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Yehuda Afek, Eli Daian, Eli Gafni, Giuliano Losa
Life in 1-consensus Yehuda Afek, Eli Daian, Eli Gafni, Giuliano Losa
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Classifying Deterministic Shared Memory Objects
Computability = the power to solve tasks wait-free = non-blocking August 16, 2019
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Deterministic Objects Task (deterministic and non-det.)
Linearizable concurrent state machine Atomic R/W Reg’s Stack, Queue, F&Add, … CAS, Input-vector Output-vector function: (0,2,5,14,1) {(2,5,5,2,5),(2,2,2,2,2)…..} ….. Consensus (agreement & validity) Set-Consensus (set-agreement) Immediate Snapshot deterministic object = DFSA
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Classifying Deterministic Shared Memory Objects
Computability = the power to solve tasks wait-free = non-blocking Standard asynchronous shared memory Shared deterministic objects Interested (only) in wait-free linearizable implementations of tasks == non-blocking! August 16, 2019
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Object O Consensus Number
Cons#(O) = C=Max number of processors that can do w-f consensus with any number of O and Reg’s
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Herlihy’s Consensus Hierarchy
Test-and-set LL/SC R/W registers Fetch-and-add m-write 1 SWAP Queue Stack Compare-and-swap 2 ∞
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Herlihy’s Consensus Hierarchy
Compare-and-swap LL/SC ∞ Test-and-set Fetch-and-add Queue 2 Stack SWAP R/W registers 1 m-write If cons#(O1)<cons#(O2) O1 cannot implement O2 Object in Lk is universal for k processes (implements any O) Is level k complete? different objects in Lk can implement each other in a system with n>k processes?
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Common 2 [AWeisbergerWeisman 1994]
Test&Set from 2-cons Yes Fetc&Add from 2-cons SWAP from 2-cons Stack from 2-cons Queue from 2-cons ?? Test-and-set Fetch-and-add 1 SWAP Queue Stack 2 ∞
![Common 2 [AWeisbergerWeisman 1994]](http://slideplayer.com./slide/17263739/100/images/8/Common+2+%5BAWeisbergerWeisman+1994%5D.jpg "Test&Set from 2-cons. Yes. Fetc&Add from 2-cons. SWAP from 2-cons. Stack from 2-cons. Queue from 2-cons. Test-and-set. Fetch-and-add. 1. SWAP. Queue. Stack. 2. ∞")
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FALSE AEllenGafni 2016 Common 2 Common 3 Common m
Consensus Hierarchy Conjecture: Deteministic Obj of consensus number n can be implemented from n-consensus for any number of processors. FALSE
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Set-Consensus Soma Chaudhuri 1990: (n,k)Set-Consensus: Agreement – output-vector is a set of at most k different values Validity – each value is an input of some process [HS, BG, SZ 1999] Atomic R/W cannot w-f solve (n,k)set-consensus, n>k>1
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Borowsky Gafni 1993 Roughly, n>l and k>j: (l,j)set-consensus implements (n,k)set-consensus if k=j*n/l (use n/l copies of (l,j)set-consensus) More precisely (dealing with remainders): ………if and only if k ≥ j, n/k ≤ m/j, and either k ≥ j ⎡n/m⎤ or k ≥ j ⎣n/m⎦ + n – m ⎣n/m⎦ .
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FALSE AEllenGafni 2016 AABBA cons#(AABBA)=2=(2,1)sc
T A2 B1 B2 A/A */A B/B */B */ AABBA cons#(AABBA)=2=(2,1)sc Set-consensus Power (AABBA) = (5,2) BG(2,1)set-consensus cannot do (5,2)set-consensus FLP (bivalent) proof cons#(AABBA)<3, =2 ! counter example for common2 !! Common2 conjecture FALSE
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AEllenGafni 2016 AABBCCBA (= (8,3)) implements AABBA
AABBA does not implement AABBCCBA Hierarchy ! Test-and-set LL/SC R/W registers Fetch-and-add m-write 1 SWAP Stack Queue Compare-and-swap 2 ∞
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Comparing the Hierarchies
Herlihy’s Hierarchy AEG Uses consensus Deterministic task Classifies deterministic objects Uses set consensus Nondeterministic task Classifies deterministic objects August 16, 2019
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Remains Open [AEG cojecture]
Are there deterministic objects O s.t.,: Reg’s < O < 2-consensus ?? Test-and-set LL/SC R/W registers Fetch-and-add m-write 1 SWAP Stack Queue Compare-and-swap 2 ∞
![Remains Open [AEG cojecture]](http://slideplayer.com./slide/17263739/100/images/15/Remains+Open+%5BAEG+cojecture%5D.jpg "Are there deterministic objects O s.t.,: Reg’s < O < 2-consensus Test-and-set. LL/SC. R/W registers. Fetch-and-add. m-write. 1. SWAP. Stack. Queue. Compare-and-swap. 2. ∞")
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T A[0], A[1], … , A[k-1] Registers, initially WRN(i,v): Write v A[i]
A[0], A[1], … , A[k-1] Registers, initially WRN(i,v): Write v A[i] Return A[(i+1) mod k] T August 16, 2019
