Presburger Arithmetic and Its Use in Verification

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CHAPTER 5. DURATION CALCULUS AND PRESBURGER FRAGMENTS Figure 5.1. Observation intervals [10]. Figure 5.2. AsimpleKripkestructure[15]. where P is in the set of state variable names, k, c i ∈ Z and ⊲⊳ ∈{}. The above fragments of DC are undecidable already; if the duration is limited in the form ∫ S⊲⊳k,thediscrete-timesatisfiabilityproblemisdecidable[10]. Withthe decidable fragments of DC, the model-checking problem is described in the following manner: models are expressed by means of automata (normally, a labelled automata like Kripke structures) and durations are calculated basing on traces of execution. Here the notion of traces is abstracted to the notion of visiting frequencies of nodes. Therefore, given a DC formula φ and a Kripke structure K, K is a model of φ when every trace of K satisfies φ, writtenK |= φ. We consider an example of Kripke structures described in Figure 5.2 and check it against the property ∫ P < 2. Certainly the answer is false because there exist some traces which visit P at least twice. When abstracting from traces, some frequencies are not corresponding to any real trace, so a counting semantic in a multiset m is introduced [10]. The multiset guarantees the consistency between visiting frequencies and traces. The modelchecking algorithm does a bottom-up marking and generates side-condition in a form of Presburger formulas. The consistency condition C(K, i 0 ,j 0 , m, e) isexpressed by linear constraints of inflow and outflow equations, and there are some 36
5.1. DURATION CALCULUS AND SIDE-CONDITION PRESBURGER FORMULAS additional constraints introduced to deal with loop structures [10]. C(K, i 0 ,j 0 , m, e) has the size of O(|V | + |E|) wheneveryloophasauniqueentrypointanditssatisfiability is the sufficient condition for existence of a consistent trace from i 0 to j 0 . The simplified marking algorithm is given in [10] as follows: sim t (⊤,i,j,m, e) = true sim f (⊤,i,j,m, e) = false sim t (Σ i∈Ω c i ∫ Si
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 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: Chapter 5 Duration Calculus and Pre
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 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 96 and 97:
APPENDIX B. SOURCE CODE OF EXPERIME
Page 98 and 99:
APPENDIX B. SOURCE CODE OF EXPERIME
Page 100:
TRITA-CSC-E 2011:108 ISRN-KTH/CSC/E
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Presburger Arithmetic and Its Use in Verification

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