Introduction to the Gödel Prize Session

© Sergio Rajsbaum; DISC/GETCO 2004

Introduction to the Gödel Prize Session

Sergio RajsbaumMath Institute

UNAM, Mexico


The Gödel Prize

• For outstanding papers in theoretical CS• Sponsored jointly by EATCS and ACM-SIGACT• Presented annually since 1993

– (alternately in ICALP and STOC)• To journal paper(s) published in the past 7 years• Past winners related to distributed computing:

– 2000: M. Vardi, P. Wolper about logic for concurrency

– 1997: J. Halpern, Y. Moses about knowledge

Sergio Rajsbaum

The Gödel Prize Committee, consisting of Kurt Mehlhorn, Amir Pnueli, Franco Preparata (chair), Janos Simon, Wolfgang Thomas, and Leslie Valiant, has selected Moshe Vardi and Pierre Wolper as the recipients of the 2000 Gödel Prize. for their paper "Reasoning about infinite computations," Information and Computation 115 (1994), pp 1-37.

Sergio Rajsbaum

The 1997 Gödel Prize for outstanding journal articles in the area of theoretical computer science will be awarded to Joseph Halpern and Yoram Moses for their paper "Knowledge and Common Knowledge in a Distributed Environment," J. ACM 37} (1990), 549-587.


2004 Gödel Prize

Maurice Herlihy, Nir Shavit

JACM 1999

Mike Saks, Fotios Zaharoglou SICOMP 2000

[HS]

[SZ]

Sergio Rajsbaum


2004 Gödel Prize

• And a third one with the same result, but not published in a journal:

Liz Borowsky, Eli Gafni [BG]

• Both conference versions in STOC 1993


The importance of the papers in a nutshell

• Lamport’s “Time, Clocks,…” paper is perhaps the first and most fundamental in distributed computing – 1st Dijkstra prize

• An execution is actually a partial order– Different total orders are indistinguishable– A connection with special relativity

The approach discovered by these paper show how to paste together multiple executions– Inherent connection with topology


The Results


The result in common to the three papers

• Wait-free: solving the problem in an asynchronous system where any number out of n processes can crash

• K-Set Agreement: While consensus requires all processors to decide on the same value, k-set agreement allows the processes to decided on at most k different values

• 0 < k < n : k = 1 is consensus, k = n is trivial

Wait-free k-set agreement is impossible


More generally, if t can crash

• t=1, k=1 : consensus impossibility [FLP] Fischer, Lynch, Paterson, 1985

• Dijkstra Award 2001

k-set agreement is possible if t < k But not if t >= k


So we have a t-solvable problem that is not t+1-solvable

• Soma Chaudhuri introduced set agreement in PODC 1990 motivated by this challenge

• It was only known FLP, Fischer, Lynch, Paterson: consensus is possible iff t < 1

• Soma described an algorithm and tried to generalize FLP’s bivalency argument for t

• Working wait-free (t=n-1) is easier than t resilient

k-set agreement is possible if t < k But not if t >= k


Why is this so important?


Why is the wait-free set agreement impossibility so important?

• First step was FLP: consensus is impossible if t ≥ 1• Then characterization for t = 1

– [BMZ’90] O. Biran, S. Moran, S. Zaks 1990, and – [MW’87] Moran, Y. Wolfstahl

• Deciding if a problem is 1-solvable is NP-hard– Biran, Moran, Zaks 1990 – And other related papers for t=1

It lead to a characterization of the problems that are wait-free solvable


Important also because

Many papers in this direction:• t - resilient• More powerful shared memory objects such as test&set• Compositions of objects to wait-free implement others• Unbounded number of processes• Synchronous and partially synchronous systems• Failure detectors• Conditions that restrict the inputs of a problem

It opened the way to characterizations of the problems that are solvable in other models


And New Relations Between Models

In any distributed computing model:• Topological properties determine problem solvability• And time complexity of the solutions• As a function of failure model and asynchrony assumptions• This has lead to new proofs with the same structure for both

synchronous and asynchronous problems, and a more uniform theoretical foundation to the field– Consensus FLP impossibility vs. t+1 synchronous lower bound

