A Hard Problem In A Boolean Asynchronous Freezing Cellular Automata - - PowerPoint PPT Presentation

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a hard problem in a boolean asynchronous freezing

A Hard Problem In A Boolean Asynchronous Freezing Cellular Automata - - PowerPoint PPT Presentation

Feb 15, 2024 136 likes •1.94k views

A Hard Problem In A Boolean Asynchronous Freezing Cellular Automata Eric Goles, Diego Maldonado, Pedro Montealegre, and Mart n R os-Wilson Facultad de Ingenier a y Ciencias International Workshop on Boolean Neworks, 9 de enero

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SLIDE 1

A Hard Problem In A Boolean Asynchronous Freezing Cellular Automata

Eric Goles, Diego Maldonado, Pedro Montealegre, and Mart´ ın R´ ıos-Wilson

Facultad de Ingenier´ ıa y Ciencias

International Workshop on Boolean Neworks, 9 de enero de 2020

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SLIDE 2

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 3

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 4

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 5

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

slide-6

SLIDE 6

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 7

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 8

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 9

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 10

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 11

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 12

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 13

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 14

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 15

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 16

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 17

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 18

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 19

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 20

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 21

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 22

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 23

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 24

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 25

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 26

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 27

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 28

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 29

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 30

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 31

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 32

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 33

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 34

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 35

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 36

Cellular automata

Recipe

Ingredients: ◮ n States ◮ 1 Grid with cells ◮ 1 Neighborhood ◮ 1 Local function Directions: Apply the local function on each cell in parallel (synchronously) once. Repeat until obtain a dynamical system.

Remark

In general, CA shows a complex behavior.

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SLIDE 37

Asynchronous Cellular Automata

An Asynchronous Cellular Automaton is a function F : QZd→ QZd: F σ(0)(x) = x; F σ(t)(x)z =

f (F σ(t−1)(x)N(z))

if z = σ(t) xz

therwise.

◮ N ⊂ Zd is called the neighborhood ◮ f is the local function ◮ σ : N → Zd is the update scheme

Remark

In asynchronous cellular automata only changes one cell at each time step.

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SLIDE 38

Neighborhoods and Boolean Network

Main neighborhoods

von Neumann neighborhood Moore neighborhood

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SLIDE 39

Neighborhoods and Boolean Network

u v w s t u v w s t von Neumann Cellular automata grid Boolean network interaction graph

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SLIDE 40

Neighborhoods and Boolean Network

u v w s t u v w s t Moore Cellular automata grid Boolean network interaction graph von Neumann Cellular automata grid p q r x p q r x

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SLIDE 41

Example: Life without Death.

Local funcation of Life without death

f : Or any permutation

Time 1 1 2 3 4

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SLIDE 42

Example: Life without Death.

Local funcation of Life without death

f : Or any permutation

Time 1 1 2 3 4 f

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SLIDE 43

Example: Life without Death.

Local funcation of Life without death

f : Or any permutation

Time 2 2 3 4 f

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SLIDE 44

Example: Life without Death.

Local funcation of Life without death

f : Or any permutation

Time 3 2 3 4 f

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SLIDE 45

Example: Life without Death.

Local funcation of Life without death

f : Or any permutation

Time 4 4 f

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SLIDE 46

Example: Life without Death.

Local funcation of Life without death

f : Or any permutation

Time 5 f

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SLIDE 47

Example: Life without Death.

Local funcation of Life without death

f : Or any permutation

Time 5

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SLIDE 48

Definitions

Definition

An ACA is freezing if f : ; ∗ →

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SLIDE 49

Definitions

Definition

An ACA is freezing if f : ; ∗ →

Definition

Given a configuration x, a cell z ∈ Z2 is unstable if ∃σ, ∃T : F σ(T)(x)z = xz

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SLIDE 50

Definitions

Definition

An ACA is freezing if f : ; ∗ →

Definition

Given a configuration x, a cell z ∈ Z2 is unstable if ∃σ, ∃T : F σ(T)(x)z = xz

Definition (AsyncUnstabilityF)

F is a FACA. INPUT: A n × n-periodic configuration x and a cell z. QUESTION: Does there exist a updating scheme σ and T > 0 such that F σ(T)(x)z = xz?

