Solvability of Colorless Tasks in Different Models - PowerPoint Presentation

Slide1

Solvability of Colorless Tasks in Different Models

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Companion slides

forDistributed ComputingThrough Combinatorial TopologyMaurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum

1

Slide2

Road Map

Overview of Models

t-resilient layered snapshot models

Layered Snapshots with k-set agreement

Adversaries

Message-Passing Systems

Distributed Computing through Combinatorial Topology

Decidability

2

Slide3

Road Map

Overview of Models

t-resilient layered snapshot models

Layered Snapshots with k-set agreement

Adversaries

Message-Passing Systems

Distributed Computing through Combinatorial Topology

Decidability

3

Slide4

Skeleton

4

C

skel

1

C

skel

0

C

boo!

Slide5

(I,O,¢) has a wait-free protocol iff

Parameter p

Distributed Computing through Combinatorial Topology

5

Model characterized by parameter p, 0 · p · n

there is a continuous map

f: |skelp I|  |O|

carried

by

¢

.

Slide6

Dimension of Skeleton map vsComputational Power

Distributed Computing through Combinatorial Topology

6

2-skeleton map

harder than

1-skeleton map

Slide7

(I,O,¢) has a wait-free protocol …

Wait-Free Layered Immediate Snapshots

Distributed Computing through Combinatorial Topology

7

Up to n out of n+1 can crash

Just can’t wait (to be king)

if and only if …

there is a continuous map

f: |skeln I|  |O|

carried by ¢.

Slide8

(I,O,¢) has a wait-free protocol …

t-resilient Layered Immediate Snapshots

Distributed Computing through Combinatorial Topology

8

Up to t out of n+1 can crash

OK to wait for n-t+1

if and only if …

there is a continuous map

f: |skelt I|  |O|

carried

by

¢

.

Slide9

(I,O,¢) has a wait-free protocol …

Wait-Free Layered Immediate Snapshot with k-set Agreement

Distributed Computing through Combinatorial Topology

9

shared black boxes that solve k-set agreement

if and only if …

there is a continuous map

f: |skelk-1 I|  |O|

carried

by

¢

.

Slide10

Equivalent Models

Distributed Computing through Combinatorial Topology

10

t

-resilient model …

wait-free + t+1-set agreement …

have

identical

computational power!

Slide11

Decidability

Distributed Computing through Combinatorial Topology

11

Is it decidable whether a task has a protocol ina model characterized by:

f: |skelp I|  |O| ?

decidable if and only if

p

·

1

!

Slide12

Road Map

Overview of Models

t-resilient layered snapshot models

Layered Snapshots with k-set agreement

Adversaries

Message-Passing Systems

Distributed Computing through Combinatorial Topology

Decidability

12

Slide13

t-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

13

shared

mem

array

0..N-1,0..n

of

Value

view := input

for

l

:= 0

to

N-1

do

do

immediate

mem

[

l

][

i

] := view;

snap := snapshot(

mem

[

l

][*])

until

|names(snap)| >= n+1-t

view := values(snap)

return

δ

(view)

Slide14

t-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

14

shared mem array 0..N-1,0..n of Valueview := inputfor l := 0 to N-1 do do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) until |names(snap)| >= n+1-t view := values(snap)return δ(view)

2-dimensional memory array

row

is clean per-layer memory

column

is

per-process

word

Slide15

t-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

15

shared mem array 0..N-1,0..n of Valueview := inputfor l := 0 to N-1 do do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) until |names(snap)| >= n+1-t view := values(snap)return δ(view)

initial view is input value

Slide16

t-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

16

shared mem array 0..N-1,0..n of Valueview := inputfor l := 0 to N-1 do do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) until |names(snap)| >= n+1-t view := values(snap)return δ(view)

run for N layers

Slide17

t-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

17

shared mem array 0..N-1,0..n of Valueview := inputfor l := 0 to N-1 do do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) until |names(snap)| >= n+1-t view := values(snap)return δ(view)

layer l : immediate write & snapshot of row l

Slide18

t-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

18

shared mem array 0..N-1,0..n of Valueview := inputfor l := 0 to N-1 do do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) until |names(snap)| >= n+1-t view := values(snap)return δ(view)

wait to hear from n+1-t processes

why is this safe?

