Verification of Parameterised FPGA Circuit Descriptions with Layout ...

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CHAPTER 4. VERIFYING CIRCUIT LAYOUTS 72 CompilerSimps Structures Functions ParallelComposition IntAlgebra QuartzLayout SeriesComposition Inbuilt Block Types Figure 4.3: The QuartzLayout theory hierarchy. Rectangular nodes develop the theory of the language itself, oval nodes define functions for size expressions and useful theorems. Arrows indicate dependencies. actual hardware generated by a Quartz description is of interest only so far as it affects the layout of the design and wiring is of no interest, since we assume that wiring resources are separate from computational resources. Our Quartz model is built up hierarchically from a number of Isabelle theories, as illustrated in Figure 4.3. In this chapter we describe how some of the language features of Quartz are modelled and identify some useful theorems. The full Isabelle theory development can be seen in Appendix B. 4.4.1 Type System Isabelle/HOL has types and polymorphism based on the Hindley/Milner system [53] - the same basic system as Quartz. It also supports overloading, although it is based on type classes [84] which are not compatible with the Quartz overloading system [64]. Overloading complicates verification since it is not necessarily possible to determine which block instance is selected and different instances can have different size expressions. In the QuartzLayout system we ignore overloading and assume it has been resolved prior to
CHAPTER 4. VERIFYING CIRCUIT LAYOUTS 73 verification. The modified Quartz compiler ensures that this occurs (see Section 4.5). Otherwise the HOL type system is generally suitable for modelling the Quartz type system. Quartz boolean types can be represented as HOL booleans and Quartz integers can be modelled as HOL integers. Isabelle/HOL also defines a theory of natural numbers with an extensive range of useful theorems and we will make some use of this when modelling recursive blocks, however in general we use the integer type which is a more accurate model and supports true arithmetic. Quartz tuples are represented using the HOL Cartesian product type, which defines tuples as nested pairs. This means that the types of (a, b, c) and (a, (b, c)) are considered equivalent, however this is not a major problem since this is more permissive than the Quartz type system and designs must already type-check using the Quartz compiler’s type processor. The Types theory defines the types of wires and of vectors. Wires are named types only, with no properties. This is a useful model for our purposes since we are actually not interested in wiring in any way and the values on wires in a generated circuit are not relevant to its compilation. Vectors are defined as functions from integers to some other type. In this model a two dimensional Quartz vector wire a[m][n] is given type int → int → wire. There are significant limitations to this model [52], essentially arising from the fact that it is not possible to identify that a vector has a fixed number of elements and if each element of two vectors are equivalent then the vectors themselves are equivalent. However, this theoretical model is quite sufficient for layout proofs although it would likely be limiting when reasoning about functional properties. 4.4.2 Blocks and Block Instantiation There are three characteristics of blocks that are of interest when reasoning about Quartz layouts: their semantic interpretation, width expression and height expression. It is necessary to model the meaning of blocks as well as purely their height and width expressions since the two are inter-related, with size expressions possibly involving internal signals defined in a block.
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Imperial College of Science, Techno
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Acknowledgements Firstly, I’d lik
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TABLE OF CONTENTS iv 2.5 Isabelle:
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TABLE OF CONTENTS vi 5.3.1 Speciali
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TABLE OF CONTENTS viii C.1.1 fst .
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Chapter 1 Introduction This thesis
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CHAPTER 1. INTRODUCTION 3 B A C Fig
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CHAPTER 1. INTRODUCTION 5 pler, all
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CHAPTER 2. BACKGROUND AND RELATED W
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CHAPTER 5. SPECIALISATION 123 a fas
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CHAPTER 5. SPECIALISATION 125 block
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CHAPTER 5. SPECIALISATION 127 y y y
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CHAPTER 5. SPECIALISATION 129 with
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CHAPTER 6. LAYOUT CASE STUDIES 131
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CHAPTER 7. CONCLUSION AND FUTURE WO
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Bibliography [1] A. Aggoun and N. B
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BIBLIOGRAPHY 177 [19] H. Gelernter.
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BIBLIOGRAPHY 179 [41] Y. Li and M.
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BIBLIOGRAPHY 181 [60] L. C. Paulson
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BIBLIOGRAPHY 183 [83] J. Voeten. On
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APPENDIX A. QUARTZ LANGUAGE GRAMMAR
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Appendix B Theoretical Basis for La
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APPENDIX B. THEORETICAL BASIS FOR L
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Appendix C Placed Combinator Librar
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APPENDIX C. PLACED COMBINATOR LIBRA
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