Algebraic Topology and Decidability in Distributed Computing Maurice

Algebraic Topology and Decidability in Distributed Computing Maurice Herlihy Brown University Joint work with Sergio Rajsbaum, Nir Shavit, and Mark Tuttle

Overview • Applications of algebraic topology – to fault-tolerant computing – especially decidability issues • Known results – focus on techniques 3/9/2021 2

Decision Tasks Before: private inputs 3/9/2021 After: private outputs 3

Example: 3 -Consensus Before: private inputs 3/9/2021 After: agree on one input 4

Example: (3, 2)-Consensus Before: private inputs 3/9/2021 After: agree on 1 or 2 inputs 5

A Vertex Point in high-dimensional Euclidean Space 3/9/2021 6

Simplexes 0 -simplex (vertex) 2 -simplex (solid triangle) 3/9/2021 1 -simplex (edge) 3 -simplex (solid tetrahedron) 7

Simplicial Complex 3/9/2021 8

Simplicial Maps • Vertex-to-vertex map • carrying simplexes to simplexes • induces piece-wise linear map 3/9/2021 9

Vertex = Process State Process id (color) 7 3/9/2021 Value (input or output) 10

Simplex = Global State 3/9/2021 11

Complex = Global States 3/9/2021 12

Initial States for Consensus 0 0 1 3/9/2021 • Processes: blue, red, green. • Independently assign 0 or 1 • Isomorphic to 2 sphere • the input complex 13

Final States for Consensus • Processes agree on 0 or 1 • Two disjoint nsimplexes • the output complex 0 0 0 1 3/9/2021 1 1 14

Problem Specification • For each input simplex S • relation D(S) – defines corresponding set of legal outputs – carries input simplex – to output subcomplex 3/9/2021 15

Consensus Specification 0 0 0 1 Simplex 3/9/2021 of all-zero inputs 1 1 16

Consensus Specification 0 0 0 1 Simplex 3/9/2021 of all-one inputs 1 1 17

Consensus Specification 0 0 0 1 Mixed-input simplex 3/9/2021 1 1 18

Protocols • • Finite program starts with input values behavior depends on model. . . halts with decision value 3/9/2021 19

Protocol Complex • Each protocol defines a complex – vertex: my view of computation – simplex: everyone’s view • Protocol complex – depends on model of computation – what did you expect? 3/9/2021 20

Simple Model: Synchronous Message-Passing 3/9/2021 Round 0 Round 1 21

Failures: Fail-Stop Partial broadcast 3/9/2021 22

Single Input: Round Zero 0 0 3/9/2021 0 • No messages sent • vertexes labeled with input values • isomorphic to input simplex 23

Round Zero Protocol Complex 0 0 1 3/9/2021 • No messages sent • vertexes labeled with input values • isomorphic to input complex 24

Single Input: Round One red fails 00 00 green fails 000 000 blue fails 3/9/2021 00 00 no one fails 25

Protocol Complex: Round One 3/9/2021 26

Protocol Complex Evolution zero 3/9/2021 two one 27

Observation • Decision map • is a simplicial map – vertexes to vertexes, but also – simplexes to simplexes – respects specification relation D 3/9/2021 28

Summary d Protocol complex Input 3/9/2021 complex D Output complex 29

New Model: Asynchronous Failures ? ? ? 3/9/2021 ? ? ? 30

What We Know already • Impossibility results • Algorithms • in various models – k-Consensus – (n, k)-consensus – renaming, etc. 3/9/2021 31

Decidability Results • Biran, Moran, & Zaks 88 – one-resilient message-passing decidable • Gafni & Koutsoupias 96 – t-resilient read/write undecidable • Herlihy & Rajsbaum 97 – lots of other models 3/9/2021 32

Robot Rendez-Vous (formerly loop agreement) • Complex • loop • three vertexes (rendez-vous points) 3/9/2021 33

One Rendez-Vous Point output input 3/9/2021 34

Two Rendez-Vous Points output input 3/9/2021 35

Three Rendez-Vous Points output input 3/9/2021 36

Contractibility contractible not contractible 3/9/2021 37

Theorem • The Robot Rendez-Vous problem – in the asynchronous – message-passing model • has a solution – if and only if – loop is contractible 3/9/2021 38

Solvable implies Contractible • Theorem: – any protocol complex • in the asynchronous message-passing model • where more than one process can fail – is connected and simply connected • path between any two vertexes • any loop is contractible – trust me! • or consult [Herlihy, Rajsbaum, Tuttle 98] 3/9/2021 39

Solvable implies Contractible d Protocol Complex 3/9/2021 v 0 All inputs v 0 Output Complex 40

Solvable implies Contractible v 0 d v 1 d All inputs 3/9/2021 v 1 41

Solvable implies Contractible d All inputs 3/9/2021 v 0 or v 1 42

Solvable implies Contractible d 3/9/2021 43

Solvable implies Contractible Protocol complex is simply connected d 3/9/2021 D E Q 44

Contractible implies Solvable f Map f is continuous 3/9/2021 45

Contractible implies Solvable f Take simplicial approximation 3/9/2021 46

Contractible implies Solvable f Approximate agreement 3/9/2021 D E Q 47

Decidability • Contractibility is undecidable – even for finite complexes – [Novikov 1955] • Reduces to – the word problem for – finitely-presented groups 3/9/2021 48

Decidability • Asynchronous message-passing – decidable for one failure – undecidable otherwise • But wait, there’s more. . . 3/9/2021 49

Decidability Results Weird or what? 3/9/2021 50

Conclusions • Decidability still an open area – word problem is actually solvable for most reasonable classes of groups – do these classes correspond to reasonable models of computation? 3/9/2021 51

Clip Art 3/9/2021 52