2.7IBRTA1-Ak102508

2.7IBRTA1-Ak102508 - 1 2.7 IBRT in A 1,A k,fA 1,fA k ⊆ In...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 1-7,11 are uniform. We now determine their witnesses sets. LEMMA 2.7.1. The inclusions in clauses 1-7 each have no witnesses. I.e., their witness sets are ∅ ....
View Full Document

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.

Page1 / 13

2.7IBRTA1-Ak102508 - 1 2.7 IBRT in A 1,A k,fA 1,fA k ⊆ In...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online