Interval Mathematics Foundations, Algebraic Structures, and Applications By Hend Dawood A Thesis Submitted to Department of Mathematics, Faculty of Science Cairo University In Partial Fulfillment of the Requirements For the Degree of Master of Science In Computer Science March 2012 c Copyright 2012 by Hend Dawood All Rights Reserved Approval Sheet for Submission Thesis Title: Interval Mathematics: Foundations, Algebraic Structures, and Applications. Name of Candidate: Hend Dawood Mohamed. This thesis has been approved for submission by the supervisors: 1. Assoc. Prof. Dr. Hassan A. M. Aly Signature: 2. Assoc. Prof. Dr. Hossam A. H. Fahmy Signature: Prof. Dr. Mohamed Zeidan Abdalla Signature: Chairman of Mathematics Department Faculty of Science, Cairo University. Addresses Hend Dawood (Author) Department of Mathematics Faculty of Science, Cairo University Giza, PO Box: 12613 Egypt. Hassan A. M. Aly (Principal Adviser) Associate Professor of Information Security and Cryptography Department of Mathematics Faculty of Science, Cairo University Giza, PO Box: 12613 Egypt Hossam A. H. Fahmy (Adviser) Associate Professor of Electronics and Communications Engineering Department of Electronics and Communications Engineering Faculty of Engineering, Cairo University Giza, PO Box: 12613 Egypt To my family. Abstract Student Name: Hend Dawood Mohamed. Thesis Title: Interval Mathematics: Foundations, Algebraic Structures, and Applications. Degree: Master of Science in Computer Science. We begin by constructing the algebra of classical intervals and prove that it is a nondistributive abelian semiring. Next, we formalize the notion of interval dependency, along with discussing the algebras of two alternate theories of intervals: modal intervals, and constraint intervals. With a view to treating some problems of the present interval theories, we present an alternate theory of intervals, namely the “theory of optimizational intervals”,and prove that it constitutes a rich -field algebra, which extends the ordinary field of the reals, then we construct an S optimizational complex interval algebra. Furthermore, we define an order on the set of interval numbers, then we present the proofs that it is a total order, compatible with the interval operations, dense, and weakly Archimedean. Finally, we prove that this order extends the usual order on the reals, Moore’s partial order, and Kulisch’s partial order on interval numbers. Keywords. Classical interval arithmetic; Machine interval arithmetic; Interval dependency; Constraint intervals; Modal intervals; Classical complex intervals; Optimizational intervals; Optimizational complex intervals; -field algebra; Ordering interval numbers. S Supervisors: 1. Assoc. Prof. Dr. Hassan A. M. Aly Signature: 2. Assoc. Prof. Dr. Hossam A. H. Fahmy Signature: Prof. Dr. Mohamed Zeidan Abdalla Signature: Chairman of Mathematics Department Faculty of Science, Cairo University. vii Publications from the Thesis During the course of the thesis work, we published the following book that provides a bit of perspective on some topics of this research. Some ideas and figures of the thesis have appeared previously in this book. Hend Dawood; Theories of Interval Arithmetic: Mathematical Foundations and Applications; LAP Lambert Academic Publishing; Saarbrücken; 2011; ISBN 978-3- 8465-0154-2. http://books.google.com/books?id=Q0jPygAACAAJ ix Acknowledgments I am deeply indebted to my teacher and supervisor Assoc. Prof. Dr. Hassan A. M. Aly, who showed a lot of interest in guiding this work, for his continual support and wise advice and for so many most fruitful discussions. My discussions with Dr. Hassan inspired me to think in a radically new way that permeated the present work. I owe great thanks to my supervisor Assoc. Prof. Dr. Hossam A. H. Fahmy for encouraging my early interests in interval arithmetic and for his fruitful discussions and valuable suggestions. Dr. Hossam spent a lot of his time introducing me to the concepts of the interval theory. It is a pleasure to acknowledge the substantial help which I have received from many emi- nent professors and dear friends in the Department of Mathematics, Faculty of Science, Cairo University, throughout the course of this thesis. Special thanks are due to Prof. Dr. Nefertiti Megahed and Prof. Dr. Mohamed A. Amer for giving me so much of their precious time and for their moral and scientific support throughout this work. I would like also to acknowledge Prof. Dr. Jürgen Wolff von Gudenberg, Lehrstuhl für Informatik II, Universität Würzburg, and Prof. Dr. G. William (Bill) Walster, Computer Sys- tems Research, Sun Microsystems Laboratories, for their constructive comments and valuable feedbacks from which I have benefited greatly. Finally, my greatest debt is to my family who has always been supportive and encouraging. Hend Dawood Cairo University, Cairo, March, 2012. [email protected] xi Notation and Conventions Most of our notation is standard, and notational conventions are characterized, in detail, on their first occurrence. However, we have many theories of intervals being discussed throughout the text, with each theory has naturally its own pe- culiar notation; along with some basic logical, set-theoretic, and order-theoretic symbols. So, for the purpose of legibility, we give here a consolidated list of symbols for the entire text. Logical Symbols = Identity (equality). Logical Negation (not). : Implication (if ..., then ...). ) Equivalence (if, and only if ). , Conjunction (and). ^ Inclusive disjunction (or). _ Universal quantifier (for all). 