A Soliton Circuit Design System by Michael Peter Groves, B.Sc. (Ilons.) A thesi,s submitted, tor the degree ot Doctor of Philosophy in the Depørtment of Computer Science Uniuersity of Adelaide June 1987 -)* \qt-l A, J . ., n -bqr à4 , N ^* CONTENTS LIST OF TABLES vi LIST OF FIGURES vii SUMMARY x DECLARATION xi ACI{NOWLEDGEMENTS xii CIIAPTER 1. INTRODUCTION I 1.1. Soliton circuits 1 1.2. Aims 2 t 1.3. Limitations r) 1.4. Surrmary of the thesis 4 CHAPTER, 2. COMPONENTS 5 2.1. Introduction b 2.2. Connectors 6 2.2.L. Polyacetylene chains 6 2.2.2. Junctions I 2.2.3. Ring junctions 10 2.2.4. Introduction to dynamic circuit diagrams L2 2.3. Soliton switches 13 2.3.L. Logical view 13 2.3.2. The hinge switch 15 2.3.3. A stereochemical switch 16 2.3.4. Conclusions on switch structures t7 2.4. Hills and seesaws 18 2.4.L. Soliton hill 18 2.4.2. Soliton seesaw 19 2.4.3. The carbonyl seesaw 20 2.5. Sids 20 2.5J. Logical view 20 2.5.2. Seesaw sids 2L 2.6. Dynamic circuit diagrams 25 2.6.L. Introduction 25 2.6.2. A symbol for inversion 26 2.6.3. The null symbol 28 2.6.4. N-arm junctions 28 2.6.5. N-arm ring junctions 30 2.6.6. Combinations of symbols 30 (Ð 2.6.7. Summary of symbols 32 2.6.8. Rules for component symbols 34 2.6.9. Rules for dashes in symbols 35 2.6.LO. Tokens 36 2.6.LL. Usefulness of dynamic circuit diagrams 37 2.7. Synthesis and verifrcation of components 39 2.7.1. Introduction 39 2.7.2. Verifi.cation of structures for components 39 2.7.3. External electrical connections 40 2.7.4. Synthesis of a switch 4L 2.7.5. Verification of a switch 43 2.7.6. Synthesis of sids 45 2.7.7. Verifi.cation of sids 46 2.7.8. From switches to soliton circuits 50 2.7.9. Conclusions on the synthesis of components 51 2.8. Conclusions 52 CIIAPTER 3. CIR,CUITS 53 3.1. Introduction 53 3.2. The single soliton switch 54 3.2.L. The problem 54 3.2.2. The structure 55 3.2.3. Logical symbol 58 3.3. Mechanisms and models 60 3.3.1. The mechanism of soliton generation 60 3.3.2. Communication models 62 3.3.3. The soliton model 63 3.3.4: The potential model 64 3.3.5. The amplifier 66 3.3.6. The state model 70 3.4. ,A,mplifiers and inverters 75 3.4.L. Introduction 75 3.4.2. Amplifiers to drive an odd number of inputs tÐ 3.4.3. External inversion 77 3.4.4. Internal inversion 79 3.4.5. Potential inverters 81 3.5. Boolean gates 83 3.5.1. Introduction 83 3.5.2. And gales 84 3.5.3. Or gates 86 3.5.4. External inversion 87 3.5.5. Internal inversion 91 3.5.6. Conclusions on gates formed from arnplifiers and inverters 91 3.5.7. Sub-gates 93 3.5.8. The erclusiue or gate 96 (ii) 3.6. Memories 99 3.6.1. Introduction 99 3.6.2. Selectors 99 3.6.3. A memory cell 101 3.6.4. The fan memory to4 3.6.5. Grid memories 107 3.6.6. The properties of these memories 109 3.7. Circuit design 110 3.7.1. Introduction 110 3.7.2. Design using boolean gates 111 3.7.3. Complex gates LTz 3.7.4. The binary adder as a complex gate 116 3.7.5. Intuitive design 118 3.7 .6. Programmable circuits L20 3.8. Input, output and clocking for soliton circuits L2L 3.8.1. Input and output LzL 3.8.2. Clocking L23 3.9. Conclusions L24 CII.A.PTER 4. MATIIEMATICAL CIRCUIT VER,IFICATION L26 4.1. Introduction L26 4.2. Mathematical state calculations 128 4.2.L. Introduction L28 4.2.2. Legal states L29 4.2.3. Legal states of components and chains L29 4.2.4. Legal states of circuits L32 4.2.5. Examples of state calculations L32 4.2.6. Numbering states 135 4.2.7. Usefulness of mathematical state calculations 138 4.3. State calculation by computer 138 4.3.1. Introduction 138 4.3.2. The algorithm 139 4.3.3. The program L4L 4.3.4. Using the program L4L 4.3.5. Output L43 4.4. Determination of loops L45 4.4.L. Introduction L45 4.4.2. Some assumptions L45 4.4.3. Loops r47 4.4.4. Graphs of circuits L47 4.4.5. Algorithms 148 4.4.6. Loops of the amplifier 151 4.5. Determining initial states L52 4.5.1. Initial states of