Essential link between: distributed computing and topology


To Understand the Topology Link

• Let’s start by looking at a k-set algorithm

• For t < k

• In a shared memory system

• Assuming snapshot operations are available (snapshots can be wait-free implemented in read/write memory)


Algorithm for k-Set agreement if t < kIn a snapshot shared-memory system:

Each process writes its input to shared memory Takes snapshots until it gets a view vector with at least n-t inputs Decides on the smallest input in its view Correctness:

• They see views with ≤ t empty entries• Views are ordered by containment• There are at most t+1 different views => they solve k = t+1 -set agreement


The Views Obtained by the Processes

• Each view is an input vector with at most t empty entries: – (1,2,3) a process saw that Pi wrote value i

• In each execution each process has its view– P1 saw (0,1,2)– P2 saw (0,1,-)

• Some processes may not distinguish between two executions


Views are Put TogetherThe case of 3 Processes, t =2

Each execution is represented by a triangle: simplexThe corners of a simplex are labeled with the views of

each of the 3 processes in the executionThe set of all the executions is the union of all simplexes:

protocol complexSome views after

1 round, starting with

the same input vector,

say (1,2,3):


Generating the ISE Protocol Complex

(1)Immediate Snapshot Executions: a snapshot goes immediately after a write

Both [BG] and [SZ] used ISE

Exec

P1: wr

P2: wr

P3: wr

p1

p2p3

ViewsP1: (1,-,-)P2: (1,2,-)P3: (1,2,3)



(2)

Exec

P1: wr

P2: wr

P3: wr

p1

p2p3

ISE: a snapshot goes immediately after a write

Each vertex represents the local state of one process, 1 round:

ViewsP1: (1,2,-)P2: (1,2,-)P3: (1,2,3)



(3)


Exec

P1: wr

P2: wr

P3: wr

p1

p2

p3


ViewsP1: (1,2,-)P2: (-,2,-)P3: (1,2,3)



(4)


Exec

P1: wr

P2: wr

P3: wrp1

p2

p3


ViewsP1: (1,2,3)P2: (1,-,-)P3: (1,2,3)



(5)


Exec

P1: wr

P2: wr

P3 wrp1

p2p3


ViewsP1: (1,2,3)P2: (1,2,3)P3: (1,2,3)


The Decision is a function only of the process’ view

Exec

P1: wr

P2: wr

P3: wr

p1

p2

p3

P1 can decide only on the input from P2 or itselfP2 can decide on its own input only

ViewsP1: (1,2,-)P2: (-,2,-)P3: (1,2,3)


Decisions induce Sperner’s coloring on the vertices

Exec

P1: wr

P2: wr

P3: wr

p1

p2

p3

• In corners: Pi can decide only on its own input• In Pi-Pj boundary: P can decide only on Pi or Pj inputs

p1

p3


An input vector with 3 different inputs

Exec

P1: wr

P2: wr

P3: wr

• P1 starts with • P2 starts with • P2 starts with

p1

p1

p2

p3

p3


Sperner’s coloring on the boundary

p1

p2

p1

p3


Sperner’s Coloring in the inside arbitrary

p1

p2

p1

p3


Sperner’s Lemma implies

p1

p2

• At least one triangle has all 3 colors.• At least one execution has 3 different values decided 2-set agreement, t=2, impossible in 1 round

p1

p3


But 2-set agreement, t=1 is possible(t<k)

• Views with 2 empty places are removed => • The corners are removed

p2 p3

p1


How to prove that k-set agreement, t ≥ k, is impossible in any number of rounds?

let’s see for wait-free…


Sperner’s lemma holds for any subdivided simplex

• Any subdivided simplex with a Sperner’s Coloring Has a simplex with all colors


Wait-free views always induce a subdivision

• More rounds create finer subdivision– One more round creates one more level of smaller triangles inside each triangle

• Topological invariant: “wait-free is like a solid sphere” – No holes (of any dimension)