![T A[0], A[1], … , A[k-1] Registers, initially WRN(i,v): Write v A[i]](http://slideplayer.com./slide/17263739/100/images/16/T+A%5B0%5D%2C+A%5B1%5D%2C+%E2%80%A6+%2C+A%5Bk-1%5D+Registers%2C+initially+WRN%28i%2Cv%29%3A+Write+v+%EF%83%A0+A%5Bi%5D.jpg "A[0], A[1], … , A[k-1] Registers, initially. WRN(i,v): Write v A[i] Return A[(i+1) mod k] T. August 16,")
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WRN(0, 14) Write Read August 16, 2019
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14 Write Read August 16, 2019
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Write Read August 16, 2019
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WRN(2, 8) Read Write August 16, 2019
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8 Read Write August 16, 2019
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Read Write August 16, 2019
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Example – 3 processes August 16, 2019
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Example – 3 processes August 16, 2019
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Example – 3 processes August 16, 2019
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Example – 3 processes August 16, 2019
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Renaming More Iterations August 16, 2019
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The Critical State August 16, 2019
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Same value is returned August 16, 2019
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DONE! August 16, 2019
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Remains Open [AEG cojecture]
FALSE WRNk is a deterministic object WRNk s.t.,: Reg’s < WRNk < 2-consensus ?? Test-and-set LL/SC R/W registers Fetch-and-add m-write 1 SWAP Stack Queue Compare-and-swap 2 ∞
![Remains Open [AEG cojecture]](http://slideplayer.com./slide/17263739/100/images/45/Remains+Open+%5BAEG+cojecture%5D.jpg "FALSE. WRNk is a deterministic object WRNk s.t.,: Reg’s < WRNk < 2-consensus Test-and-set. LL/SC. R/W registers. Fetch-and-add. m-write. 1. SWAP. Stack. Queue. Compare-and-swap. 2. ∞")
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Building an Infinite Hierarchy
(If p is elected, then p elects itself) August 16, 2019
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Building from Strong Set Election
August 16, 2019
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Implementation Directions
[BG 1993] Already shown In the paper August 16, 2019
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Infinite hierarchy in L1
An infinite hierarchy Test-and-set LL/SC R/W registers Fetch-and-add m-write 1 SWAP Stack Queue Compare-and-swap 2 ∞
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Set-Consensus below 2-consensus
processes 2-setCons 3-setConse 4-setCons 5-setCons 6-setCons 7-setCons 3 4 5 6 7 8 Reg’s=1-consensus
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Set-Consensus below 2-consensus
processes 2-setCons 3-setConse 4-setCons 5-setCons 6-setCons 7-setCons 3 WRN3 4 5 6 7 8 Reg’s=1-consensus
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Set-Consensus below 2-consensus
processes 2-setCons 3-setConse 4-setCons 5-setCons 6-setCons 7-setCons 3 WRN3 4 WRN4 5 WRN5 6 WRN6 7 WRN7 8 WRN8 Reg’s=1-consensus
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Set-Consensus below 2-consensus
processes 2-setCons 3-setConse 4-setCons 5-setCons 6-setCons 7-setCons 3 WRN3 4 WRN4 5 WRN5 6 2xWRN3 WRN6 7 WRN7 8 WRN8 Reg’s=1-consensus
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Set-Consensus below 2-consensus
processes 2-setCons 3-setConse 4-setCons 5-setCons 6-setCons 7-setCons 3 WRN3 4 WRN4 5 WRN5 6 2xWRN3 WRN6 7 WRN7 8 2xWRN4 WRN8 Reg’s=1-consensus
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Set-Consensus below 2-consensus
processes 2-setCons 3-setConse 4-setCons 5-setCons 6-setCons 7-setCons 3 WRN3 4 WRN4 5 WRN5 6 2xWRN3 WRN6 7 WRN7 8 2xWRN4 WRN8 Reg’s=1-consensus ??? OPEN ???
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Consensus vs. Set Consensus Power
August 16, 2019
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Delporte-Gallet, Fauconnier, Gafni, Kuznetsov
Set-consensus power vector (Obj)= (k1, k2, k3, …..) where Obj and registers can w-f implement (ki,i)set-consensus and NOT (ki+1,i)set-consensus.
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Set Consensus Power Vector
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Remaining Open Questions
Thank you! August 16, 2019
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A Ellen Gafni Test-and-set LL/SC R/W registers Fetch-and-add m-write
1 SWAP Queue Stack Compare-and-swap 2 ∞
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Spoiler Test-and-set R/W registers Fetch-and-add SWAP LL/SC 1 CAS
Stack Queue 2 ∞
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Spoiler Test-and-set R/W registers Fetch-and-add SWAP LL/SC 1 CAS
Stack Queue 2 ∞
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The Open Question ??? August 16, 2019
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