Slide19

t-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

19

shared mem array 0..N-1,0..n of Valueview := inputfor l := 0 to N-1 do do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) until |names(snap)| >= n+1-t view := values(snap)return δ(view)

new view is set of values seen

Slide20

t-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

20

shared mem array 0..N-1,0..n of Valueview := inputfor l := 0 to N-1 do do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) until |names(snap)| >= n+1-t view = values(snap)return δ(view)

finally apply decision map to final view

Slide21

19

19

21

21

k

-set Agreement

k

= 2

Slide22

(t+1)-Set Agreement

Distributed Computing through Combinatorial Topology

22

view := input

snap:

array

of

Value =

;

do

immediate

mem

[0][

i

] := view;

snap :=

snapshot(

mem

[0][*])

until

|names(snap)|

>= n+1-t

return

min

(values(view))

Slide23

(t+1)-Set Agreement

Distributed Computing through Combinatorial Topology

23

view := inputsnap: array of Value = ;do immediate mem[0][i] := view; snap := snapshot(mem[0][*]) until |names(snap)| >= n+1-treturn min(values(view))

write input and take snapshot

Slide24

(t+1)-Set Agreement

Distributed Computing through Combinatorial Topology

24

view := inputsnap: array of Value = ;do immediate mem[0][i] := view; snap := snapshot(mem[0][*]) until |names(snap)| >= n+1-treturn min(values(view))

wait to hear from n+1-t processes

Slide25

(t+1)-Set Agreement

Distributed Computing through Combinatorial Topology

25

view := inputsnap: array of Value = ;do immediate mem[0][i] := view; snap := snapshot(mem[0][*]) until |names(snap)| >= n+1-treturn min(values(view))

return least value in view

can miss at most

t lesser values

most

t+

1

values returned

Slide26

Informal Skeleton Lemma

Distributed Computing through Combinatorial Topology

26

If

And

We have a protocol for a task ...

Then

WLOG, we can “pre-process” with k-set agreement.

A protocol for

k

-set agreement …

Slide27

Skeleton Lemma

Distributed Computing through Combinatorial Topology

27

If

And

protocol (I, P, ¥) solves task (I,O,¢)

Then

The composition of k-set agreement with (I,P,¥) also solves (I,O,¢).

There is a

k

-set agreement protocol for

I

Slide28

Informal Protocol Complex Lemma

Distributed Computing through Combinatorial Topology

28

WLOG

We can assume that any protocol complex is a barycentric subdivision of the input complex.

Slide29

Informal Protocol Complex Lemma

Distributed Computing through Combinatorial Topology

29

WLOG

We can assume that any protocol complex is a barycentric subdivision of the input complex.

Slide30

Protocol Complex Lemma

Distributed Computing through Combinatorial Topology

30

If

There is a t-resilient layered protocol for (I,O,¢) ...

Then

Then there is a protocol (I,P,¥) such that …

P = BaryN(skelt I)

¥

(

¾

)

=

Bary

N



skel

t

(

¾

).

Slide31

Theorem

Distributed Computing through Combinatorial Topology

31

The colorless task (I,O,¢) has a t-resilient layeredsnapshot protocol …

if and only if …

there is a continuous map

f: |skelt I|  |O|

carried by ¢.

I

skel

1

(

I

)

Slide32

Protocol Implies Map

Distributed Computing through Combinatorial Topology

32

May assume protocol complex is P = BaryN skelt I.

decision map

d: BaryN skelt I  O

carried by ¢.

|d|: |BaryN skelt I|  |O|

|d|: |skelt I|  |O|

Slide33

Simplicial Approximation Theorem

Given a continuous mapthere is an N such that f has a simplicial approximation

17-Feb-15

33

Slide34

Map Implies Protocol

Distributed Computing through Combinatorial Topology

34

f: |skelt I|  |O|

Á: BaryN skelt I  O

Solve using …

barycentric agreement

t-set agreement

carried by ¢.

Slide35

Road Map

Overview of Models

t-resilient layered snapshot models

Layered Snapshots with k-set agreement

Adversaries

Message-Passing Systems

Distributed Computing through Combinatorial Topology

Decidability

35

Slide36

Motivation

Distributed Computing through Combinatorial Topology

36

Today …

Practically all modern multiprocessors provide synchronization more powerful than read-write …

Like …

test-and-set, compare-and-swap, ….

