27 lines
No EOL
1.8 KiB
Markdown
27 lines
No EOL
1.8 KiB
Markdown
Which objects allow for a wait free implementation of (binary) consensus? The answer depends on the number of participants
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The **consensus number** of an object of type T is the greatest number n such that it is possible to wait free implement a consensus object in a system of n processes by only using objects of type T and atomic R/W registers.
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For all T, CN(T) > 0; if there is no sup, we let CN(T) := +∞
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**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).
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*Proof:*
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- 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.
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- 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.
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- 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.
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- contradiction with CN(T1) < n
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### Schedules and Configurations
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**Schedule:** sequence of operation invocations issued by processes.
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**Configuration:** the global state of a system at a given execution time (values of the shared memory + local state of every process).
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Given a configuration C and a schedule S, we denote with S(C) the configuration obtained starting from C and applying S.
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Let's consider binary consensus implemented by an algorithm A by using base objects and atomic R/W registers; let us call $S_A$ a schedule induced by A.
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A configuration C obtained during the execution of all A is called:
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- **v-valent** if $S_A(C)$ decides v, for every $S_A$
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- **monovalent**, if there exists $v \in \{0,1\}$ s.t. C is v-valent
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- **bivalent**, otherwise. |