Presburger Arithmetic and Its Use in Verification

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CHAPTER 7. PARALLEL EXECUTION OF DECISION PROCEDURES parallel elimination is based on following rules where formulas are in NNF: ∃ x 1 ...x n . ∨ i ∀ x 1 ...x n . ∧ i F i ≡ ∨ i F i ≡ ∧ i ∃ x 1 ...x n .F i (7.1) ∀ x 1 ...x n .F i (7.2) However, for arbitrary NNF formulas, the degree of parallelism may be limited because of small disjunctions (or conjunctions). One way to enhance concurrency is converting formulas into DNF. Normally a DNF formula is in a form of a huge disjunction of many inner conjunctions and quantifier elimination can be distributed to inner conjunctions immediately. Because conversion into DNF causes the formula’s size to grow very fast, we only do DNF conversion once at the outermost level of quantifiers. Because there are still some cases where DNF formulas do not expose enough concurrency, we seek concurrency inside the procedure. First, due to recursive nature of Presburger formulas, these formulas are represented by tree data structures. Certainly we can manipulate these formulas by doing operations on tree in aparallelmanner,forexample,doingparallelevaluationoftreebranches.Second, certain parts in quantifier elimination could be done in parallel. As can be seen from Figure 7.1, Get Coefficients, Get A-Terms and Get B-Terms have no order of execution. We are able to create three Tasks to run them concurrently. Similarly Least Satisfying Assignment and Small Satisfying Assignments could be calculated in parallel. The figure preserves relative ratios between sizes of different tasks where Small Satisfying Assignments is the largest task (it consists of |B| (or |A|) timessubstitutionofavariablebyatermintheformula)andEliminate Variable is a very lightweight task where we assemble different results. Assuming that the number of B-Terms are always smaller than that of A-Terms, we have aroughestimationofparallelismbasedontheDAG model of multithreading as follows: •AssumethatEliminate Variable is an insignificant task and omit it. •EstimateeachtaskbythenumberoftraversalsthroughthewholePresburger formula. • Work = 1 + 1 + 1 + 1 + 1 + |B| = |B| + 5 • Span = 1 + 1 + 1 = 3 • Parallelism Factor =(|B| + 5)/3 Roughly speaking, Parallelism Factor is bounded by (|B| + 5)/3, sowecanhardly achieve a good speedup if the number of B-Terms (or A-Terms) is too small. Actually the number of terms is quite small due to symbolic representation of big disjuncts and the choice of the smallest number of terms in projection which leads to limited concurrency in each elimination step. One interesting thing is after each 50
7.1. A PARALLEL VERSION OF COOPER’S ALGORITHM Figure 7.1. Task graph of an elimination step in Cooper’s algorithm. elimination step, a big disjunct consists of 1 + |B| inner formulas which are subject to parallel elimination. Again good concurrency requires a big number of terms. 7.1.2 Design and Implementation The algorithm we use follows ideas of Cooper’s algorithm discussed in Section 4.2 and there are some differences related to eliminating blocks of quantifiers and using heterogeneous coefficients as discussed in Section 6.3. Here we focus on data representation which will be shown to have a big influence on performance later on. The type Term is employed to represent a linear term c + Σ i a i x i : •Itconsistsofaconstantc and a collection of variables x i and corresponding coefficients a i . •Variablescannotbeduplicated,andeachvariablehasexactlyonecoefficient. •Arithmeticoperationssuchasaddition,subtractionbetweenTerms, unary minus of Terms;multiplicationanddivisionbyaconstantarealsosupported. AnaturalrepresentationofTerm in F# is the following type: type Term = struct val Vars: Map val Const: int end 51
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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 10 and 11: 5.2 Simplification of Presburger fo
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: Chapter 7 Parallel execution of dec
Page 63 and 64: 7.1. A PARALLEL VERSION OF COOPER
Page 65 and 66: 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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