Presburger Arithmetic and Its Use in Verification

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CHAPTER 4. THEORY OF PRESBURGER ARITHMETIC δ∨ F ′′ (x) = j=1(F ′ +∞ (j) ∨ ∨ F ′ (a − j)) a∈A Construction of infinite formulas F ′ −∞(j) andF ′ +∞(j) isbasedonsubstitution of every literal α by α −∞ and α +∞ respectively: α α −∞ α +∞ 0 = x + t ′ ⊥ ⊥ 0 ≠ x + t ′ ⊤ ⊤ 0 < x+ t ′ ⊥ ⊤ 0 < −x + t ′ ⊤ ⊥ α α α Table 4.1. Construction of infinite formulas. The inner big disjunct of formulas is constructed by accumulating all literals which are A-Terms or B-Terms: Literals B A 0 = x + t ′ −t ′ − 1 −t ′ +1 0 ≠ x + t ′ −t ′ −t ′ 0 < x+ t ′ −t ′ − 0 < −x + t ′ − t ′ α − − Table 4.2. Construction of A-Terms and B-Terms [6]. There are some bottlenecks of Cooper’s algorithm which need to be carefully addressed. First, coefficients of big disjuncts become quite huge after some quantifier alternation. One way to avoid this problem is keeping big disjuncts symbolically and only expanding them when necessary. Actually, symbolic representation of big disjuncts is easy to implement and quite efficient because inner formulas demonstrate all properties of big disjuncts. Another optimization proposed by Reddy et al. is only introducing a new divisibility constraint after selecting a literal for substitution. This modification avoids calculating lcm of all coefficients of a variable which is often a huge value so that each substitution results in a new coefficient smaller than lcm [28]. The authors argued that if quantifier alternation is done for aseriesofexistentialquantifiersonly,heterogeneous coefficients of different divisibility constraints are not multiplied together; therefore, coefficients are not exploded. Another bottleneck of Cooper’s algorithm is a large number of A-Terms and B- Terms occurring while doing quantifier alternation. Due to symmetric projections, we can always choose an infinite projection with the least number of literals. This heuristic minimizes the number of disjuncts inside the bounded formula. However, when quantifiers are eliminated, increase of the number of literals is unavoidable. 30
4.2. DECISION PROCEDURES FOR PRESBURGER ARITHMETIC Another technique to minimize the number of literals inside quantifier alternation is eliminating blocks of quantifiers [5]. Based on the following rule: δ∨ ∃x 1 ...x n−1 . j=1(F ′ −∞(x 1 ...x n−1 , j) ∨ ∨ F ′ (x 1 ...x n−1 , b + j)) b∈B δ∨ ≡ 1 ...x n−1 .F j=1(∃x ′ −∞(x 1 ,...,x n−1 , j) ∨ ∨ ∃x 1 ...x n−1 .F ′ (x 1 ...x n−1 , b + j)) b∈B we can see that after each quantifier alternation, only 1 + |B| formulas need to be examined in the next iteration. Therefore, removing a series of quantifiers does not increase sizes of quantifying formulas. Due to the huge time complexity of PA, many heuristics are incorporated to deal with subsets of PA. For example, for PA fragments with universal quantifiers only, aheuristicisinvalidating∀x.F(x) withaparticularfalse instance F(c) [17];some ground values of the formula have been instantiated to quickly decide the formula. AvariantofCooper’sprocedurewiththissimpletechniqueistestedin10000randomly generated formulas and shown to outperform some other decision procedures. Another heuristic is solving a set of divisibility constraints [5]. When eliminating quantifiers, formulas could be represented symbolically, but they have to be expanded for evaluation. And this expansion is expensive on even a small problem; the idea of solving divisibility constraints is instead of expanding big disjuncts by all possible assignments, only assignments which satisfy divisibility constraints are instantiated. This heuristic is relevant because each quantifier alternation generates anewdivisibilityconstraintinbigdisjuncts. 4.2.2 The Omega Test The Omega Test proposed by Pugh [27] is an extension of Fourier-Motzkin variable elimination to check dependence analysis in a production compiler. It consists of aprocedureforeliminatingequalitiesandelementsfordecidingsatisfiabilityofa conjunction of weak inequalities. Three elements of the Omega Test are the Real Shadow, theDark Shadow and the Gray Shadow which are summarized in Figure 4.1. Real Shadow The Real shadow is an overapproximating projection of the problem, and it is used for checking unsatisfiability [18]: ∃x. β≤ bz ∧ cz ≤ γ =⇒ cβ ≤ bγ If the Real shadow i.e. the equation cβ ≤ bγ does not hold, there is no real solution, hence there is no integer solution for the problem. 31
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: 4.2. DECISION PROCEDURES FOR PRESBU
Page 43 and 44: 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 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 90 and 91:
APPENDIX B. SOURCE CODE OF EXPERIME
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TRITA-CSC-E 2011:108 ISRN-KTH/CSC/E
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Presburger Arithmetic and Its Use in Verification

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