Presburger Arithmetic and Its Use in Verification

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5.2 Simplification of Presburger formulas . . . . . . . . . . . . . . . . . . 38 5.2.1 Guarded Normal Form . . . . . . . . . . . . . . . . . . . . . . 38 5.2.2 Equation-based simplification . . . . . . . . . . . . . . . . . . 39 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 Experiments on Presburger Arithmetic 43 6.1 Generation of Presburger fragments . . . . . . . . . . . . . . . . . . 43 6.2 Simplification of Presburger formulas . . . . . . . . . . . . . . . . . . 45 6.3 Optimization of Cooper’s algorithm . . . . . . . . . . . . . . . . . . 46 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7 Parallel execution of decision procedures 49 7.1 A parallel version of Cooper’s algorithm . . . . . . . . . . . . . . . . 49 7.1.1 Finding concurrency . . . . . . . . . . . . . . . . . . . . . . . 49 7.1.2 Design and Implementation . . . . . . . . . . . . . . . . . . . 51 7.1.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 53 7.2 A parallel version of the Omega Test . . . . . . . . . . . . . . . . . . 55 7.2.1 Finding concurrency . . . . . . . . . . . . . . . . . . . . . . . 56 7.2.2 Design and Implementation . . . . . . . . . . . . . . . . . . . 57 7.2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . 58 7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8 Conclusions 61 References 63 A Examples of multicore parallelism in F# 67 A.1 Source code of π calculation . . . . . . . . . . . . . . . . . . . . . . . 67 A.2 Source code of MergeSort . . . . . . . . . . . . . . . . . . . . . . . . 69 B Source code of experiments 71 B.1 Utilities.fs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 B.2 Term.fs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 B.3 Formula.fs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 B.4 PAGenerator.fs (excerpt) . . . . . . . . . . . . . . . . . . . . . . . . 77 B.5 Cooper.fs (excerpt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 B.6 OmegaTest.fs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 B.7 OmegaTestArray.fs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Chapter 1 Introduction 1.1 Background In 1965, Gordon Moore proposed Moore’s Law predicting the number of transistors on an integrated circuit would double every 18-24 months resulting in a corresponding increase of processing power in the same time period [32]. Software used to get free extra speed whenever hardware manufacturers released newer and faster mainstream systems. However, when clock frequency of a single processor reached their peak a few years ago, the free lunch was almost over. Due to power consumption problem, instead of making more powerful and faster processors, CPU makers tried to integrate many cores into a single processor. In theory, CPU power still grows according to Moore’s Law, but single-threaded applications cannot benefit from extra CPU resources anymore. Multicore trend helps chip designers to avoid serious problems such as heat losses and leakage current but still achieve good processing power at an economical price. However, multicore computing brings in a new challenge for software architects and developers, they need to reconsider the way which systems are built to be able to have good utilization of CPU resources. Although parallel computing is a hinder for software development because of requiring so much time and effort for development, it brings significant opportunities for organizations being able to exploit parallelism. Organizations will have new experiences of fast service delivery and efficient products leading to potential business opportunities. Some areas obviously requiring parallel computing are product design simulation where manufacturers are able to quickly prototype virtual products and financial modeling thanks to which financial companies can offer customers powerful modeling tools with rich analysis of financial scenarios. Also parallel programming is expected to brings benefits to numerical software products where heavy computation of numerical and symbolic algorithms requires efficient use of computing power. Procedural and object-oriented programs are difficult to parallelize due to the problem of shared states and imperative data structures. This problem does not occur in functional programming, in this thesis we shall investigate how to exploit 1
Page 1 and 2: Presburger Arithmetic and Its Use i
Page 3 and 4: Abstract Today, when every computer
Page 5 and 6: Preface The thesis is a part of the
Page 7: List of Abbreviations CPU DAG DC DN
Page 12 and 13: CHAPTER 1. INTRODUCTION functional
Page 15 and 16: Chapter 2 Multicore parallelism on
Page 17 and 18: 2.1. MULTICORE PARALLELISM: A BRIEF
Page 19 and 20: 2.2. MULTICORE PARALLELISM ON .NET
Page 29 and 30: Chapter 3 Functional paradigm and m
Page 31 and 32: 3.1. PARALLEL FUNCTIONAL ALGORITHMS
Page 33 and 34: 3.2. SOME PITFALLS OF FUNCTIONAL PA
Page 35 and 36: 3.2. SOME PITFALLS OF FUNCTIONAL PA
Page 37 and 38: Chapter 4 Theory of Presburger Arit
Page 39 and 40: 4.2. DECISION PROCEDURES FOR PRESBU
Page 45 and 46: Chapter 5 Duration Calculus and Pre
Page 47 and 48: 5.1. DURATION CALCULUS AND SIDE-CON
Page 49 and 50: 5.2. SIMPLIFICATION OF PRESBURGER F
Page 51: 5.3. SUMMARY occur in equation do n
Page 54 and 55: CHAPTER 6. EXPERIMENTS ON PRESBURGE
Page 56 and 57: CHAPTER 6. EXPERIMENTS ON PRESBURGE
Page 59 and 60: Chapter 7 Parallel execution of dec
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7.1. A PARALLEL VERSION OF COOPER
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7.1. A PARALLEL VERSION OF COOPER
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7.2. A PARALLEL VERSION OF THE OMEG
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Chapter 8 Conclusions In this work,
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References [1] Nikolaj Bjørner. Li
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[28] C. R. Reddy and D. W. Loveland
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APPENDIX A. EXAMPLES OF MULTICORE P
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APPENDIX A. EXAMPLES OF MULTICORE P
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APPENDIX B. SOURCE CODE OF EXPERIME
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TRITA-CSC-E 2011:108 ISRN-KTH/CSC/E
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Presburger Arithmetic and Its Use in Verification

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