Which objects allow for a wait free implementation of (binary) consensus? The answer depends on the number of participants

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.

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).

*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.

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.

A configuration C obtained during the execution of all A is called:
- **v-valent** if $S_A(C)$ decides v, for every $S_A$
- **monovalent**, if there exists $v \in \{0,1\}$ s.t. C is v-valent
- **bivalent**, otherwise.

### Fundamental theorem
If A wait-free implements binary consensus for n processes, then there exists a bivalent initial configuration.

*Proof:*
![[Pasted image 20250401083747.png]]

### CN(Atomic R/W registers) = 1
**Thm:** There exists no wait-free implementation of binary consensus for 2 processes that uses atomic R/W registers.

*Proof:*
Assume by contradiction A wait-free, with processes p and q.

By the previous result, it has an initial bivalent configuration C
-  let S be a sequence of operations s.t. C’ = S(C) is maximally bivalent (i.e., p(C') is 0-valent and q(C') is 1-valent, or viceversa)
	- partendo da C' posso ancora avere due possibili computazioni dove una decide 0 e una decide 1, ma è l'ultima configurazione in cui è possibile. Quelle successive sono monovalenti.

p(C’) can be R1.read() or R1.write(v) and q(C’) can be R2.read() or R2.write(v’)

1. if R1 != R2
	- Whatever operations p and q issue, we have that q(p(C’)) = p(q(C’)) But q(p(C’)) is 0-val (because p(C’) is) whereas p(q(C’)) is 1-val