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SLIDE 51

History

◮ Prediction problem. Input: F, z, x, q, t, question: F t(x)z = q?. (Banks 1971)

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SLIDE 52

History

◮ Prediction problem. Input: F, z, x, q, t, question: F t(x)z = q?. (Banks 1971) ◮ Given a FCA and a n × n conf. x, F O(n2)(x) is a fixed point. (Goles, Ollinger, Theyssier 2015)

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SLIDE 53

History

◮ Prediction problem. Input: F, z, x, q, t, question: F t(x)z = q?. (Banks 1971) ◮ Given a FCA and a n × n conf. x, F O(n2)(x) is a fixed point. (Goles, Ollinger, Theyssier 2015) ◮ Unstability problem. Input: F, z, x, question: F O(n2)(x)z = xz?. (Goles, M, Montealegre, Ollinger 2017)

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SLIDE 54

History

◮ Prediction problem. Input: F, z, x, q, t, question: F t(x)z = q?. (Banks 1971) ◮ Given a FCA and a n × n conf. x, F O(n2)(x) is a fixed point. (Goles, Ollinger, Theyssier 2015) ◮ Unstability problem. Input: F, z, x, question: F O(n2)(x)z = xz?. (Goles, M, Montealegre, Ollinger 2017) ◮ We study boolean FCA with von Neumann neighborhood and we found one “complex”.

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SLIDE 55

History

◮ Prediction problem. Input: F, z, x, q, t, question: F t(x)z = q?. (Banks 1971) ◮ Given a FCA and a n × n conf. x, F O(n2)(x) is a fixed point. (Goles, Ollinger, Theyssier 2015) ◮ Unstability problem. Input: F, z, x, question: F O(n2)(x)z = xz?. (Goles, M, Montealegre, Ollinger 2017) ◮ We study boolean FCA with von Neumann neighborhood and we found one “complex”. f2: Or any permutation ◮ Is f2 complex in others context too? (today, Asynchronous)

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SLIDE 56

History

◮ Prediction problem. Input: F, z, x, q, t, question: F t(x)z = q?. (Banks 1971) ◮ Given a FCA and a n × n conf. x, F O(n2)(x) is a fixed point. (Goles, Ollinger, Theyssier 2015) ◮ Unstability problem. Input: F, z, x, question: F O(n2)(x)z = xz?. (Goles, M, Montealegre, Ollinger 2017) ◮ We study boolean FCA with von Neumann neighborhood and we found one “complex”. f2: Or any permutation ◮ Is f2 complex in others context too? (today, Asynchronous)

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SLIDE 57

Computational Complexity

Definition

A decision problem A is NP-Complete if A is in NP and for each problem B in P B can by reduced to A, i.e. there is a φ polynomial s.t. ∀x, B(x) = true ⇔ A(φ(x)) = true

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SLIDE 58

Computational Complexity

Definition

A decision problem A is NP-Complete if A is in NP and for each problem B in P B can by reduced to A, i.e. there is a φ polynomial s.t. ∀x, B(x) = true ⇔ A(φ(x)) = true ◮ If A is NP-complete and P NP, then A ∈ P ◮ Boolean satisfiability problem (SAT) and Circuit SAT are NP-complete (x1 ∨ x2) ∧ (x1 ∨ ¬x2) x1 x2 ∨ ∨ ¬ ∧

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SLIDE 59

Computational Complexity

Definition

A decision problem A is NP-Complete if A is in NP and for each problem B in P B can by reduced to A, i.e. there is a φ polynomial s.t. ∀x, B(x) = true ⇔ A(φ(x)) = true ◮ If A is NP-complete and P NP, then A ∈ P ◮ Boolean satisfiability problem (SAT) and Circuit SAT are NP-complete (x1 ∨ x2) ∧ (x1 ∨ ¬x2) x1 x2 ∨ ∨ ¬ ∧ ∨ ¬ ∨

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SLIDE 60

The Problem

We will study the following FACA with von Neumann neighborhood: f2: Or any permutation von Neumann neighborhood

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SLIDE 61

The Problem

We will study the following FACA with von Neumann neighborhood: f2: Or any permutation von Neumann neighborhood Unstability (synchronous version of AsyncUnstability) is P-complete for f2, then the question is :

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SLIDE 62

The Problem

We will study the following FACA with von Neumann neighborhood: f2: Or any permutation von Neumann neighborhood Unstability (synchronous version of AsyncUnstability) is P-complete for f2, then the question is :

Problem

Is AsyncUnstability NP-complete for f2?

