Task Computability in Unreliable Anonymous Networks Nayuta - - PowerPoint PPT Presentation

Task Computability in Unreliable Anonymous Networks

Nayuta Yanagisawa DeNA Co., Ltd. Petr Kuznetsov Telecom ParisTech

I am here to talk about ... the computability issues of failure-prone anonymous broadcast models.

Anonymous Model

Anonymous model: Processes have no unique IDs, i.e. processes execute an identical program.

• Some sensor networks and peer-to-peer file sharing

systems are anonymous by design or may choose anonymity for the sake of privacy.

• It is theoretically interesting to study how the existence
• f unique identifiers affects the computational power of

distributed systems.

Existing Works

t = 0 t > 0 Shared-memory Attiya et al. ’02 Guerraoui and Ruppert ’07 Yanagisawa ’17 Delporte et al. ’18a, b Message-passing Angluin ‘80 Aspnes et al ’06

Existing works concerning the anonymous and asynchronous computation, where t is the number of failures:

Anonymous Broadcast Model

• A distributed system consists of a set of anonymous

and asynchronous processes.

• At most t<n/2 processes are prone to crash failures.
• Processes communicate by broadcasting messages

via a fully connected reliable FIFO network. We assume that a process cannot detect through which link a message has been received.

Overview of Our Results

• Impossibility: Any non-trivial object type has no

linearizable implementation if t>0.

• Possibility: A sequentially-consistent add-only set
• bject is implementable if 0≦t<n/2.
• Characterization: A colorless task is t-resiliently

solvable in the anonymous broadcast model iff it is t-resiliently solvable in the non-anonymous broadcast model, where 0≦t<n/2.

Impossibility

Any non-trivial object type has no linearizable implementation if t>0.

Non-commuting Operation

Definition 1 Let T = (Q,q0,O,R,∆) be an object type. We say that

• perations o1,o2 ∈ O are weakly-non-commutative if,

for all responses r1,r2 ∈ R s.t. o1r1o2r2 is legal,

• 1r1o2r2o1r1 is not legal.

[ ] [ ] [ ]

read()

write(1)

ack read()

legal

not legal

Impossibility

Theorem 1 There is no 1-resilient linearizable implementation

• f an object type with weakly-non-commutative
• perations in the anonymous broadcast model.

[Aspnes et al. ’06] An atomic register has 0-resilient linearizable implementation in the anonymous broadcast model.

Proof Sketch

[ ]

read()

[ ]

write(1) ack

p1 pn

[ ]

read()

… p2

zzz...

…

Possibility

A sequentially-consistent add-only set

• bject is implementable if 0≦t<n/2.

Add-Only Set

Definition 2 An add-only set stores a set of values, initially ∅, and exports operations add(v) (v∈V) and get(). add(v) adds a value v to the set. get() returns the current snapshot of the set. add(v) and get() are weakly-non-commutative and thus the add-only set has no 1-resilient linearizable implementation.

Sequential-Consistency

[ ]

read()

linearizable sequentially-consistent

[ ] [ ]

read()

write(1)

ack

[ ]

read()

[ ] [ ]

read()

write(1)

ack

Possibility

Theorem 2 An add-only set has a sequentially-consistent t- resilient implementation in the anonymous broadcast model, where 0≦t<n/2.

Collorary 1 A snapshot object has a sequentially-consistent implementation in the anonymous broadcast model, where 0≦t<n/2.

Implementation

• Each process keeps its local estimate of the content
• f the set and a round number.
• To add a value v, a process insert v into its local

estimate, broadcast it with the round number, and increment the round number.

• To get the content of the set, a process collects

more than n/2 of other processes’ estimate with the same round number. If all the estimated sets are identical, then the process return the set. Otherwise, it increments the round number and tries to collect

• ther processes’ estimated sets again.

Characterization

A colorless task is t-resiliently solvable in the anonymous broadcast model if and only if it is t-resiliently solvable in the non-anonymous model, where 0≦t<n/2.

Colorless Task

Definition 3 A colorless task is defined through a set I of input sets, a set O of output sets, and a total relation 
 ∆ : I ︎→ 2O that associates each input set with a set

• f possible output sets.

Consensus, k-set agreement, and loop agreement are all colorless tasks.

Characterization

Theorem 3 A colorless task T is t-resiliently solvable in the anonymous broadcast model if and only if it is t-resiliently solvable in the non-anonymous broadcast model.

Proof Sketch: If Part

T is t-resiliently solvable in the non-anonymous
 broadcast model.
 ⇒ T is t-resilient solvable in the non-anonymous
 shared-memory model [Attiya et al. ’95]. ⇒ T is t-resilient solvable in the anonymous shared- 
 memory model [Delporte et.al. ’18]. ⇒ T is t-resilient solvable in the anonymous
 broadcast model (Collorary 1).

Existing Works

t = 0 t > 0 Shared-memory Attiya et al. ’02 Guerraoui and Ruppert ’07 Yanagisawa ’17 Delporte et al. ’18a Delporte et al. ’18b Message-passing Angluin ‘80

Impossibility Possibility Characterization

Aspnes et al ’06

Future Work

• Extending our characterization to a general class
• f distributed decision tasks.
• Consistency conditions stronger than sequential-

consistency.