Presburger Arithmetic and Its Use in Verification

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CHAPTER 7. PARALLEL EXECUTION OF DECISION PROCEDURES Figure 7.4. Speedup factors in Cooper evaluation. 7.2.1 Finding concurrency As mentioned in Section 4.2.2, converting Presburger formulas into DNF is inevitable for the Omega Test. DNF is a significant bottleneck of the algorithm because formulas may grow exponentially. Take a formula ∧ m i=1 L i ∧ ∧ n j=1 (L1 j ∨ L2 j ) as an example, its DNF version is a disjunction of 2 n conjunctions of m + n literals. DNF is a challenge for employing the Omega Test, but there is a chance of massive concurrency in this procedure. In the above example, 2 n conjunctions are totally independent and ready for being resolved in parallel. To do so we rely on the following rule (which is similar to Equation 7.2): ∃ x 1 ...x n . ∨ ∧ L ij ≡ ∨ ∃ x 1 ...x n . ∧ L ij (7.3) i j i j Using DNF, we can easily divide a formula into many smaller ones and apply the Omega Test independently. The parallel pattern used here is Data Parallelism, when we have the same operation working on a big collection of data. As discussed in Section 2.2, F# provide very good constructs for data parallelism such as Parallel.For(Each), Array.Parallel and PSeq module. Looking into details of the Omega Test procedure, it is easy to recognize that their three basic blocks could be executed in parallel because there is no requirement for order of executions among Real Shadow, Dark Shadow and Gray Shadow. We can easily spawn three independent Tasks, butthereisalittlecoordinationwhich should be done between these tasks. First of all, if Real Shadow’s task completes first and returns false, the other tasks should be terminated immediately. Similarly, if Dark Shadow’s task returns true, the other running tasks should be stopped right away. In the worst-case scenario, the result is returned from Gray Shadow’s task. It is likely that this task is the longest running one so total execution time is comparable to execution time of this task. This idea of three parallel coordinated 56
7.2. A PARALLEL VERSION OF THE OMEGA TEST tasks can be easily implemented in F# using Task and CancellationTokenSource constructs. We have sketched our ideas of parallelizing Omega Test. These two ideas can be incorporated into a full procedure with a sensible way of controlling the degree of parallelism. We recognize that exponential growth of formulas’ size is the biggest hindrance, especially Gray Shadow happens quite often and causes formulas to be normalized into DNF again and again. Noticing that consistency predicates (see Section 5.1) consist of literals with coefficients zero or one, we might think that Exact Shadow is a good choice for quickly resolving formulas of this form. The original form of Exact Shadow (see Equation 4.1) is adapted in a form of strict comparisons: ∃x. β
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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 and 64: 7.1. A PARALLEL VERSION OF COOPER
Page 65: 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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