• Each paper used different techniques to prove it, and new techniques have been proposed since then that are easier to apply to other models

Main result of BG, HS, SZ :


Topological invariance induces the set agreement impossibility

p2

• Any wait-free protocol induces a subdivision (as below) • Decisions on an input vector with 3 values define a Sperner’s coloring• Sperner’s lemma implies Set agreement is impossible for any number of rounds

p1

p3


A Sperner’s Lemma Proof (there are many)

• The coloring says how to map a solid triangle into another triangle, sending boundary to boundary• Impossible to map it into a hollow triangle, no matter how much it is stretched (subdivided) or bent


1-dimensional Sperner’s lemma

• A connected complex cannot be mapped to a disconnected complex• => consensus is impossible with t > 0

p2

p1


Sperner’s lemma fundamental role

• We saw Sperner’s lemma => set agreement unsolvable• The opposite is also true !

– (using approximate agreement protocol)

• Equivalent to Brouwer’s Fixed Point Theorem– Shake a glass of water, one point remains fixed

• One of the most used theorems of mathematics• Used for all of the distributed computing problems, except for symmetric version of renaming


The same framework is used for any decision problem


The picture can be generalized from one input vector ( ) to another along the boundary

p2

• Change the input of p3 from to• Along the boundary p1 and p2:

– do not know p3’s input and then– cannot distinguish from

p1

p3

p2

p1

p3


Initial states for consensus

0 • Processes: blue, red, orange.

• Independently assign 0 or 1

• Isomorphic to 2-sphere• This is the input complex

0

01

1


States after 1 round, starting in the initial states for consensus

0 • Running a wait-free protocol creates a subdivision of each input simplex

• Topology of the input complex is preserved

0

01

1


What about more powerful models?

Synchronous systems, asynchronous with more powerful primitives

More problems are solvable because holes are introduced into the subdivision

Let’s see a 1-resilient asynchronous model…


States after 1 round (input 2 not shown)

0 • Running a 1-resilient protocol creates holes

• Topology of the input complex is not exactly preserved

2-set agreement is possible

0

01

1


2-set agreement is possible because now there is a map from the protocol

complex into the hollow simplex0

0

01

1

0

1 2


Synchronous Model

• In t-resilient computation, t >1 there are holes, but do not change their type with the number of runs

• In synchronous computation yes…


Synchronous protocol complex evolution

zero

two

one

Connected butnot 1-connected

Disconnected


Summarizing the importance of the papers


The Topology Link

• Any algorithm in any model approx. preserves the

topology of the inputs to a problem• To solve a problem we need to create “holes”

in the topology (to be able to map inputs to outputs)• Stronger models create more “holes” than

weaker models; wait-free does not create any holes

Why a problem is solvable after more rounds?

1. Finer subdivisions of the inputs are produced, and

2. in some models also holes


Problem solvability is undecidable

By reduction to a classic topology problem, equivalent to a classic algebra problem:

Can a given loop be contracted in a complex?

For a distributed computing model, is there an algorithm solving a given problem?

No: wait-free and in other models [KG,HR] (yes in an asynchronous system if t=1, Moran et al)


Contractibility is undecidable

not contractible

contractible

contractible


What does it mean?

The Turing theory of computation has been extraordinary successful in providing the foundations for theoretical computer science

Lenore Blum

The previous fundamental limitation is about distribution of resources, and due to uncertainty from delays and failures− NOT from the limitations of a Turing Machine


Conclusions

• The 2004 Gödel Award papers have been greatly influential, motivating a lot of subsequent work, as well as changing the way we understand distributed computing

• There are many open problems about:– Proving topological properties of specific models

– Establishing consequences of these

– E.g. Byzantine failures, randomized , self-stabilization, network topology


We have seen a “cut” perspective of the computation

• Considering how each execution is deformed into another, we get an orthogonal topology theory

• Introduced by Eric Goubault in ~93, with follow up by CONCUR community

Considered the states at the end of a run


Congratulations to the authors of the three papers !

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Introduction to the Gödel Prize Session