TwoProcess Systems Companion slides for Distributed Computing Through

Two-Process Systems Companion slides for Distributed Computing Through Combinatorial Topology Maurice Herlihy & Dmitry Kozlov & Sergio Rajsbaum Distributed Computing through Combinatorial Topology

Two-Process Systems Two-process systems can be captured by elementary graph theory gentle introduction to more general structures needed later for larger systems Distributed Computing through 2 Combinatorial Topology

Road Map Elementary Graph Theory Tasks Models of Computation Approximate Agreement Task Solvability Distributed Computing through Combinatorial Topology 3

Road Map Elementary Graph Theory Tasks Models of Computation Approximate Agreement Task Solvability Distributed Computing through Combinatorial Topology 4

A Vertex Distributed Computing through Combinatorial Topology 5

A Vertex Combinatorial: an element of a set. Distributed Computing through Combinatorial Topology 6

A Vertex Combinatorial: an element of a set. Geometric: a point in Euclidean Space Distributed Computing through Combinatorial Topology 7

An Edge 8

An Edge Combinatorial: a set of two vertexes. 9

An Edge Combinatorial: a set of two vertexes. Geometric: line segment joining two points 10

A Graph Distributed Computing through Combinatorial Topology 11

A Graph Combinatorial: a set of sets of vertices. Distributed Computing through Combinatorial Topology 12

A Graph Combinatorial: a set of sets of vertices. Geometric: points joined by line segments Distributed Computing through Combinatorial Topology 13

Graphs finite set V with a collection G of subsets of V, Distributed Computing through Combinatorial Topology 14

Graphs vertices finite set V with a collection G of subsets of V, simplices (singular: simplex) Distributed Computing through Combinatorial Topology 15

Graphs finite set V with a collection G of subsets of V, (1) If X 2 G, then |X| · 2 Distributed Computing through Combinatorial Topology 16

Graphs finite set V with a collection G of subsets of V, (1) If X 2 G, then |X| · 2 vertex: |X| = 1 edge: |X|= 2 Distributed Computing through Combinatorial Topology 17

Graphs finite set V with a collection G of subsets of V, (1) If X 2 G, then |X| · 2 (2) for all v 2 V, {v} 2 G 18

Graphs finite set V with a collection G of subsets of V, (1) If X 2 G, then |X| · 2 (2) for all v 2 V, {v} 2 G (3) for all X 2 G, and Y ½ X, Y 2 G 19

Dimension dimension 0 dimension 1 dim(X) = |X|-1. Distributed Computing through Combinatorial Topology 20

Pure Graphs pure of dim 0 pure of dim 1 Distributed Computing through Combinatorial Topology 21

Graph Coloring Distributed Computing through Combinatorial Topology 22

Graph Coloring : G ! C Distributed Computing through Combinatorial Topology 23

Graph Coloring : G ! C for each edge (s 0, s 1) 2 G, (s 0) (s 1). Distributed Computing through Combinatorial Topology 24

Graph Coloring : G ! C for each edge (s 0, s 1) 2 G, (s 0) (s 1). chromatic graphs usually process names Distributed Computing through Combinatorial Topology 25

Graph Labeling 0 1 Distributed Computing through Combinatorial Topology 26

Graph Labeling 0 1 f: G ! L 0 1 Distributed Computing through Combinatorial Topology 27

Graph Labeling 0 1 f: G ! L usually values from some domain Distributed Computing through Combinatorial Topology 28

Labeled Chromatic Graph 1 0 name(s) = (s) view(s) = f(s) Distributed Computing through Combinatorial Topology 29

Simplicial Maps Vertex-to-vertex map … that also sends edges to edges. Distributed Computing through Combinatorial Topology 30

Rigid Simplicial Maps A simplicial map can send an edge to a vertex … Distributed Computing through Combinatorial Topology 31

Rigid Simplicial Maps A simplicial map can send an edge to a vertex … A simplicial map that sends distinct vertices to distinct vertices is rigid. Distributed Computing through Combinatorial Topology 32

A Path Between two Vertices Distributed Computing through Combinatorial Topology 33

A Path Between two Vertices A graph is connected if there is a path between every pair of vertices Distributed Computing through Combinatorial Topology 34

Not Connected A graph is connected if there is a path between every pair of vertices Distributed Computing through Combinatorial Topology 35

Theorem The image of a connected graph under a simplicial map is connected. Á Distributed Computing through Combinatorial Topology 36

Carrier Maps For graphs G, H, a carrier map ©: G ! 2 H © Carries each simplex of G to a subgraph of H … Distributed Computing through Combinatorial Topology 37