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SLIDE 63

Previous Works

Theorem (Goldschlager 1977)

Planar circuit value problem is P-complete.

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SLIDE 64

Previous Works

Theorem (Goldschlager 1977)

Planar circuit value problem is P-complete. 1 2 3 4 5

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SLIDE 65

Previous Works

Theorem (Goldschlager 1977)

Planar circuit value problem is P-complete. 1 2 3 4 5 1 2 3 4 5

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SLIDE 66

Previous Works

Theorem (Goldschlager 1977)

Planar circuit value problem is P-complete. 1 2 3 4 5 1 2 3 4 5 a b ⊕ ⊕ ⊕ b a

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SLIDE 67

Previous Works

Theorem (Goldschlager 1977)

Planar circuit value problem is P-complete. 1 2 3 4 5 1 2 3 4 5 a b ⊕ ⊕ ⊕ b a

{

C

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SLIDE 68

Lemma I

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0, C and Si, where Si sends a value for the South output and the opposite value for the East output and C is a crossing gate.

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SLIDE 69

Lemma I

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0, C and Si, where Si sends a value for the South output and the opposite value for the East output and C is a crossing gate.

Si xi ¬xi

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SLIDE 70

Lemma I

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0, C and Si, where Si sends a value for the South output and the opposite value for the East output and C is a crossing gate. x1 x2 ∨ ¬ ∨ ¬ ∨ φ(x1, x2) = ¬x1 ∨ x2

1 2 3 4 5 6 7

S1 C ⊤ C C C C C S2 C C ⊤ C C C C ∨ C C C C ⊢ C C ∨ C C C C C C C ∨ C ⊤ C ⊢ C C C ∨ C C C C ⊢ C C ∨

1 2 3 4 5 6 7 1 2 3 4 5 6 7

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SLIDE 71

Lemma I

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0, C and Si, where Si sends a value for the South output and the opposite value for the East output and C is a crossing gate. x1 x2 ∨ ¬ ∨ ¬ ∨ φ(x1, x2) = ¬x1 ∨ x2

1 2 3 4 5 6 7

S1 C ⊤ C C C C C S2 C C ⊤ C C C C ∨ C C C C ⊢ C C ∨ C C C C C C C ∨ C ⊤ C ⊢ C C C ∨ C C C C ⊢ C C ∨

1 2 3 4 5 6 7 1 2 3 4 5 6 7

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SLIDE 72

Lemma I

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0, C and Si, where Si sends a value for the South output and the opposite value for the East output and C is a crossing gate. x1 x2 ∨ ¬ ∨ ¬ ∨ φ(x1, x2) = ¬x1 ∨ x2

1 2 3 4 5 6 7

S1 C ⊤ C C C C C S2 C C ⊤ C C C C ∨ C C C C ⊢ C C ∨ C C C C C C C ∨ C ⊤ C ⊢ C C C ∨ C C C C ⊢ C C ∨

1 2 3 4 5 6 7 1 2 3 4 5 6 7

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SLIDE 73

Lemma I

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0, C and Si, where Si sends a value for the South output and the opposite value for the East output and C is a crossing gate. x1 x2 ∨ ¬ ∨ ¬ ∨ φ(x1, x2) = ¬x1 ∨ x2