8 Existential quantifier (there exists). 9 Q , A quantifier variable symbol (with or without subscripts). 2 f8 9g A quantification matrix, (Q1x1) ... (Qnxn), where x1, ..., xn are vari- Q able symbols and each Qi is or . 8 9 n xk The universal quantification matrix, ( x1) ... ( xn). 8k=1 8 8 n xk The existential quantification matrix, ( x1) ... ( xn). 9k=1 9 9 ' A quantifier-free formula. A prenex sentence, ', where is a quantification matrix and ' Q Q is a quantifier-free formula. xiii NOTATION AND CONVENTIONS Set-Theoretic and Order-Theoretic Symbols ? The empty set. , , Set variable symbols (with or without subscripts). S T U } ( ) The powerset of a set . S S n The n-th Cartesian power of a set . Sh i S The set membership relation. 2 The set inclusion relation. The set intersection operator. \ The set union operator. [ n k The finitary set intersection 1 ... n. \k=1S S \ \S n k The finitary set union 1 ... n. [k=1S S [ [S A relation variable symbol (with or without subscripts). < dom ( ) The domain of a relation . < < ran ( ) The range of a relation . < < fld ( ) The field of a relation . < < The converse of a relation . < < Idb The identity relation on a set . S S -inf ( ) The infimum of a set relative to an ordering relation . < S S < -sup ( ) The supremum of a set relative to an ordering relation . < S S < g The lattice binary join operator. f The lattice binary meet operator. Interval-Theoretic Symbols R The set of real numbers. x, y, z Real variable symbols (with or without subscripts, and with or with- out lower or upper hyphens). xiv NOTATION AND CONVENTIONS a, b, c Real constant symbols (with or without subscripts, and with or with- out lower or upper hyphens). +, A binary algebraic operator. 2 f g , 1 A unary algebraic operator. 2 f g [R] The set of classical interval numbers. [R]p The set of point (singleton) classical interval numbers [x, x]. [R] The set of symmetric classical interval numbers [ x, x]. s [R]0 The set of zeroless classical interval numbers. X, Y, Ze Interval variable symbols (with or without subscripts). A, B, C Interval constant symbols (with or without subscripts). inf (X) The infimum of an interval number X. sup (X) The supremum of an interval number X. w (X) The width of an interval number X. r (X) The radius of an interval number X. m (X) The midpoint of an interval number X. X The absolute value of an interval number X. j j d (X, Y ) The distance (metric) between two interval numbers X and Y . M R The set of machine-representable real numbers. Mn The set of machine real numbers with n significant digits. [M] The set of machine interval numbers. Downward rounding operator. 5 Upward rounding operator. 4 Outward rounding operator. If (X1, ..., Xn) The image of the real closed intervals X1, ..., Xn under a real-valued function f. Y DX The interval variable Y is dependent on the interval variable X. Y X The interval variable Y is independent on the interval variable X. = xv NOTATION AND CONVENTIONS n (Xk) All the interval variables X1, ..., Xn are pairwise mutually independent. =k=1 The set of modal intervals. M The set of existential (proper) modal intervals. M9 The set of universal (improper) modal intervals. M8 mX, mY, mZ Modal interval variable symbols (with or without subscripts). mA, mB, mC Modal interval constant symbols (with or without subscripts). mode (mX) The mode of a modal interval mX. set (mX) The set of a modal interval mX. dual (mX) The dual of a modal interval mX. proper (mX) The proper of a modal interval mX. improper (mX) The improper of a modal interval mX. inf (mX) The infimum of a modal interval mX. sup (mX) The supremum of a modal interval mX. t [R] The set of constraint intervals. t [R]p The set of point constraint intervals. t [R]s The set of symmetric constraint intervals. t [R]0 The set of zeroless constraint intervals. o [Re] The set of optimizational interval numbers. o [R]p The set of point optimizational interval numbers. o [R]s The set of symmetric optimizational interval numbers. o [R]0 The set of zeroless optimizational interval numbers. Ce The set of ordinary complex numbers. i = p 1 The ordinary imaginary unit. x, y, z Complex variable symbols (with or without subscripts). a, b, c Complex constant symbols (with or without subscripts). [C] The set of classical complex intervals. [C]p The set of point classical complex intervals.