Carrier Maps For graphs G, H, a carrier map ©: G ! 2 H © Carries each simplex of G to a subgraph of H … satisfying monotonicity: for all ¾, ¿ 2 G, if ¾µ¿, then Distributed Computing through Combinatorial Topology ©(¾)µ©(¿). 38

Strict Carrier Maps Monotonicity For all ¾, ¿ 2 G, if ¾µ¿, then ©(¾)µ©(¿). Distributed Computing through Combinatorial Topology 39

Strict Carrier Maps Monotonicity For all ¾, ¿ 2 G, if ¾µ¿, then ©(¾)µ©(¿). Equivalent to … ©(¾Å¿) µ ©(¾) Å ©(¿) Distributed Computing through Combinatorial Topology 40

Strict Carrier Maps Monotonicity For all ¾, ¿ 2 G, if ¾µ¿, then ©(¾)µ©(¿). Equivalent to … ©(¾Å¿) µ ©(¾) Å ©(¿) Definition © is strict if ©(¾Å¿) = ©(¾) Å ©(¿) Distributed Computing through Combinatorial Topology 41

Road Map Elementary Graph Theory Tasks Models of Computation Approximate Agreement Task Solvability Distributed Computing through Combinatorial Topology 42

Two Processes Hello! I’m Bob Alice Distributed Computing through Combinatorial Topology 43

Informal Task Definition Processes start with input values … They communicate … They halt with output values … legal for those inputs. Distributed Computing through Combinatorial Topology 44

Formal Task Definition Input graph I all possible assignments of input values Distributed Computing through Combinatorial Topology 45

Formal Task Definition Input graph I all possible assignments of input values Output graph O all possible assignments of output values Distributed Computing through Combinatorial Topology 46

Formal Task Definition Input graph I all possible assignments of input values Output graph O all possible assignments of output values Carrier map : I ! 2 O all possible assignments of output values for each input Distributed Computing through Combinatorial Topology 47

Task Input Graph: Consensus 0 1 I 0 Distributed Computing through Combinatorial Topology 1 48

Task Input Graph 0 1 I 0 1 Distributed Computing through Combinatorial Topology 49

Task Input Graph 1 1 Pure Colored by process names Labeled by input values 0 0 Distributed Computing through Combinatorial Topology 50

Task Output Graph 1 1 O 0 0 Distributed Computing through Combinatorial Topology 51

Task Carrier Map 1 1 : I ! 2 O I 0 1 O 0 Distributed Computing through Combinatorial Topology 0 52

Task Carrier Map 1 1 : I ! 2 O I 0 1 O 0 0 If Bob runs alone with input 1 … then he decides output 1. Distributed Computing through Combinatorial Topology 53

Task Carrier Map 1 1 : I ! 2 O I 0 1 O 0 0 If Bob and Alice both have input 1 … then they both decide output 1. Distributed Computing through Combinatorial Topology 54

Task Carrier Map 1 1 : I ! 2 O I 0 1 O 0 0 If Bob has 1 and Alice 0 … then they must agree, on either one. Distributed Computing through Combinatorial Topology 55

Example: Coordinated Attack Alice Bob Enemy Alice and Bob win If they both attack together Distributed Computing through Combinatorial Topology 56

Input Graph Attack at dawn! I Attack at noon! Indifferent Distributed Computing through Combinatorial Topology 57

Output Graph failed! dawn! noon! O Distributed Computing through Combinatorial Topology 58

Carrier Map I O Distributed Computing through Combinatorial Topology 59

Carrier Map dawn! I O failed! Distributed Computing through Combinatorial Topology 60

Carrier Map I O failed! Distributed Computing through Combinatorial Topology noon! 61

Carrier Map dawn! I O Distributed Computing through Combinatorial Topology 62

Example: Coordinated Attack Alice Bob Enemy Distributed Computing through Combinatorial Topology 63

Example: Coordinated Attack Alice Bob Enemy Alice and Bob realize that they do not need to agree on an exact time … Distributed Computing through Combinatorial Topology 64

Example: Coordinated Attack Alice Bob Enemy Alice and Bob realize that they do not need to agree on an exact time … they will win if attack times are sufficiently close. Distributed Computing through Combinatorial Topology 65

Coordinated Attack Graphs I 0 0 1/5 1 2/5 3/5 4/5 1 O Distributed Computing through Combinatorial Topology 66

Coordinated Attack Graphs I 0 0 1/5 1 2/5 3/5 4/5 1 O Distributed Computing through Combinatorial Topology 67

Coordinated Attack Graphs I 0 0 1/5 1 2/5 3/5 4/5 1 O Distributed Computing through Combinatorial Topology 68

Coordinated Attack Graphs I 0 0 1/5 1 2/5 3/5 4/5 1 O Distributed Computing through Combinatorial Topology 69