Here …

we

consider protocols constructed by

composing

layered

snapshot protocols with

k

-set

agreement protocols.

Slide37

Wait-Free Layered Set Agreement Protocol

Distributed Computing through Combinatorial Topology

37

shared

mem

array

0..N-1,0..n

of

Value

shared

SA

array

0..N-1

of

SetAgree

view := input

for

l

:= 0

to

N-1

do

view: View := SA[

l

].decide(view)

immediate

mem

[

l

][

i

] := view;

snap := snapshot(

mem

[

l

][*])

view := values(snap)

return

δ

(view)

Slide38

Wait-Free Layered Set Agreement Protocol

Distributed Computing through Combinatorial Topology

38

shared mem array 0..N-1,0..n of Valueshared SA array 0..N-1 of SetAgreeview := inputfor l := 0 to N-1 do view: View := SA[l].decide(view) immediate mem[l][i] := view; snap := snapshot(mem[l][*]) view := values(snap)return δ(view)

per-level shared memory

Slide39

Wait-Free Layered Set Agreement Protocol

Distributed Computing through Combinatorial Topology

39

shared mem array 0..N-1,0..n of Valueshared SA array 0..N-1 of k-SetAgreeview := inputfor l := 0 to N-1 do view: View := SA[l].decide(view) immediate mem[l][i] := view; snap := snapshot(mem[l][*]) view := values(snap)return δ(view)

per-level k-set agreement object

Slide40

Wait-Free Layered Set Agreement Protocol

Distributed Computing through Combinatorial Topology

40

shared mem array 0..N-1,0..n of Valueshared SA array 0..N-1 of k-SetAgreeview := inputfor l := 0 to N-1 do view: View := SA[l].decide(view) immediate mem[l][i] := view; snap := snapshot(mem[l][*]) view := values(snap)return δ(view)

initial view is input value

Slide41

Wait-Free Layered Set Agreement Protocol

Distributed Computing through Combinatorial Topology

41

shared mem array 0..N-1,0..n of Valueshared SA array 0..N-1 of k-SetAgreeview := inputfor l := 0 to N-1 do view: View := SA[l].decide(view) immediate mem[l][i] := view; snap := snapshot(mem[l][*]) view := values(snap)return δ(view)

do k-set agreement with others at this level

Slide42

Wait-Free Layered Set Agreement Protocol

Distributed Computing through Combinatorial Topology

42

shared mem array 0..N-1,0..n of Valueshared SA array 0..N-1 of k-SetAgreeview := inputfor l := 0 to N-1 do view: View := SA[l].decide(view) immediate mem[l][i] := view; snap := snapshot(mem[l][*]) view := values(snap)return δ(view)

then do immediate snapshot

Slide43

Wait-Free Layered Set Agreement Protocol

Distributed Computing through Combinatorial Topology

43

shared mem array 0..N-1,0..n of Valueshared SA array 0..N-1 of k-SetAgreeview := inputfor l := 0 to N-1 do view: View := SA[l].decide(view) immediate mem[l][i] := view; snap := snapshot(mem[l][*]) view := values(snap)return δ(view)

new view is set of values seen

Slide44

Protocol Complex Lemma

Distributed Computing through Combinatorial Topology

44

If (I,P,¥) is a k-set layered snapshot protocol …

then P is equal to BaryN skelk-1 I, …

for

some

N

¸

0

.

Slide45

Theorem

Distributed Computing through Combinatorial Topology

45

The colorless task (I,O,¢) has a wait-free k-set layeredsnapshot protocol …

if and only if …

there is a continuous map

f: |skelk-1 I|  |O|

carried

by

¢

.

Slide46

Theorem

Distributed Computing through Combinatorial Topology

46

The colorless task (I,O,¢) has a wait-free k-set layeredsnapshot protocol …

if and only if …

there is a continuous map

f: |skelk-1 I|  |O|

carried by ¢.

k

-1

skeleton, not

t

-skeleton!

