Scholars' Mine Masters Theses Student Theses and Dissertations 1970 The dinary numerical system and its applications to modern parallel processing computers Hai Chi Koo Follow this and additional works at: https://scholarsmine.mst.edu/masters_theses Part of the Electrical and Computer Engineering Commons Department: Recommended Citation Koo, Hai Chi, "The dinary numerical system and its applications to modern parallel processing computers" (1970). Masters Theses. 5470. https://scholarsmine.mst.edu/masters_theses/5470 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. THE DINARY NUMERICAL SYSTEM AND ITS APPLICATIONS TO MODERN PARALLEL PROCESSING COMPUTERS BY HAl CHI KOO, 1944- A THESIS submitted to the faculty ot THE UNIVERSITY OF MISSOURI-ROLLA ' . in partial fulfillment of the requirements tor the Degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEEBING Rolla. Missouri 1970 Approved by (advisor) ()a,.l,~ ii ABSTRACT The Dinary Numerical System (DNS) is an extension or Binary Numerical System. By utilizing the newly developed Large-Scale-Integration technology, DNS can be imple­ mented as a new concept of computer organization which has the merits or parallel processing. DNS consists or three rield elements: they are -1, 0, and +1. Since DNS representation is not necessarily unique, new algorithms to generate the number or repre­ sentations and all equivalent representations are pre­ sented in this paper. A Block-Oriented-Computer (BOC) is developed as an array or computing systems. For each of the systems. the arithmetic unit is essentially a modified Digital Difrer­ ential Analyzer. The BOC uses binary and dinary numbers as inputs. Arter being manipulated by special schemes. the BOC which operates rrom left to right contrary to the conventional computers generates dinary numbers as outputs. The merit or the lert-to-right algorithm is that the most signiricant digits are available for further operations while the least signiricant digits are still being processed. iii ACKNOWLEDGEMENTS The author wishes to express gratitude to his advisor Dr. Darrow F. Dawson of the University of Missouri at Rolla for his assistance and guidance throughout the course of this thesis. The author is greatly indebted to Dr. Paul Stigall of the University of Missouri at Rolla for his valuable advice and aid throughout this research program. 1v TABLE OF COi<fTENTS Page ABSTRACT ••••••••••••• • • • • • • • • • • • • • • • • • • • • • • • • • • • 1i ACKNOWLEDGEMENTS ••••• I I I I I I I I I I I I I I I I I I I I I I I I I I I ii1 LIST OF ILLUSTRATIONS ••••• • • • • • • • • • • • • • • • • • • • • • • vii LIST OF TABLES ••••••• • • • • • • • • • • • • • • • • • • • • • • • • • • • viii I • INTRODUCTION •• I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 II. REVIEW OF LITERATURE••••••••••••••••••••• 3 A. Dinary Numerical System •••••••••••••• 3 B. Implementation of Dinary Numerical System ••••••••••••••••••••••••••••••• 3 1. Digital Differential Analyzer •••• 3 2. Dinary Power Increment ••••••••••• 3 c. Block-Oriented-Computer •••••••••••••• 4 III. DINARY NUMERICAL SYSTEM•••••••••••••••••• 5 A. Introduction ••••••••••••••••••••••••• 5 B. Dinary String •••••••••••••••••••••••• 5 c. Fundamental Properties of Dinary Numerical System ••••••••••••••••••••• 11 IV. THE NUMBER OF REPRESENTAT! ONS AND EQUIVALENT REPRESENTATIONS FOR DINARY NUMERICAL SYSTEMS••••••••••••••••• 21 A. Tabulation of the Number of DNS Repr es en ta ti ons ••••••••••••••••••• • • • 21 v B. A Computer Program for Tabulating Number of DNS Representations ••••••• 26 c. Algorithm for Generating Equivalent DNS Representations ••••••••••••••••• 28 D. A Computer Program for Generating Equivalent DNS Representations •••••• 32 E. Conclusion •••••••••••••••••••••••••• 32 V. THE THEORY OF DINARY POWER INCREMENT COMPUTING ••••••••••••••••••••••••••••••• 35 A. Introduction •••••••••••••••••••••••• 35 B. Dinary Computing Principles ••••••••• 35 c. Recursion Formula ••••••••••••••••••• 38 VI. THE IMPLEMENTATION OF DNS ON DIGITAL COMPUTERS • ••••• , •••••••••••••••••••••••• 42 A. Binary to Dinary Conversion ••••••••• 42 B. Dinary Multiplication ••••••••••••••• 44 VII. THE BLOCK- ORIENTED- COMPU'rER ••••••••••••• 54 A. Dinary Numerical System ••••••••••••• 54 B. Structure ••••••••••••••••••••••••••• 55 c. The Arithmetic Unit Element ••••••••• 58 D. Parallel Operation •••••••••••••••••• 60 VIII. CONCLUSION •••••••••••••••••••••••••••••• 62 A. Rev1 ew • ••••••••••••••••• , ••••••••••• 62 B. Ob j e c t i v e s • • • • • • • • • • • • • • • • • • • • • • • • • o 62 c. Suggestion •••••••••••••••••••••••• • • 63 IX. APPEN"DICES ••• , ••••••••••••••••• • • • • • • • • • 64 vi A. A Computer Program for Tabulating Number of DNS Representations ••••••• B. A Computer Program for Generating Equivalent Representations •••••••••• 66 x. BIBLIOGRAPHY ••••••••••••••••••••.