Road Map Elementary Graph Theory Tasks Models of Computation Approximate Agreement Task Solvability Distributed Computing through Combinatorial Topology 70

Protocols Models of Computation Distributed Computing through Combinatorial Topology 71

Alice’s Protocol shared mem array 0. . 1 of Value view: Value : = my input value; for i: int : = 0 to L do mem[A] : = view; view : = view + mem[A] + mem[B]; return δ(view) Finite program Bob’s protocol is symmetric Distributed Computing through Combinatorial Topology 72

Alice’s Protocol shared mem array 0. . 1 of Value view: Value : = my input value; shared two-element memory for i: int : = 0 to L do mem[A] : = view; view : = view + mem[B]; return δ(view) Distributed Computing through Combinatorial Topology 73

Alice’s Protocol shared mem array 0. . 1 of Value view: Value : = my input value; for i: int : = 0 to L do Start with input value mem[A] : = view; view : = view + mem[B]; return δ(view) Distributed Computing through Combinatorial Topology 74

Alice’s Protocol shared mem array 0. . 1 of Value view: Value : = my input value; for i: int : = 0 to L do mem[A] : = view; Run for L layers view : = view + mem[B]; return δ(view) Distributed Computing through Combinatorial Topology 75

Alice’s Protocol shared mem array 0. . 1 of Value view: Value : = my input value; for i: int : = 0 to L do mem[A] : = view; view : = view + mem[B]; return Alice writes her value, read Bob’s value, δ(view) and concatenate it to my view Distributed Computing through Combinatorial Topology 76

Alice’s Protocol shared mem array 0. . 1 of Value view: Value : = my input value; for i: int : = 0 to L do mem[A] : = view; view : = view + mem[B]; return Alice writes her value, read Bob’s value, δ(view) and concatenate it to my view ) l o c o t o r p n o i t a m r o f in l l u f ( Distributed Computing through Combinatorial Topology 77

Alice’s Protocol shared mem array 0. . 1 of Value view: Value : = my input value; for i: int : = 0 to L do mem[A] : = view; view : = view + mem[B]; return δ(view) finally, apply task-specific decision map to view Distributed Computing through Combinatorial Topology 78

Formal Protocol Definition Input graph I all possible assignments of input values Distributed Computing through Combinatorial Topology 79

Formal Protocol Definition Input graph I all possible assignments of input values Protocol graph P all possible process views after execution Distributed Computing through Combinatorial Topology 80

Formal Protocol Definition Input graph I all possible assignments of input values Protocol graph P all possible process views after execution Carrier map ¥: I ! 2 P all possible assignments of views Distributed Computing through Combinatorial Topology 81

One-Round Protocol Graph I 0 1 ¥ 0? 01 01 ? 1 P Distributed Computing through Combinatorial Topology 82

One-Round Protocol Graph P 0? 01 01 ? 1 Colored by process names Labeled with final views Distributed Computing through Combinatorial Topology 83

One-Round Protocol Graph P 0? 01 01 ? 1 Alice finishes before Bob starts, doesn’t see his value Distributed Computing through Combinatorial Topology 84

One-Round Protocol Graph P 0? 01 01 ? 1 Alice and Bob run together, she sees his value. Distributed Computing through Combinatorial Topology 85

One-Round Protocol Graph P 0? 01 01 ? 1 Alice finishes, then Bob starts Distributed Computing through Combinatorial Topology 86

One-Round Protocol Graph P 0? 01 01 ? 1 Alice and Bob run together Distributed Computing through Combinatorial Topology 87

One-Round Protocol Graph P 0? 01 01 ? 1 Bob can’t tell whether Alice saw him Distributed Computing through Combinatorial Topology 88

Execution Carrier Map I 0 1 ¥ 0? 01 01 ? 1 P Distributed Computing through Combinatorial Topology 89

Execution Carrier Map I 0 ¥: I ! 2 P 0? 1 ¥ 01 01 P ? 1 strict carrier map ¥(¾) Å ¥(¿) = ¥(¾ Å ¿) Distributed Computing through 90 Combinatorial Topology

The Decision Map 0? 01 01 ? 1 2/3 1 Protocol graph δδ 0 1/3 Distributed Computing through Combinatorial Topology Output graph 91

All Together 0 1 I ¥ 0? 01 01 ? 1 P 0 1/3 2/3 1 O δ Distributed Computing through Combinatorial Topology 92

Definition Decision map δ is carried by carrier map if for each input vertex s, δ(¥(s)) µ (s) for each input edge ¾, δ(¥(¾)) µ (¾). Distributed Computing through Combinatorial Topology 93

