Presburger Arithmetic and Its Use in Verification

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CHAPTER 7. PARALLEL EXECUTION OF DECISION PROCEDURES Figure 7.2. Speedup factors with different workloads. Test No. Pigeons Holes Variables Quantifiers Literals 1 21 1 21 3 483 2 22 1 22 3 528 3 23 1 23 3 575 4 24 1 24 3 624 5 25 1 25 3 675 6 26 1 26 3 728 7 27 1 27 3 783 8 28 1 28 3 840 9 29 1 29 3 899 10 30 1 30 3 960 Table 7.1. Test set for Cooper elimination. Each formula contains 32 smaller formulas which can be resolved in parallel. Experimental results are presented in Figure 7.3. As can be seen from the figure, speedup factors are around 4 − 5×; thesesuboptimalspeedupscanbeexplainedby afast-growingnumberofcachemisseswhentheproblemsizeincreases. Someother reasons of suboptimal speedups could be found in Section 3.2. Cooper evaluation is done with some Pigeon-Hole Presburger formulas. They are chosen in a way that their truth values are false, so it ensures that the algorithm has to iterate through the whole search space to search for an answer. Some characteristics of tests formulas are presented in Table 7.2. Experimental results are shown in Figure 7.4. Speedup factors are good, ranging from 5x to 8x, because search spaces are huge and the algorithm has been traversed 54
7.2. A PARALLEL VERSION OF THE OMEGA TEST Figure 7.3. Speedup factors in Cooper elimination only. Test No. Pigeons Holes Variables Quantifiers Literals 1 4 2 8 8 56 2 8 1 8 8 80 3 9 1 9 9 99 4 5 2 10 10 80 5 10 1 10 10 120 6 11 1 11 11 143 7 4 3 12 12 96 8 6 2 12 12 108 9 12 1 12 12 168 10 13 1 13 13 195 Table 7.2. Test set for Cooper evaluation. through them to validate results. Detailed source code of Cooper elimination and evaluation can be found in Appendix B.5. 7.2 A parallel version of the Omega Test This section is about our concern of parallelism in the Omega Test. The section starts with discussion about concurrency occurring in the algorithm and ideas about how to exploit it. Thereafter we present our design, implementation and experimental results. 55
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Presburger Arithmetic and Its Use i
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Abstract Today, when every computer
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Preface The thesis is a part of the
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List of Abbreviations CPU DAG DC DN
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5.2 Simplification of Presburger fo
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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
Page 61 and 62: 7.1. A PARALLEL VERSION OF COOPER
Page 63: 7.1. A PARALLEL VERSION OF COOPER
Page 67 and 68: 7.2. A PARALLEL VERSION OF THE OMEG
Page 69 and 70: 7.2. A PARALLEL VERSION OF THE OMEG
Page 71 and 72: Chapter 8 Conclusions In this work,
Page 73 and 74: References [1] Nikolaj Bjørner. Li
Page 75: [28] C. R. Reddy and D. W. Loveland
Page 78 and 79: APPENDIX A. EXAMPLES OF MULTICORE P
Page 80 and 81: APPENDIX A. EXAMPLES OF MULTICORE P
Page 82 and 83: APPENDIX B. SOURCE CODE OF EXPERIME
Page 100: TRITA-CSC-E 2011:108 ISRN-KTH/CSC/E
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