Slide47

Road Map

Overview of Models

t-resilient layered snapshot models

Layered Snapshots with k-set agreement

Adversaries

Message-Passing Systems

Distributed Computing through Combinatorial Topology

Decidability

47

Slide48

Wait-Free

All but one can fail

Distributed Computing through Combinatorial Topology

Slide49

t

-resilient

·

t

can fail

Distributed Computing through Combinatorial Topology

Slide50

Irregular Failures

Same server

Different servers

Distributed Computing through Combinatorial Topology

Slide51

http://pixabay.com/en/chess-figures-game-play-strategy-145184/

Adversaries

17-Feb-15

Distributed Computing through Combinatorial Topology

Walt Disney

Slide52

Faulty Sets

These can fail

Or this …

Distributed Computing through Combinatorial Topology

Slide53

Faulty Sets Closed under Containment

If both can fail …

So can only one

Never

require

failures

Distributed Computing through Combinatorial Topology

Slide54

Failure Complex

17-Feb-15

54

Vertex per process

Distributed Computing through Combinatorial Topology

Slide55

Failure Complex

17-Feb-15

55

Vertex per process

Simplex = faulty set

Distributed Computing through Combinatorial Topology

Slide56

Irregular Failure Complex

17-Feb-15

56

Distributed Computing through Combinatorial Topology

Slide57

Wait-Free Failure Complex

17-Feb-15

57

(

n

-1

)-skeleton

Distributed Computing through Combinatorial Topology

Slide58

t-resilient Failure Complex

17-Feb-15

58

(

t-

1)

-skeleton

Distributed Computing through Combinatorial Topology

Slide59

Cores

17-Feb-15

59

Minimal set of processes that cannot all fail

Safe to wait for at least one member of a particular core to show up

Distributed Computing through Combinatorial Topology

Slide60

Cores & Failure Complex

17-Feb-15

60

Minimal simplex not in failure complex

Distributed Computing through Combinatorial Topology

Slide61

Irregular Failure Complex

17-Feb-15

61

Distributed Computing through Combinatorial Topology

Slide62

Wait-Free Failure Complex

17-Feb-15

62

All

n

+1

processes only core

Distributed Computing through Combinatorial Topology

Slide63

t-resilient Failure Complex

17-Feb-15

63

Any

t

-simplex is a core

Any set of

t

+1

processes is a core

Distributed Computing through Combinatorial Topology

Slide64

Cores

17-Feb-15

64

For many models,

Completely determines adversary’s power to solve any colorless task!

minimum core size…

So adversaries with same min core size solve the same colorless tasks

Distributed Computing through Combinatorial Topology

Slide65

Survivor Sets

17-Feb-15

65

Minimal set of processes that might all survive

Safe to wait for all members of some survivor set to show up

Dual to cores: each one determines the other

Distributed Computing through Combinatorial Topology

Slide66

Survivor Sets in Failure Complex

17-Feb-15

66

Complementary simplex in failure complex

Distributed Computing through Combinatorial Topology

Slide67

Irregular Failure Complex

17-Feb-15

67

Survivor Sets

Distributed Computing through Combinatorial Topology

Slide68

Wait-Free Failure Complex

17-Feb-15

68

Any vertex

Any individual process is a survivor set

Distributed Computing through Combinatorial Topology

Slide69

t-resilient Failure Complex

17-Feb-15

69

Any (

n-t

)-simplex

Any set of

n-t+1

processes is a survivor set

Distributed Computing through Combinatorial Topology

Slide70

A-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

70

shared

mem

array

0..N-1,0..n

of

Value

view := input

for

l

:= 0

to

N-1

do

do

immediate

mem

[

l

][

i

] := view;

snap := snapshot(

mem

[

l

][*])

until

names(snap)

µ

survivor set

view := values(snap)

return

δ

(view)

Slide71

A-Resilient Layered Immediate Snapshot Protocol

Distributed Computing through Combinatorial Topology

71

shared mem array 0..N-1,0..n of Valueview := inputfor l := 0 to N-1 do do immediate mem[l][i] := view; snap := snapshot(mem[l][*]) until names(snap) µ survivor set view := values(snap)return δ(view)

wait to hear from a survivor set

why is this safe?

