Discrete Mathematics Using a Computer John O’Donnell, Cordelia Hall and Rex Page Discrete Mathematics Using a Computer Second Edition John O’Donnell, PhD Cordelia Hall, PhD Computing Science Department, University of Glasgow, Glasgow G12 8QQ, UK Rex Page, PhD School of Computer Science, University of Oklahoma, Norman, Oklahoma, USA British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2005935334 ISBN-10: 1-84628-241-1 ISBN-13: 978-1-84628-241-6 Printed on acid-free paper © Springer-Verlag London Limited 2006 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed in the United States of America (HAM) 987654321 Springer Science+Business Media springer.com This book is dedicated to our parents. Preface to the Second Edition Computer science abounds with applications of discrete mathematics, yet stu- dents of computer science often study discrete mathematics in the context of purely mathematical applications. They have to figure out for themselves how to apply the ideas of discrete mathematics to computing problems. It is not easy. Most students fail to experience broad success in this enterprise, which is not surprising, since many of the most important advances in science and engineering have been, precisely, applications of mathematics to specific science and engineering problems. To be sure, most discrete math textbooks incorporate some aspects applying discrete math to computing, but it usually takes the form of asking students to write programs to compute the number of three-ball combinations there are in a set of ten balls or, at best, to implement a graph algorithm. Few texts ask students to use mathematical logic to analyze properties of digital circuits or computer programs or to apply the set theoretic model of functions to understand higher-order operations. A major aim of this text is to integrate, tightly, the study of discrete mathematics with the study of central problems of computer science. Concepts in discrete mathematics are illustrated through the solution of problems that arise in software development, hardware design, and other fun- damental domains of computer science. The text introduces discrete math concepts and immediately applies them to computing problems. Applications of mathematical logic in design and analysis of hardware and software is an especially strong theme. The goal in this part of the material is to prepare stu- dents for a world that places a high value on the correct operation of computing systems in safety-critical, security-sensitive, and embedded systems and recog- nizes that formal methods based in mathematical logic are the primary tools for ensuring that computing systems function properly in such environments. The emphasis, here, is on preparation. In commercial applications, mecha- nized logic engines are essential to the enterprise of applying logic to the design and implementation of computing hardware and software. This text introduces students to mechanized logic in the form of propositional proof checking, and, vii viii Preface through numerous paper-and-pencil exercises in applying logic to mathematical verification of hardware and software artifacts, gives students experience with the fundamental notions used by engineers who apply mechanized logic engines to the design of commercial computing systems. We believe these skills will be of increasing value in computer and software engineering, and our experience suggests that such skills contribute positively, even in the short run, to the ability of students to successfully design and implement software. The text is organized in four parts: reasoning with equations, formal logic, set theory, and applications. The principle of induction is introduced early, for reasoning with equations, and applied to problems throughout the text. Reasoning with equations covers examples in several domains, including natural numbers of course, but also including sequences and sets. The logic portion of the text discusses two frameworks for formal reasoning: the natural deduction format of Gentzen and another syntax-based reasoning system based in Boolean algebra. Propositional logic is introduced first, then predicate logic, both in a natural deduction and Boolean algebra setting. Set theory discusses the usual basics, and illustrates many of the concepts by applying induction to define the integers. The set theoretic definitions of relations and functions are discussed, along with the usual properties that categorize them and allow them to be combined and manipulated. The applications portion of the text covers two extended examples, one concerning the design of a circuit for n-bit, ripple-carry addition, the other on the implementation of AVL tree operations. These augment the many, smaller examples that occur throughout the text and, together, help students understand how discrete mathematics contributes to the solution of difficult and important problems in computing. A website for the text contains a collection of tools for experimenting with most of the concepts introduced. Included among these is a proof-checking system for propositional calculus. Students can use this system to make sure their proofs are correct and, more importantly, to experience the notion that proofs can be entirely formal and, therefore, useful in verifying the correctness of software and digital circuits. Other tools allow experimentation with set operations, Boolean formulas, and the notions of predicate calculus. These tools are expressed in Haskell, and the various operations for experimentation, including proofs, are expressed using Haskell syntax. In addition, Haskell is used to express the software and hardware designs that illustrate practical uses of logic and other aspects of discrete mathematics in computer science. We feel that Haskell is an ideal notational choice for these examples be- cause of its close affinity with customary algebraic notation. The compactness of software and hardware artifacts expressed in Haskell is another important advantage. Haskell serves both as a formal, mathematical notation, and as a practical and powerful programming language. This helps to strengthen the tight connection between mathematics and applications. Thus Haskell is used in the text on an equal footing with other mathematical notations. Students see Haskell in its role as a programming language, as well as a hardware description Preface ix language, and the emphasis in this book is on reasoning about programs and circuits, not just programming. We hope that students will find the experience of learning about logic, sets, mathematical induction, and other concepts of discrete mathematics and its applications to computing problems interesting and enjoyable, and that they will be able to use these ideas in subsequent studies and professional work in computer science. Software Tools for Discrete Mathematics A central part of this book is the use of the computer to help learn the discrete mathematics. The software (which is free; see below) provides many facilities that aid the student in learning the material: • Logic and set theory have many operators that are used to build mathe- matical expressions. The software allows the user to type in such expres- sions interactively and experiment with them. • Predicate logic expressions with quantifiers can be expanded into propo- sitional logic expressions, as long as the universe is finite and reasonably small. This makes the meaning of the quantifiers more concrete and helps the development of intuition. • Students frequently misuse expressions in logic and set theory; a typical error that arises frequently is to write an expression that treats A ⊆ B as a set rather than a Boolean value. The software tools will immedi- ately flag such mistakes as type errors. Teaching experience shows that many students will have long-lasting misconceptions about basic nota- tions without immediate feedback. • A formal proof checker for natural deduction is provided. This allows students to find errors in their proofs before handing in exercises, and it also provides a quick and effective way for the instructor to check the validity of large numbers of proofs. Furthermore, the automated proof checker underscores the nature of formal proof; vague or ill-formed proofs are not acceptable. • Using a proof checker gives a deeper appreciation of the relationship be- tween discrete mathematics and computer science. The experience of debugging a proof is much like debugging a computer program; the proof checker is itself a computer