circuits L52 (iii) 4.5.2. Initial states of circuit elements 155 4.5.3. Initial states of the amplifier 156 4.6. Determining final states 158 4.6.L. Introduction 158 4.6.2. Final states 158 4.6.3. The soliton function 159 4.6.4. Implementation of the soliton function 161 4.6.5. Properties of the soliton function L62 4.6.6. Final states for the amplifier 163 4.6.7. Determining the final states directly t64 4.6.8. The inverse of the soliton function 167 4.6.9. Crossed loops 168 4.6.10. Generalizing the soliton function 169 4.7. Soliton matrices 170 4.7.L. The soliton relation L70 4.7.2. Sets of legal states L7t 4.7.3. Consecutive solitons L73 4.7.4. Alternative solitons 173 4.7.5. Obtaining the domain and range L75 4.7 .6. Special matrices L75 4.7.7. Soliton matrices of the amplifier 176 4.8. Energetic considerations t77 4.8.1. A simple model L77 4.8.2. The energy model 178 4.8.3. The state of hills 180 4.8.4. Input potentials 180 4.8.5. External blocking of loops 181 4.8.6. Using the potential model L82 4.8.7. Energetics of the amplifier 183 4.8.8. The FE and ZE matrices 184 4.8.9. Equivalent energy states 185 4.8.10. Modelling equivalent states 188 4.9. Sets of states that a circuit can enter 190 4.9.1. Introduction 190 4.9.2. Defi.ning the mathematical verification problem 190 4.9.3. Possible and probable states 191 4.9.4. Probable states fo¡ the amplifier 193 4.9.5. Steady and sticky states L94 4.9.6. Steady states of the amplifier 196 4.9.7. Possible sticky states and probable steady states 197 4.9.8. Probable steady states of the amplifier 200 4.9.9. Crossed loops 20L 4.9.10. Short circuits 203 4.9.11. Summary of the states that a circuit can enter 205 4.10. The simulator 206 4.10.1. The simulation process 206 (iu) 4.LO.2. Programming the simulations 208 4.10.3. Simulating an and gate 209 4.LO.4. The equivalent states of the and gate 2L5 4.10.5. Simulation of a binary adder 216 4.10.6. The usefulness of the simulator 2L8 4.11. Mechanisms for sids 2L9 4.12. Other approaches 223 4.13. Conclusions 226 CI{Á.PTER 5. CONCLUSIONS AND FURTHER TI¡ORK 230 5.1. Summary 230 5.2. Further work 23t 5.3. Conclusions 232 .t., ., REFERENCES .JJ Addendum 235 (") LIST OF TABLES Table 4.1. The initial states of the amplifier. 158 Table 4.2. The initial and final state pairs for the amplifier. L64 Table 4.3. The potenergy for the loops of the amplifier. 184 Table 4.4. The details for each of the remaining loops of the and gate. 2L3 (ui) LIST OF FIGURES Figure 2.1. Trans-polyacetylene. I Figure 2.2. The three states of a junction. I Figure 2.3. L ring junction in its four stable states. 11 Figure 2.4. The symbol for the soliton junction, in its three states. t2 Figure 2.5. The symbol for the ring junction. 13 Figure 2.6. The three states of a soliton switch. L4 Figure 2.7. A hinge switch. 16 Figure 2.8. stereochemical switch. 17 Figure 2.9. ^Chemical structure and logical symbol for the soliton hill. 18 Figure 2.10. The soliton seesaw. 19 Figure 2.11. The analogy between a seesa\á and a soliton seesaw. 19 Figure 2.L2. A structure for a carbonyl seesaw. 20 Figure 2.13. The logical symbol for a sid. 2L Figure 2.L4. A seesaw and a hill combined to form a seesaw sid. 22 Figure 2.15. Carter's soliton generator. 23 Figure 2.16. The presence and absence of inversion between bonds on a polyacetylene chain. 27 Figure 2.17. The logical symbols for inverting and non-inverting chains. 28 Figure 2.18. The null symbol. 28 Figure 2.19. The four states of two normal junctions connected by an inverting chain and the equivalent states of a four-arm junction. 29 Figure 2.2O. TJrre four-arm ring junction is formed by connecting two normal two normal ring junctions. 30 Figure 2.2L. Two unconnected chains. 30 Figure 2.22. The combined symbols for two connected switches. 31 Figure 2.23. The three states of a combined switch and sid symbol. 32 Figure 2.24.