Thesis (M. Ed.)--University of Hong Kong, 2005. / Title proper from title frame. Also available in printed format.

Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1991. / Vita. Abstract. Includes bibliographical references (leaves 71-72). Also available via the Internet.

"Let P(n) be a statement for every positive integer n. We denote the set of all positive integers by N and consider G = {n [is an element of] N [such that] P(n) is true}. The principle of mathematical induction can now be stated as follows: If [(i) 1 [is an element of] G and, (ii) for all k [is an element of] N if k [is an element of] G, then k + 1 [is an element of] G], then G = N. Now symbolize this statement as follows: P: 1 [is an element of] G. R: k [is an element of] G. S: k + 1 [is an element of] G. Q: G = N. Therefore the statement of the principle of mathematical induction can be seen in the following form. If [P and, [for all] k [is an element of] N (if R, then S)], then Q. One strategy for teaching this principle is to explain that in order to apply the principle of mathematical induction and assert Q, one must appeal to the logical rule of modus ponens (the law of detachment). That is, we must affirm the antecedent [P and, [for all] k [is an element of] N (if R, then S)], and then we can assert Q. Therefore the research hypothesis for this study was that if people have the prerequisite knowledge of logic, and that if they are taught the principle of mathematical induction in terms of logic, then they will perform better on a criterion test over the principle of mathematical induction than people who are not taught in terms of logic"--Introduction. / Typescript. / "June, 1972." / "Submitted to the Department of Mathematics Education in partial fulfillment of the requirements for the degree of Doctor of Education in Mathematics Education." / Advisor: E. D. Nichols, Professor Directing Dissertation. / Includes bibliographical references.

published_or_final_version / abstract / Education / Master / Master of Education

This thesis proposes improved methods for the automatic generation of proofs by structural induction in a formal system. The main application considered is proving properties of programs. The theorem-proving problem divides into two parts: (1) a formal system, and (2) proof generating methods. A formal system is presented which allows for a typed language; thus, abstract data types can be naturally defined in it. Its main feature is a general structural induction rule using a lexicographic ordering based on the substructure ordering induced by type definitions. The proof generating system is carefully introduced in order to convince of its consistency. It is meant to bring solutions to three problems. Firstly, it offers a method for generalizing only certain occurrences of a term in a theorem; this is achieved by associating generalization with the selection of induction variables. Secondly, it treats another generalization problem: that of terms occurring in the positions of arguments which vary within function definitions, besides recursion controlling arguments. The method is called indirect generalization, since it uses specialization as a means of attaining generalization. Thirdly, it presents a sound strategy for using the general induction rule which takes into account all induction subgoals, and for each of them, all induction hypotheses. Only then are the hypotheses retained and instantiated, or rejected altogether, according to their potential usefulness. The system also includes a search mechanism for counter-examples to conjectures, and a fast simplification algorithm.

The goal of Inductive Learning is to produce general rules from a set of seen examples, which can then be applied to other unseen examples. ID3 is an inductive learning algorithm that can be used for the classification task. The input to the algorithm is a set of tuples of description and class. The ID3 algorithm learns a decision tree from these input examples, which can then be used for classifying unseen examples given their descriptions. ID3 faces a problem called the replication problem. An algorithm called the Expert-Gate algorithm is presented in this thesis. The aim of the algorithm is to tackle the replication problem. We discuss the various issues involved with each step of the algorithm and present results corroborating our choices. The algorithm was tested on various artificially created problems as well as on a real life problem. The performance of the algorithm was compared with that of Fringe. The algorithm was found to give excellent results on the artificially created problems. The Expert-Gate algorithm gave satisfactory results on the NETtalk problem. Overall, we believe the algorithm is a good candidate for testing on other real life domains. / Graduation date: 1993

Thesis (doctoral)--Technische Universität, Kaiserslautern, 2003. / "'infix' is a joint imprint of Akademische Verlagsgesellschaft Aka GmbH and IOS Press BV (Amsterdam)"--T.p. verso. Includes bibliographical references and index.

The coverage of a learning algorithm is the number of concepts that can be learned by that algorithm from samples of a given size for given accuracy and confidence parameters. This thesis begins by asking whether good learning algorithms can be designed by maximizing their coverage. There are three questions raised by this approach: (i) For given sample size and other learning parameters, what is the largest possible coverage that any algorithm can achieve? (ii) Can we design a learning algorithm that attains this optimal coverage? (iii) What is the coverage of existing learning algorithms? This thesis contributes to answering each of these questions. First, we generalize the upper bound on coverage given in [Dietterich 89]. Next, we present two learning algorithms and determine their coverage analytically. The coverage of the first algorithm, Multi-Balls, is shown to be quite close to the upper bound. The coverage of the second algorithm, Large-Ball, turns out to be even better than Multi-Balls in many situations. Third, we considerably improve upon Dietterich's limited experiments for estimating the coverage of existing learning algorithms. We find that the coverage of Large-Ball exceeds the coverage of ID3 [Quinlan 86] and FRINGE [Pagano and Haussler 90] by more than an order of magnitude in most cases. Nevertheless, further analysis of Large-Ball shows that this algorithm is not likely to be of any practical help. Although this algorithm learns many concepts, these do not seem to be very interesting concepts. These results lead us to the conclusion that coverage maximization alone does not appear to yield practically-useful learning algorithms. The results motivate considering the biased-coverage under which different concepts are assigned different weight or importance based on given background assumptions. As an example of the new setting, we consider learning situations where many of the features present in the domain are irrelevant to the concept being learned. These situations are often encountered in practice. For this problem, we define and study the MIN-FEATURES bias in which hypotheses definable using a smaller number of features involved are preferred. We prove a tight bound on the number of examples needed for learning. Our results show that, if the MIN-FEATURES bias is implemented, then the presence of many irrelevant features does not make the learning problem substantially harder in terms of the needed number of examples. The thesis also introduces and evaluates a number of algorithms that implement or approximate the MIN-FEATURES bias. / Graduation date: 1993

This thesis provides statistical analysis methods and a validation procedure for conducting this statistical analysis, under the common random number (CRN) correlation-induction strategy. The proposed statistical analysis provides estimates for the unknown parameters that are needed for validating the model. While conducting this statistical analysis, we make some key assumptions. Validation comprises of a three-stage statistical procedure. The first stage tests for the multivariate normality,the second stage tests the structure of the covariance matrix between responses, and the third stage tests for the adequacy of the proposed model. The statistical analysis and validation procedures are illustrated with an example of a hospital simulation study. / Master of Science

Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.