The Topological Structure of Asynchronous Computability Nir Shavit

The Topological Structure of Asynchronous Computability Nir Shavit Tel-Aviv University and Sun Microsystems Labs (Joint work with Maurice Herlihy of Brown University) © Herlihy and Shavit 2004 1

From the New York Times … May 7 th, 2004 – “Intel said on Friday that it was scrapping its development of two microprocessors, a move that is a shift in the company's business strategy…. ” May 8 th, 2004 – “Intel … [has] decided to focus its development efforts on “dual core” processors …. with two engines instead of one, allowing for greater efficiency because the processor workload is essentially shared. ” 2

The Future of Computing § Speeding up uniprocessors is harder and harder § Intel, Sun, AMD, now focusing on “multi-core” architectures § Soon, every computer will be a multiprocessor 3

What has this got to do with theory? Alan Mathison Turing. Forefather of Computer Science. Inventor of the Turing Machine. 4

1936 - Turing Computability Turing-Machine f r # a r a zrw 4 p t l r s r 0 1 g a k 0 3 0 A mathematical model that captures what is and is not computable on uniprocessors… 5

Time and Asynchrony “Time is Nature’s way of making sure that everything doesn’t happen all at once. ” (Anonymous, circa 1970) Real world asynchrony: no “clock in the sky” 6

Asynchronous Computability 10011 Shared Memory/Network A mathematical model capturing what is and is not computable by multiple processors in an asynchronous environment 7

Asynchrony Complicates Life Synchrony Asynchrony A single execution A multitude of concurrent executions Seek help from modern mathematics! 8

FLP § Fischer Lynch and Paterson § Showed that Asynchronous computability ≠ Turing computability § Consensus • Trivial in uniprocessor • Impossible with 1 asynchronous failure § Reasoned directly about executions 9

Graph Theory § Biran, Moran, Zachs 1988 § Single asynchronous failure § Coordination problem • is a graph § Problem is asynchronously computable • iff graph is connected 10

More Graph Theory § Fischer Lynch 82, Dolev Strong 83, Merritt 85, Dwork Moses 90, … § Synchronous crash failures § Computation state • is a graph § Consensus not computable • while graph is connected 11

Limitations of Graph Theory § Asynchronous model • Multiple failures? § Synchronous model • Problems beyond consensus? § Need a more general notion of connectivity 12

The Topological Approach § § Borowsky Gafni STOC 93 Herlihy Shavit STOC 93, JACM 1999 Saks Zaharoglou STOC 93, SICOMP 2000 Showed k-set agreement impossible § Generalization of consensus [Chaudhuri 90] § Open problem for several years 13

The Topological Approach § Computations as geometric objects § Use topological methods to show existence of “bad” executions. § Borowsky & Gafni § Sperner’s Lemma § Herlihy & Shavit § Simplicial complexes & homology § Saks & Zaharoglou § Brouwer’s fixed-point theorem 14

Example: Autonomous Air-Traffic Control 36, 000 f t DL 227 35, 000 f t 34, 000 f t 33, 000 f t AL 991 32, 000 f t AA 082 31, 000 f t Pick your own altitude. How many slots do we need to allow safe coordination? 15

Renaming § Process has input name (flight #) § Must generate output name (altitude) § Interested in comparison-based protocols: • Equality: A=B? • Order: A<B? • Nothing else (rules out trivial solutions) 16

History § Proposed by Attiya, Bar-Noy, Dolev, Peleg, and Reischuk § They showed • Solution for 2 n-1 names • Impossibility for n+1 names • Intermediate values? § Long-standing open problem … • Topological methods finally showed … • Intermediate values also impossible 17

A Vertex Point in high-dimensional Euclidean Space 18

Simplexes n o i s en 2 m i D 3 solid tetrahedron 1 solid triangle 0 line point 19

Complexes Glue simplexes to form complexes 20

A Cycle OK to think of it as oriented path Also works in higher dimensions 21

A Boundry A cycle is a boundry if it goes around a “solid” region Encompases no holes 22