••••••• 69 XI. VITA • •••••••••• • ••••••• • ••••••••• • • • • • • • 70 vii LIST OF ILLUSTRATIONS Figures Page 1. A Flow Chart for Tabulating Number of DNS Representations••••••••••••••••••••••••• 27 2. A Flow Chart for Generating Equivalent DNS Representations ••••••••••••••••••••••••• JJ J. The Digital Integrator of Digital Differential Analyzer ••••••••••••••••••••••• 36 4. Distributed Logic Processor Organization of the Block-Oriented-Computer •••••••••••••• 57 5. The Arithmetic Unit Element ••••••••••••••••• 59 viii LIST OF TABLES Tables Page I. The Distribution of DNS Representations ••••••••••••••••••••••• 8 II. The Number of DNS Representations with 1 Digit •••••••••••••••••••••••••• 22 III. The Number of DNS Representations with 2 Digits ••••••••••••••••••••••••• 2.3 IV. The Number of DNS Representations with 3 Digits ••••••••••••••••••••••••• 24 v. The Number of DNS Representations with 4 D1gi ts ••••••••••••••••••••••••• 25 VI. The Conversion Rules for Binary to D1nary Numerical System ••••••••••••••• 29 VII. The Dinary Form Development ••••••••••• 45 VIII. The First Iteration for Dinary Multiplication •••••••••••••••••••••••• 48 IX. The Second Iteration for Dinary Multiplication •••••••••••••••••••••••• X. The Third Iteration for Dinary Multiplication•••••••••••••••••••••••• 50 XI. The Fourth Iteration for Dinary Multiplication •••••••••••••••••••••••• 51 XII. The Fifth Iteration for Dinary Multiplication •••••••••••••••••••••••• 52 XIII. The Sixth Iteration for Dinary Multiplication•••••••••••••••••••••••• 53 XIV. The Alternation Rules ••••••••••••••••• 55 1 I. INTRODUCTION The majority of the present-day computers have several arithmetic units. and most of the operations are on the base of right-to-left algorithms. Some degree of parallel processing and time sharing has been accom­ plished. New technology allows additional arithmetic units at low cost. New techniques of parallel processing using many arithmetic units can now be considered. Parallel processing requires considerable hardware, connections, and controls. With the tremendous advance in the field of Large-Scale-Integration, hardware and connections are no longer the main concern for the designer. The prime design problem is how to operate or control so that maximum parallelism can be accomplished by using hundreds of identical arithmetic units. One of the methods used to obtain highly parallel processing is to establish left-to-right algorithms. This makes the most significant digits available for other operations while the rest of the digits are still under processing. The Dinary Numerical System (DNS} is an extension of the Binary Numerical System (BNS}. It consists of three field elements; they are -1, 0, and +1. The representations of BNS are always unique. The represen­ tations of DNS are not necessarily unique because of the 2 additional field element -1. Litton Industries is currently developing a computer which operates with left-to-right algorithms by using dinary numbers as its output instead of binary numbers. The design and theory of this computer system is based on the modification of a Digital Differential Analyzer. The DNS because of its application to parallel processing is attracting more attention. Chapter III deals with the fundamental properties of DNS. Chapter IV presents two DNS algorithms which are based upon DNS characteristics. One of the algorithms is to generate the number of representations. The other is to generate the DNS equivalent representations from a binary value. Both algorithms are followed by computer programs. Chapter V shows the theory of dinary power increment computing. It is operated on a modified arithmetic unit. Chapter VI presents the implementation of DNS on such arithmetic units. The binary to dinary conversion and the dinary multiplication are also illustrated in this chapter. Chapter VII contains a discussion of the Block­ Oriented-Computer. The Block-Oriented-Computer is a new design concept for highly parallel processing computers. J II. REVIEW OF LITERATURE A. Dinary Numerical System Discarded nearly two decades ago. Dinary Numerical System (DNS) had neither attracted attention nor had been implemented on a computer. This was due to the hardware limitations of the existing electronic devices and magnetic cores which prevented the implementation of DNS. DNS is not only an extension of Binary Numerical System (with an extra field element -1) but also