Presburger Arithmetic and Its Use in Verification

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CHAPTER 7. PARALLEL EXECUTION OF DECISION PROCEDURES elimination based on Exact Shadow is very effective; it is able to eliminate all quantifiers for consistency predicates from 3-sequence and 4-sequence automata. In the case of 5-sequence automaton, the procedure stops with three quantifiers left, we think that even this partial elimination could help to simplify Presburger formulas. Detailed source code of the Omega Test can be found in Appendix B.6 and B.7. Automaton Quantifiers before Quantifiers after 3-seq 3 0 4-seq 6 0 5-seq 10 3 Table 7.4. Results of the Omega Test. 7.3 Summary We have discussed parallelism aspects of Cooper’s algorithm and the Omega Test. Each discussion follows with some experiments, reasonable speedups have been obtained. It is easy to come up with a correct implementation of these parallel algorithms; however, it is not free to achieve good efficiency. The method we use for implementation has some interesting points: •Weemployatrial-and-errorprocess.Eachattemptisevaluatedbyprofiling applications with some specific inputs. •Somelow-leveltoolsareusedincludingCLR Profiler and ILSpy. Theformer provides more information regarding memory and garbage collection and the latter helps to translate F# to other .NET languages for understanding low-level implementation and optimizing F# code. •Wefollowtheidiomoflittlememoryallocationandlittlegarbagecollection; therefore, code is optimized so that their memory footprints are small and unnecessary garbage collection is avoided. F# has provided a very convenient environment for parallel programming. We can start prototyping algorithms without worrying about sophisticated synchronization constructs and F# interpreter has been a good indicator for parallelism, so that we can decide when to start benchmarking. However, it is still difficult to optimize algorithms for parallelism because memory footprints of F# programs are quite large and garbage collection has a bad influence on performance in an unpredictable way. 60
Chapter 8 Conclusions In this work, we have studied multicore parallelism by working on some small examples in F#. The result is promising; the functional paradigm makes parallel processing really easy to start with. We have also studied theory of Presburger Arithmetic, especially their decision procedures. We have applied the functional paradigm in a simplification algorithm for Presburger formulas and two important decision procedures: Cooper’s algorithm and the Omega Test. Actually the paradigm is helpful because formulation in a functional programming language is really close to problems’ specification in logic and the like. The outcome is clear; we can prototype these decision algorithms in a short time period and experiment with various design choices of parallelism in a convenient way. Some concrete results of this work are summarized as follows: •Proposeanewsimplificationalgorithmwhichreduces4.5% ofnumbersof quantifiers and 10.5% ofnestinglevelsofquantifierscomparedtotheold method. •Showthateliminating blocks of quantifiers and using heterogeneous coefficients of literals are helpful, which contribute up to 20× performance gain compared to the baseline implementation. •PresentparallelismconcernsinCooper’salgorithm. Wehavegot4 − 5× speedup in elimination of formulas consisting of 32 disjunctions. The evaluation phase is easier to parallelize, and we have achieved 5 − 8× speedup for the test set of Pigeon-Hole Presburger formulas. •ExplorethechanceofparallelismintheOmegaTest.Wehaveimplemented apartialprocessusingExact Shadow, andhavegot5 − 7× speedup for the test set of consistency predicates. While doing the thesis, we have learned many interesting things related to functional programming, parallelism and decision procedures: 61
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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
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Chapter 2 Multicore parallelism on
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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 and 64: 7.1. A PARALLEL VERSION OF COOPER
Page 65 and 66: 7.2. A PARALLEL VERSION OF THE OMEG
Page 67 and 68: 7.2. A PARALLEL VERSION OF THE OMEG
Page 69: 7.2. A PARALLEL VERSION OF THE OMEG
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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