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Unformatted text preview: 1 2.7. IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ . In this section, we analyze IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ on (SD,INF), (ELG ∩ SD,INF), (ELG,INF), (EVSD,INF), and (MF,INF). We show that for all k ≥ 1, IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ on each of (SD,INF), (ELG ∩ SD,INF), (ELG,INF), (EVSD,INF) is RCA secure. We show that IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ on (MF,INF) is ACA' secure (see Definition 1.4.1). We also show that the only correct format for IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ on (SD,INF), (ELG ∩ SD,INF), (ELG,INF), (EVSD,INF) is ∅ . This is not true on (MF,INF). We begin with (MF,INF), for some fixed k ≥ 1. We need to analyze all statements of the form #) ( ∃ f ∈ MF)( ∀ A 1 ,...,A k ∈ INF)(A 1 ⊆ ... ⊆ A k → ϕ ). where ϕ is an A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ format. Recall that the instances of #) are Boolean equivalent to the assertions of IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ , and the negations of the statements in IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ . Recall the list of all A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ elementary inclusions that were used in section 2.6: 1. A i = ∅ . 2. fA i = ∅ . 3. A i ∩ fA j = ∅ . 4. A i = N. 5. fA i = N. 6. A i ∪ fA j = N. 7. A i ⊆ A j , j < i. 8. A i ⊆ fA j . 9. A i ⊆ A j ∪ fA p , j < i. 10. fA i ⊆ A j . 11. fA i ⊆ fA j , j < i. 12. fA i ⊆ A j ∪ fA p , p < i. 13. A i ∩ fA j ⊆ A p , p < i. 14. A i ∩ fA j ⊆ fA p , p < j. 15. A i ∩ fA j ⊆ A p ∪ fA q , p < i and q < j. For each of these elementary inclusions, ρ , we will provide a useful description of the witness set for ρ , in the following sense: The set of all f ∈ MF such that ( ∀ A 1 ,...,A k ∈ INF)(A 1 ⊆ ... ⊆ A k → ρ ). 2 To analyze formats, we analyze the intersections of these witness sets, determining which intersections are nonempty. I.e., a format is correct if and only if the intersection of the set of witnesses of each element is nonempty (in IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ on (MF,INF)). We also use this technique for the other four BRT settings. Thus a format is correct if and only if the intersection of the set of witnesses of each element meets V (in IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , ⊆ on (V,INF), V ⊆ MF)). Each numbered entry in the list represents several inclusions. In some numbered entries, all of the inclusions will have the same witness set. We call such an entry uniform. Unfortunately, some of the numbered entries are not uniform. We shall see that entries 17,11 are uniform. We now determine their witnesses sets. LEMMA 2.7.1. The inclusions in clauses 17 each have no witnesses. I.e., their witness sets are ∅ ....
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math

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