1 2 3 4 5 6 7

S1 C ⊤ C C C C C S2 C C ⊤ C C C C ∨ C C C C ⊢ C C ∨ C C C C C C C ∨ C ⊤ C ⊢ C C C ∨ C C C C ⊢ C C ∨

1 2 3 4 5 6 7 1 2 3 4 5 6 7

slide-74

SLIDE 74

Lemma I

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0, C and Si, where Si sends a value for the South output and the opposite value for the East output and C is a crossing gate. x1 x2 ∨ ¬ ∨ ¬ ∨ φ(x1, x2) = ¬x1 ∨ x2

1 2 3 4 5 6 7

S1 C ⊤ C C C C C S2 C C ⊤ C C C C ∨ C C C C ⊢ C C ∨ C C C C C C C ∨ C ⊤ C ⊢ C C C ∨ C C C C ⊢ C C ∨

1 2 3 4 5 6 7 1 2 3 4 5 6 7

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SLIDE 75

Lemma I

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0, C and Si, where Si sends a value for the South output and the opposite value for the East output and C is a crossing gate. x1 x2 ∨ ¬ ∨ ¬ ∨ φ(x1, x2) = ¬x1 ∨ x2

1 2 3 4 5 6 7

S1 C ⊤ C C C C C S2 C C ⊤ C C C C ∨ C C C C ⊢ C C ∨ C C C C C C C ∨ C ⊤ C ⊢ C C C ∨ C C C C ⊢ C C ∨

1 2 3 4 5 6 7 1 2 3 4 5 6 7

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SLIDE 76

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

slide-77

SLIDE 77

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

slide-78

SLIDE 78

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 79

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 80

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

slide-81

SLIDE 81

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 82

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 83

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 84

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 85

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 86

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 87

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 88

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 89

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 90

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

X X a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

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SLIDE 91

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

a b s1 s2 a′ b′ ↓ ↓ ↓ → → ↓ → → 1 ↓ ↓ 1 1 ↓ → 1 → ↓ 1 → → 1 ↓ ↓ 1 ↓ → 1 → ↓ 1 → → 1 1 1 ↓ ↓ 1 1 1 ↓ → 1 1 1 1 → ↓ 1 1 → → 1

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SLIDE 92

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

a b ∧ ∧ ∧ s ∨ s ∧ ∧ b′ a′ ∨ ∨

(3, 0) (0, 3) (2, 4) (4, 2) (3, 3) (4, 1) (4, 3) (1, 4) (6, 4) (4, 6) (7, 4) (4, 7) (3, 4) (4, 4)

1 2 3 4 5 6 7 ∨ 1 ∨ S ∨ ∨ 2 ∨ ∧ ∨ 3 ∨ ∨ ∨ ∧ ∨ ∨ 4 S ∧ ∨ ∨ ∨ ∧ ∨ 5 ∨ ∨ 6 ∨ ∨ ∨ ∧ 7 ∨

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SLIDE 93

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

1 2 3 4 5 6 7 ∨ 1 ∨ S ∨ ∨ 2 ∨ ∧ ∨ 3 ∨ ∨ ∨ ∧ ∨ ∨ 4 S ∧ ∨ ∨ ∨ ∧ ∨ 5 ∨ ∨ 6 ∨ ∨ ∨ ∧ 7 ∨

Problem!!!

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SLIDE 94

Lemma II

South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

1 2 3 4 5 6 7 ∨ 1 ∨ S ∨ ∨ 2 ∨ ∧ ∨ 3 ∨ ∨ ∨ ∧ ∨ ∨ 4 S ∧ ∨ ∨ ∨ ∧ ∨ 5 ∨ ∨ 6 ∨ ∨ ∨ ∧ 7 ∨ 1 2 3 4 5 6 7 ∨ 1 ∨ S ∨ ∨ 2 ∨ ∧ ∨ 3 ∨ ∨ ∨ ∧ ∨ ∨ 4 S ∧ ∨ ∨ ∨ ∧ ∨ 5 ∨ ∨ 6 ∨ ∨ ∨ ∧ 7 ∨

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SLIDE 95

Gates ∨, ∧, 0 and S

∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∧ ∨ ∨ ∨ ∨ ∨ ∨ S ∨ ∨ ∨ ∨ ∨ ∨ ∨

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SLIDE 96

G

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SLIDE 97

G G

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SLIDE 98

S ∧ ∨ C

S ∧ ∨ C

Therefore South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

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SLIDE 99

Finally...