Slide72

Road Map

Overview of Models

t-resilient layered snapshot models

Layered Snapshots with k-set agreement

Adversaries

Message-Passing Systems

Distributed Computing through Combinatorial Topology

Decidability

72

Slide73

Message Passing

Distributed Computing through Combinatorial Topology

73

that send and receive messages …

There are n+1 asynchronous processes …

via a fully-connected communication network.

Message

delivery is

reliable

and

FIFO

Slide74

Message-Passing Protocols

Distributed Computing through Combinatorial Topology

74

decide after finite # steps

but protocol

forwards messages …

forever!

Slide75

Communication Syntax

Distributed Computing through Combinatorial Topology

75

send(P, v0, …, vl) to Q

send(P, v0, …, vl) to all

upon

receive

(P,

v

0

, …,

v

l

)

do

...

//

handle message

Slide76

Forwarding

Distributed Computing through Combinatorial Topology

76

background

// forward messages forever

upon

receive

(

P

j

,v

)

do

send

(

P

i

,v

)

to

all

Slide77

Get Values from n+1-t Processes

Distributed Computing through Combinatorial Topology

77

getQuorum

():

Set of

Value

V:

Set of

Value :=

;

q:

int

:=

0

do

upon

receive(

Q,v

)

do

V

:= V

[

{v}

q := q + 1

until

q = n+1-t

return

V

Slide78

Get Values from n+1-t Processes

Distributed Computing through Combinatorial Topology

78

getQuorum(): Set of Value V: Set of Value := ; q: int := 0 do upon receive(Q,v) do V := V [ {v} q := q + 1 until q = n+1-t return V

Initially, nothing.

Slide79

Get Values from n+1-t Processes

Distributed Computing through Combinatorial Topology

79

getQuorum(): Set of Value V: Set of Value := ; q: int := 0 do upon receive(Q,v) do V := V [ {v} q := q + 1 until q = n+1-t return V

remember values and count

Slide80

Get Values from n+1-t Processes

Distributed Computing through Combinatorial Topology

80

getQuorum(): Set of Value V: Set of Value := ; q: int := 0 do upon receive(Q,v) do V := V [ {v} q := q + 1 until q = n+1-t return V

safe to wait for n+1-t values

Slide81

Get Values from n+1-t Processes

Distributed Computing through Combinatorial Topology

81

getQuorum(): Set of Value V: Set of Value := ; q: int := 0 do upon receive(Q,v) do V := V [ {v} q := q + 1 until q = n+1-t return V

return values when enough received

Slide82

Protocol for (t+1)-Set Agreement

Distributed Computing through Combinatorial Topology

82

SetAgree

(v

i

):

value

send(P, v

i

)

to

all

V:

Set of

Value :=

getQuorum

()

return

min(V)

Slide83

Protocol for (t+1)-Set Agreement

Distributed Computing through Combinatorial Topology

83

SetAgree(v): value send(P, v) to all V: Set of Value := getQuorum() return min(V)

broadcast my value

Slide84

Protocol for (t+1)-Set Agreement

Distributed Computing through Combinatorial Topology

84

SetAgree(v): value send(P, v) to all V: Set of Value := getQuorum() return min(V)

get values from all but t

Slide85

Protocol for (t+1)-Set Agreement

Distributed Computing through Combinatorial Topology

85

SetAgree(v): value send(P, v) to all V: Set of Value := getQuorum() return min(V)

return min value received

possible to “miss” only

t

lesser values

Slide86

86

Barycentric Agreement

Distributed Computing through Combinatorial Topology

I

Bary

I

Slide87

Barycentric Agreement Protocol

Distributed Computing through Combinatorial Topology

87

BaryAgree

(v

i

:

Vertex):

set of

Vertex

V

i

:

set of

Vertex := {

v

i

}

count:

int

:= 0

while

count < n+1-t do

send(P

i

, V

i

)

to all

on

receive(

P

j

,

V

j

)

do

if

V

i

=

V

j

then

count

:= count + 1

else

if

V

j

n

V

i



;