Not Every Cycle is a Boundry Goes around a hole 23

Connectivity § A complex is n-connected if § Every cycle of dimension n or less § Is also a boundary § No “holes” in any dimension § Fundamental group is trivial § Higher homology groups trivial 24

Chromatic Complexes Each (n-1)-simplex colored by n distinct colors Corresponding to process ids 25

Chromatic Simplicial Map Color-preserving vertex-to-vertex map That also carries simplexes to simplexes 26

Chromatic Simplicial Maps Preserve Connectivity Cannot map boundries around holes 27

Vertex = Process State Process id (color) 2 Value (input or output) 28

Simplex = Global State 1 1 2 2 3 3 29

Complex = Global States 4 4 1 1 2 3 30

A Decision Task 2 Planes, 2 Slots input complex (all possible flights) output complex (all consistent slot choices) Task spec relation 1 2 2 1 AL 991 AA 082 31

3 Planes, 3 Slots output complex input complex 1 AL 991 2 3 1 1 DL 227 AA 082 2 3 P, 6 1 1 3 2 1 32

3 Planes, 4 Slots input complex output complex 3 1 2 3 R 1 2 4 2 1 3 3 1 4 4 1 2 3 4 1 4 24 3 4 2 1 1 3 2 4 21 1 33

Shared Memory Computation § Asynchronous § arbitrary delays § e. g. , interrupts, page faults, etc. § Wait-free § Processes can fail or be slow § Communication by reading and writing shared memory 34

Asynchronous Computability Theorem task spec Input complex output complex Simplicial map A task has a wait-free solution if and only if one can chromatically subdivide its input complex so that there exists a chromatic simplicial map to its output complex that refines the task specification map. 35

Asynchrony and Continuity 1 AL 112 AL 991: post(1) look() decide (1 , ) AA 081 2 Houston, we have a problem… (1 , 1 ) AA 082: post(1) look() decide ( , 1) d decide 11 2? 11 If there were a continuous map…problem goes away 36

Three Process Protocol Complex DL 227 runs solo (Some simplexes omitted for clarity) AA 081 and AL 112 run solo 37

In General d Protocol complex AL 991 2 R Task def 3 1 4 Input complex 2 4 DL 227 AA 082 3 1 4 4 3 2 1 Output complex 38

2 Plane 3 Slot Renaming P, 3 Q, 2 Q, 1 P, 1 If view = (my_id, ) decide 1 else if my_id < his_id decide 2 else decide 3 P, 2 Q, 3 simplicial map P Q Q P single chromatic subdivision of input complex A comparison based algorthm: only compares planes ids. 39

3 Planes 4 Slots § Impossibility by reduction § Assume a protocol for 3 airplanes, 4 slots § Choose § 0 if your name is even § 1 if your name is odd § Result § Not all odd § Not all even 40

Output Complex (3 processes) 1 0 0 0 1 1 41

Protocol Complex (schematic) 42

Boundary = 2 -Process Executions 43

Protocol Complex for One Process Execution P( ) decides 1 WLOG P( ) decides 1 by symmetry 44

2 -Process execution might be mapped this way … boundary protocol Wraps around -1 times output 45

2 -Process execution might be mapped that way … boundary protocol Wraps around +2 times output 46

In general … boundary protocol Wraps around hole 3 k-1 ≠ 0 times output 47

But a Simplicial Cannot Map a Boundary Around a Hole…QED! Hole is an obstruction 48

So 3 Planes Need At Least 5 Slots! DL 227 35, 000 f t 34, 000 f t 33, 000 f t AL 991 32, 000 f t AA 082 31, 000 f t 49

The Power of the Asynchronous Computability Theorem Asynchrony Concurrency Arguments about geometric objects A multitude of concurrent executions Topological Representation 50

The Glorious Future § Our work • Asynchronous, wait-free, one-shot, RW memory § Open problems • • Long-lived computations Other kinds of memory (compare&swap) Randomization Other progress conditions … 51