Theorem

AsyncUnstability is NP-complete. Remember: South-East grid-embedded circuit SAT is NP-complete, with gates ∧, ∨, 0 and Si.

∨ ∧ S

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SLIDE 100

OR gate in f2

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SLIDE 101

OR gate in f2

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SLIDE 102

OR gate in f2

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SLIDE 103

OR gate in f2

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SLIDE 104

OR gate in f2

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SLIDE 105

OR gate in f2

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SLIDE 106

OR gate in f2

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SLIDE 107

OR gate in f2

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SLIDE 108

OR gate in f2

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SLIDE 109

OR gate in f2

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SLIDE 110

OR gate in f2

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SLIDE 111

OR gate in f2

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SLIDE 112

OR gate in f2

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SLIDE 113

OR gate in f2

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SLIDE 114

OR gate in f2

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SLIDE 115

OR gate in f2

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SLIDE 116

OR gate in f2

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SLIDE 117

OR gate in f2

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SLIDE 118

OR gate in f2

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SLIDE 119

OR gate in f2

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SLIDE 120

OR gate in f2

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SLIDE 121

OR gate in f2

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SLIDE 122

OR gate in f2

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SLIDE 123

OR gate in f2

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SLIDE 124

OR gate in f2

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SLIDE 125

OR gate in f2

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SLIDE 126

AND gate in f2

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SLIDE 127

AND gate in f2

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SLIDE 128

AND gate in f2

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SLIDE 129

AND gate in f2

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SLIDE 130

AND gate in f2

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SLIDE 131

AND gate in f2

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SLIDE 132

AND gate in f2

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SLIDE 133

AND gate in f2

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SLIDE 134

AND gate in f2

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SLIDE 135

AND gate in f2

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SLIDE 136

AND gate in f2

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SLIDE 137

AND gate in f2

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SLIDE 138

AND gate in f2

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SLIDE 139

AND gate in f2

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SLIDE 140

AND gate in f2

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SLIDE 141

AND gate in f2

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SLIDE 142

AND gate in f2

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SLIDE 143

AND gate in f2

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SLIDE 144

AND gate in f2

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SLIDE 145

AND gate in f2

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SLIDE 146

AND gate in f2

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SLIDE 147

AND gate in f2

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SLIDE 148

AND gate in f2

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SLIDE 149

AND gate in f2

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SLIDE 150

AND gate in f2

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SLIDE 151

AND gate in f2

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SLIDE 152

AND gate in f2

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SLIDE 153

AND gate in f2

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SLIDE 154

AND gate in f2

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SLIDE 155

AND gate in f2

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SLIDE 156

AND gate in f2

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SLIDE 157

Si gate in f2 I

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SLIDE 158

Si gate in f2 I

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SLIDE 159

Si gate in f2 I

X

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SLIDE 160

Si gate in f2 I

X

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SLIDE 161

Si gate in f2 I

X

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SLIDE 162

Si gate in f2 I

X

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SLIDE 163

Si gate in f2 I

X

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SLIDE 164

Si gate in f2 I

X

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SLIDE 165

Si gate in f2 I

X

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SLIDE 166

Si gate in f2 I

X

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SLIDE 167

Si gate in f2 II

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SLIDE 168

Si gate in f2 II

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SLIDE 169

Si gate in f2 II

X

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SLIDE 170

Si gate in f2 II

X

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SLIDE 171

Si gate in f2 II

X

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SLIDE 172

Si gate in f2 II

X

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SLIDE 173

Si gate in f2 II

X

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SLIDE 174

0 gate in f2 II

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SLIDE 175

0 gate in f2 II

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SLIDE 176

0 gate in f2 II

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SLIDE 177

0 gate in f2 II

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SLIDE 178

Concluding remarks

◮ We have a very restrictive NP-complete problem for (A)CA ◮ We simple local rule with a complex behavior in synchronous and asynchronous update ◮ To explore the “one way ” (A)CA

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SLIDE 179

Gracias!