. If a function has such a definition, and can live in a Hardy field, and satisfies some auxiliary technical conditions, one can obtain in No a substantial analogue of real analysis for that function. One example is the signchange property, and this (applied to polynomials) gives an alternative treatment of the basic fact that No is real closed. I discuss the analogue for the exponential. Using these ideas one can define a generalisation of Riemann integration (the indefinite integral falling under the recursion scheme). The new integral is linear, monotone, and satisfies integration by parts. For some classical functions (e.g. polynomials) the integral yields the traditional formulae of analysis. There are, however, anomalies for the exponential function. But one can show that the logarithm, defined as the inverse of the exponential, is the integral of 1/x as usual. Acknowledgements I wish to express my gratitude to my supervisor Angus Macintyre for his constant support and assistance. Thanks to the examiners A. Maciocia and D. Richardson for their useful suggestions. Thanks also to my colleagues and my landlord for their willingness to tolerate my company. My gratitude goes to my family for their understanding and support. I wish to thank also the Engineering and Physical Sciences Research Council and
Xiangqi and Combinatorial Game Theory
, 2002
"... We explore whether combinatorial game theory (CGT) is suitable for analyzing endgame positions in Xiangqi (Chinese Chess). We discover some of the game values that can also be found in the analysis of International Chess, but we also experience the limitations of CGT when applied to a loopy and non ..."
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Cited by 1 (1 self)
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We explore whether combinatorial game theory (CGT) is suitable for analyzing endgame positions in Xiangqi (Chinese Chess). We discover some of the game values that can also be found in the analysis of International Chess, but we also experience the limitations of CGT when applied to a loopy and nonseparable game like Xiangqi.
Differential Equations as Enablers of Qualitative Reasoning Using Dimensional Analysis
, 1992
"... Dimensional analysis has been used for qualitative reasoning about simple physical systems such as the spring and for complex physical systems such as stars, heat exchangers and nuclear reactors [1, 2, 11]. The technique appears promising because its central representation is fundamental to the lang ..."
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Dimensional analysis has been used for qualitative reasoning about simple physical systems such as the spring and for complex physical systems such as stars, heat exchangers and nuclear reactors [1, 2, 11]. The technique appears promising because its central representation is fundamental to the language of physics; nevertheless, its use has been considered problematic or even mysterious. Most research in qualitative reasoning starts out with modeling constructs e.g. qualitative differential equations [10] and confluences [5]. These modeling constructs are then used to specify the particular problem that one wishes to solve, The dimensional approach uses as its modeling constructs, physical variables and their dimensional representations. At the surface, this approach to modeling might appear to be impoverished or knowledgefree; however, dimensional representations provide a compact and qualitative encoding of physical knowledge that has been captured by numerical laws. There are philosophical questions that have troubled physicists and others