Meaning δ(¥(s)) µ (s) process strarts in state s 94

Meaning δ(¥(s)) µ (s) runs the protocol to completion 95

Meaning δ(¥(s)) µ (s) makes a decision … 96

Meaning δ(¥(s)) µ (s) decision is permitted by task carrier map 97

Solving a Task Definition The protocol (I, P, ¥) solves the task (I, O, ) Distributed Computing through Combinatorial Topology 98

Solving a Task Definition The protocol (I, P, ¥) solves the task (I, O, ) if there is … Distributed Computing through Combinatorial Topology 99

Solving a Task Definition The protocol (I, P, ¥) solves the task (I, O, ) if there is … a simplicial decision map δ: P ! O Distributed Computing through Combinatorial Topology 100

Solving a Task Definition The protocol (I, P, ¥) solves the task (I, O, ) if there is … a simplicial decision map δ: P ! O such that δ is carried by. (δ agrees with ) Distributed Computing through Combinatorial Topology 101

Layered Read-Write Model 102

Layered Read-Write Protocol shared mem array 0. . 1, 0. . L of Value view: Value : = my input value; for i: int : = 0 to L do mem[i][A] : = view; view : = view + mem[A] + mem[B]; return δ(view) Distributed Computing through Combinatorial Topology 103

Layered Read-Write Protocol shared mem array 0. . 1, 0. . L of Value view: Value : = my input value; for i: int : = 0 to L do mem[i][A] : = view; As before, run for L layers view : = view + mem[A] + mem[B]; return δ(view) Distributed Computing through Combinatorial Topology 104

Layered Read-Write Protocol shared mem array 0. . 1, 0. . L of Value view: Value : = my input value; for i: int : = 0 to L do mem[i][A] : = view; view : = view + mem[A] + mem[B]; return δ(view) Each layer uses a distinct, “clean” memory Distributed Computing through Combinatorial Topology 105

Layered R-W Protocol Graph I 0 1 ¥ 0? 01 01 ? 1 P Distributed Computing through Combinatorial Topology 106

Layered R-W Protocol Graph I ¥ P P is always a subdivision of I Distributed Computing through Combinatorial Topology 107

Road Map Elementary Graph Theory Tasks Models of Computation Approximate Agreement Task Solvability Distributed Computing through Combinatorial Topology 108

Alice’s 1/3 -Agreement Protocol mem[A] : = 0 other : = mem[B] if other == ? then decide 0 else decide 1/3 Distributed Computing through Combinatorial Topology 109

Alice’s 1/3 -Agreement Protocol mem[A] : = 0 if mem[B] == ? then Alice writes her value to memory decide 0 else decide 1/3 Distributed Computing through Combinatorial Topology 110

Alice’s 1/3 -Agreement Protocol mem[A] : = 0 if mem[B] == ? then decide 0 else If she doesn’t see Bob’s value, decide her own. decide 1/3 Distributed Computing through Combinatorial Topology 111

Alice’s 1/3 -Agreement Protocol mem[A] : = 0 if mem[B] == ? then decide 0 else decide 1/3 If she see’s Bob’s value, jump to the middle Distributed Computing through Combinatorial Topology 112

0 1/3 2/3 1 Distributed Computing through Combinatorial Topology 113

One-Layer 1/3 -Agreement Protocol 0 1 I ¥ 0? 01 01 ? 1 P 0 1/3 2/3 1 O δ Distributed Computing through Combinatorial Topology 114

No 1 -Layer 1/5 -Agreement Protocol 0 1 I ¥ 0? δ 0 01 01 ? 1 P (no map possible) 1/5 2/5 3/5 Distributed Computing through Combinatorial Topology 4/5 1 O 115

2 -Layer 1/5 -Agreement 1 0 I ¥ 0? 01 layer 1 01 ? 1 layer 2 ¥ δ 0 1/5 2/5 3/5 Distributed Computing through Combinatorial Topology 4/5 O 1 116

Fact In the layered read-write model, The 1/K-Agreement Task Has a dlog 3 Ke–layer protocol Distributed Computing through Combinatorial Topology 117

Road Map Elementary Graph Theory Tasks Models of Computation Approximate Agreement Task Solvability Distributed Computing through Combinatorial Topology 118

Fact The protocol graph for any L-layer protocol with input graph I is a subdivision of I, where each edge is subdivided 3 L times. Distributed Computing through Combinatorial Topology 119

Main Theorem The two-process task (I, O, ) is solvable in the layered read-write model if and only if there exists a connected carrier map ©: I ! 2 O carried by. Distributed Computing through Combinatorial Topology 120

Corollary The consensus task has no layered read-write protocol Distributed Computing through Combinatorial Topology 121

Corollary Any ²–agreement task has a layered read-write protocol Distributed Computing through Combinatorial Topology 122

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