From 61173a6bc10f592ba1b22c3374f1fa665fbfb913 Mon Sep 17 00:00:00 2001 From: Marco Realacci Date: Tue, 1 Apr 2025 08:29:49 +0200 Subject: [PATCH] vault backup: 2025-04-01 08:29:49 --- .../notes/10 - Consensus Implementation.md | 13 ++++++++++++- 1 file changed, 12 insertions(+), 1 deletion(-) diff --git a/Concurrent Systems/notes/10 - Consensus Implementation.md b/Concurrent Systems/notes/10 - Consensus Implementation.md index b8fd0c7..2672b66 100644 --- a/Concurrent Systems/notes/10 - Consensus Implementation.md +++ b/Concurrent Systems/notes/10 - Consensus Implementation.md @@ -4,5 +4,16 @@ The **consensus number** of an object of type T is the greatest number n such th For all T, CN(T) > 0; if there is no sup, we let CN(T) := +∞ -**Thm:** let CN(T1) < CN(T2), then there exists no wait free implementation of T2 that only uses objects of type T1 and atomic R/W registers for all n s.t. CN(T1) < n <= CN(T2). +**Thm:** let CN(T1) < CN(T2), then there exists no wait free implementation of T2 that only uses objects of type T1 and atomic R/W registers, for all n s.t. CN(T1) < n <= CN(T2). +*Proof:* +- Fix such an n; by contr., there exists a wait free implementation of objects of type T2 in a system of n processes that only uses objects of type T1 and atomic RW reg.s. +- Since n ≤ CN(T2), by def. of CN, there exists a wait free implementation of consensus in a system of n processes that only uses objects of type T2 and atomic RW reg.s. +- Hence, there exists a wait free implementation of consensus in a system of n processes that only uses objects of type T1 and atomic RW reg.s. + - contradiction with CN(T1) < n + +### Schedules and Configurations +**Schedule:** sequence of operation invocations issued by processes +**Configuration:** the global state of a system at a given execution time (values of the shared memory + local state of every process). + +Given a configuration C and a schedule S, we denote with S(C) the configuration obtained starting from C and applying S. \ No newline at end of file