then

V

i

:=

V

i

[

V

j

count := 0

return

V

i

Slide88

Barycentric Agreement Protocol

Distributed Computing through Combinatorial Topology

88

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

Set of messages Pi has received

Slide89

Barycentric Agreement Protocol

Distributed Computing through Combinatorial Topology

89

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

keep track of confirmations received so far

Slide90

Barycentric Agreement Protocol

Distributed Computing through Combinatorial Topology

90

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

get confirmation from each non-faulty process

Slide91

Barycentric Agreement Protocol

Distributed Computing through Combinatorial Topology

91

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

broadcast message set received

Slide92

Barycentric Agreement Protocol

Distributed Computing through Combinatorial Topology

92

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

collect responses

Slide93

Barycentric Agreement Protocol

Distributed Computing through Combinatorial Topology

93

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

remember if message confirms my view

Slide94

Barycentric Agreement Protocol

Distributed Computing through Combinatorial Topology

94

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

otherwise learned something new, start over

Slide95

Barycentric Agreement Protocol

Distributed Computing through Combinatorial Topology

95

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

return when enough agree

Slide96

Wait, There’s More!

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96

background upon receive(Pj, Vj) do Vi := Vi [ Vj send(Pi, Vi) to all

the operating system runs forever …

Slide97

Wait, There’s More!

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background upon receive(Pj, Vj) do Vi := Vi [ Vj send(Pi, Vi) to all

keep forwarding new values

Slide98

Lemma: Protocol Terminates

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98

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

Suppose BWOC Pi runs forever …

Eventually Vi assumes final value V …

Non-faulty Pj, where Vj = V’, receives V

Pj must have sent Vj to Pi

Vj ½ V

Vj = V

OR

Pj will sent V to Pi

P

j

has sent

V

to

P

i

Slide99

All Vi, Vj Totally Ordered

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99

BaryAgree(vi: Vertex): set of Vertex Vi: set of Vertex := {vi} count: int := 0 while count < n+1-t do send(Pi, Vi) to all on receive(Pj, Vj) do if Vi = Vj then count := count + 1 else if Vj n Vi  ; then Vi := Vi [ Vj count := 0 return Vi

If Pi broadcasts V(0), …, V(k), then V(i) ½ V(i+1)

To decide …

P

i received Vi from X, |X| ¸ n+1-t

Pj received Vj from Y, |Y| ¸ n+1-t

some Pk 2 X Å Y sent both Vi, Vj

so

V

i

,

V

j

are ordered.

Slide100

Theorem

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100

For 2t < n+1, colorless task (I,O,¢) has a t-resilient message-passing protocol …

if and only if …

there is a continuous map

f: |skelt I|  |O|

carried

by

¢

.

Slide101

Theorem

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101

For 2t < n+1, colorless task (I,O,¢) has a t-resilient message-passing protocol …

if and only if …

there is a continuous map

f: |skelt I|  |O|

carried by ¢.

same as snapshot when

2t < n+1

!

Slide102

Road Map

Overview of Models

t-resilient layered snapshot models

Layered Snapshots with k-set agreement

Adversaries

Message-Passing Systems

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Decidability

102

Slide103

Automatic Proofs?

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103

What if we could program a Turing machineto tell whether a task has a protocol?

In wait-free read-write memory?

Or other models?

We could …

automatically generate conference papers

No need for grad students

Slide104

Alas no

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104

Whether a protocol exists for a task in …

Read-write memory for 3+ processes …

Read-write memory & k-set agreement …

Is undecidable.

for

k

> 2

Slide105

Loop Agreement

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105

Complex

Loop

Three

rendez-vous

points

Slide106

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106

One Rendez-Vous Point

Input

Output

Slide107

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107

Two

Rendez-Vous

Points

Input

Output

Any simplex on path between

Slide108

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108

Three

Rendez-Vous

Points

Input

Output

Any simplex

Slide109

Contractibility

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109

not contractible

contractible

Slide110

Solvable Iff Loop Contractible

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110

InputComplex

Output Complex

Slide111

Undecidability

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111

But Contractiblity is undecidable …

even for finite complexes!

(reduces to the word problem for finitely-presented groups)

Undecidable

whether a task has a protocol in wait-free read-write memory

Slide112

Other Models

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112

Wait-free read-write memory plus k-set agreement , for k > 2

Implies contractible, for k > 2

Undecidable whether a task has a protocol in wait-free read-write memory plus k-set agreement , for k > 2

Solvable iff

f

:

skel

k-1

I

*



O

*

exists …

Slide113

113

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Distributed Computing through Combinatorial Topology