Proposition C. 510. A. The parties recognize the requirement under Charter Section A8.409-9 to negotiate cost- sharing provisions that produce comparable savings and costs to the City and County as are produced through the Charter's SFERS employee contribution rate adjustment formulae. The parties intend this Section to effectuate the cost sharing provisions of San Francisco Charter Section A8.409-9. The parties further acknowledge that: (i) the annual SFERS employer contribution rate is determined by the SFERS actuary and approved by the SFERS Board for each fiscal year; and (ii) the annual employer contribution rate for SFERS for FY 2012-13 is 20.71%.
Proposition C. 6 (Uniform Total Order.) If two processes p and q both A-deliver the messages m and mj, then p A-delivers m before mj if and only if q A-delivers m before mj. Proof (sketch): The proof follows from Lemma C.4. Q Proposition C.7 (Uniform Integrity.) Every message is A-delivered at most once, and only if it was previously A-broadcast. Proof (sketch): Immediate from Algorithm 3. Q Proposition C.8 (Validity.) If a correct process A-broadcasts message m, it A-delivers m. Proof (sketch): By contradiction. From Algorithms 3, once a process includes a message in its estimate sequence, the message is either never removed from it, although the message may change its rank in estimate. Let p be a correct process that A-broadcasts m. So, p includes m in estimatep (line 6) and W-ABroadcasts estimatep (line 9). Since m is not A-delivered—by the contradiction hypothesis, not all processes W-ADeliver (−, estimatep) as the first message, but from the validity of the ordering oracle, all correct processes W-ADeliver (−, estimatep). Thus, there is a round after which for every correct processes q, m ∈ estimateq. Since faulty processes eventually crash, there is a round that no faulty process executes—that is, all faulty processes crash before that round. Thus, there is a round r that only correct processes execute and, for each correct process q, m is in estimateq. Let rj > r be a round where the k-WAB property holds. It follows that at rj, m is A-delivered. Q We now prove the correctness of Algorithm 4. Algorithm 4 modifies Algorithm 3 in two ways, and each one of these modifications leads to changes in the proofs as presented next. • In Algorithm 4, if there is a time after which messages are not broadcast, processes even- tually stop exchanging messages (lines 23–24). As for the proofs, this means that the proof presented for Lemma C.1 no longer holds. We restate Lemma C.1 next as Lemma C.9 and prove it correct. • Processes executing Algorithm 4 do not always have to send all the messages they already have delivered. Therefore, processes reduce their estimate sequences by removing messages they know the other processes have already delivered (lines 19, and 21). But to keep Algorithm 4 as similar as possible to Algorithm 3, sequences are “rebuilt” when they are received by a process (lines 14 and 16). We prove in Lemma C.10 that all the messages removed by a process p before sending its estimate are reintroduced back by any other process q that receives it. Therefore, we indire...
Proposition C. 0.1. Let Ma,b be the agreement matrix for annotators a and b defined in eq. (4) and M^a,b be the estimated agreement matrix defined in eq. (8). For every ϵ > 0 it holds that And Pn(|(M a,b)ij ) − (M^a,b)ij | < ϵ) ≥ 1 − 2e−2ϵ n. Pn ∀i, j ∈ {1, C}2 |(M a,b)ij ) − (M^a,b)ij | < ϵ ≥ 1 − 2C2e−2ϵ2n. where Pn denotes the probability according to which the n training samples are distributed, i.e. we are assuming that the samples are independently drawn according the probability P. To simplify the notation we will omit the dependency from the annotators in the matrices: M = Ma,b and M^ = M^a,b n
