L ECTURE 11: C ELLULAR A UTOMATA 1 / D ISCRETE -T IME D YNAMICAL S - - PowerPoint PPT Presentation

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LECTURE 11: CELLULAR AUTOMATA 1 / DISCRETE-TIME DYNAMICAL SYSTEMS 3

INSTRUCTOR: GIANNI A. DI CARO

15-382 COLLECTIVE INTELLIGENCE – S18

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STATE VARIABLES SPATIALLY BOUNDED ON LATTICES: CELLULAR AUTOMATA

• State variable ↔ State of a Spatial location / Cell
• Cell in a 1D, or 2D, or 3D Lattice

1D 2D

𝑦1 𝑦2 𝑦3 𝑦𝑗 𝑦𝑗−1 𝑦𝑗+1 𝑦𝑙 Example of selected neighborhood of 𝑦𝑗, represented by the set {𝑦𝑗−1, 𝑦𝑗+1} 𝑦1 𝑦2 𝑦3 𝑦𝑙 𝑦𝑙+1 𝑦𝑙+2 𝑦𝑙+3 𝑦𝑙+𝑛 Example of selected neighborhood of 𝑦𝑙+𝑗, represented by the set {𝑦𝑙+𝑗−1, 𝑦𝑙+𝑗+1, 𝑦𝑗, 𝑦𝑗+1, 𝑦𝑗−1, 𝑦2𝑙+𝑗, 𝑦2𝑙+𝑗+1, 𝑦2𝑙+𝑗−1}

𝑦𝑙+𝑗

𝑦2𝑙

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CAS ARE LATTICE MODELS

• Regular 𝒐-dimensional discretization of a continuum
• E.g., an 𝑜-dimensional grid
• Periodic (toroidal) or non periodic structure
• More abstract definition: Regular tiling of a space by a primitive cell

Bethe lattice, ∞-connected cycle-free graph where each node is connected to 𝑨 neighbors, where 𝑨 is called the coordination number

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CELLULAR AUTOMATA 1D

1D Lattice

𝑦1 𝑦2 𝑦3 𝑦𝑗 𝑦𝑗−1 𝑦𝑗+1 𝑦𝑙

𝑦𝑜+1

1

= 𝑔

1(𝑦𝑜 1, 𝑦𝑜 2, … . 𝑦𝑜 𝑙)

𝑦𝑜+1

2

= 𝑔

2(𝑦𝑜 1, 𝑦𝑜 2, … . 𝑦𝑜 𝑙)

𝑦𝑜+1

𝑙

= 𝑔

𝑙(𝑦𝑜 1, 𝑦𝑜 2, … . 𝑦𝑜 𝑙)

….. 𝑦𝑜+1

1

= 𝑔

1(𝑦𝑜 1, 𝑦𝑜 2)

𝑦𝑜+1

2

= 𝑔

2(𝑦𝑜 1, 𝑦𝑜 2, 𝑦𝑜 3)

𝑦𝑜+1

𝑗

= 𝑔

𝑗(𝑦𝑜 𝑗−1, 𝑦𝑜 𝑗 , 𝑦𝑜 𝑗+1)

….. 𝑦𝑜+1

𝑙

= 𝑔

𝑙(𝑦𝑜 𝑙−1, 𝑦𝑜 𝑙)

….. 𝑦𝑜+1

1

= 𝑔

1(𝑦𝑜 𝑙, 𝑦𝑜 1, 𝑦𝑜 2)

𝑦𝑜+1

𝑗

= 𝑔

𝑗(𝑦𝑜 𝑗−1, 𝑦𝑜 𝑗 , 𝑦𝑜 𝑗+1)

….. 𝑦𝑜+1

𝑙

= 𝑔

𝑙(𝑦𝑜 𝑙−1, 𝑦𝑜 𝑙, 𝑦𝑜 1)

….. Toroidal boundary conditions 𝑦𝑜+1

1

= 𝑔(𝑦𝑜

𝑙, 𝑦𝑜 1, 𝑦𝑜 2)

𝑦𝑜+1

𝑗

= 𝑔(𝑦𝑜

𝑗−1, 𝑦𝑜 𝑗 , 𝑦𝑜 𝑗+1)

….. 𝑦𝑜+1

𝑙

= 𝑔(𝑦𝑜

𝑙−1, 𝑦𝑜 𝑙, 𝑦𝑜 1)

Single map ….. …..

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CA: A FORMAL DEFINITION

• We can give a definition of CAs aside the general framework of DTDS
• CAs are defined by:
• Components/Cells (Connected FSMs)
• Lattice (Geometry + Topology)
• Schedule (Time + Synchronization)

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CA: COMPONENTS

• A set of 𝑁 automata (cells) 𝑏𝑗, 𝑗 = 1, … 𝑁: finite-state machines

(in a more general sense, each cell could a function)

• Each machine has a specified set of possible states, 𝑇𝑗 = {𝑡1, 𝑡2, … , 𝑡𝑤}
• For each machine 𝑏𝑗, state transitions are defined by a local state

transition function, that depends on the current state of 𝑏𝑗 and the state of the 𝑜𝑗 cells that are in 𝑏𝑗’s neighborhood, 𝒪(𝑏𝑗), 𝐺𝑗: 𝑇𝑗 ∪ 𝒪(𝑏𝑗) → 𝑇𝑗

• At discrete time 𝑢 = 0, each cell has an initial state, where the vector
• f all initial states define the initial condition of the CA

𝐺𝑗 𝑡𝑗 = 3 𝑏𝑗

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CA: LATTICE

• Cells are defined on a lattice, that induces a topology structure
• Associated to the topology, is the neighborhood map, 𝒪, of a cell 𝑏𝑗,

that associates to 𝑏𝑗 a set of neighbors, 𝒪 𝑏𝑗 = {𝑏𝑘 ∶ 𝑏𝑘 𝑗𝑡 𝑜𝑓𝑗𝑕ℎ𝑐𝑝𝑠 𝑝𝑔𝑏𝑗}

• Neighborhood  Range for a cell to be influenced by other cells,

range of influence of a cell

• Boundary conditions define how the notion of topological neighborhood

includes the boundaries, if any, of the lattice

• Infinite vs. Finite lattices (Hard boundaries vs. soft boundaries)

1D

𝑏1 𝑏2 𝑏3 𝑏𝑗−1 𝑏𝑗 𝑏𝑗+1 𝒪 𝑏𝑗 = {𝑏𝑗−1, 𝑏𝑗+1}

2D Regular grid

Von Neumann Moore Extended Moore

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CA: LATTICE, BOUNDARIES

• Infinite/adaptive lattice
• The grid grows as the pattern propagates
• Finite lattice
• Hard boundary: reflective, leftmost (rightmost) cell only diffuse right (left)
• Soft boundary: periodic boundary conditions, edges wrap around
• Hard boundary: fixed, edge cells have a fixed state

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CA: LATTICE, BOUNDARIES

• Edge wraps around
• 1D is a ring
• 2D is torus
• Weird(er) topologies with a twist: Moebius bands, Klein bottles

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CA: SCHEDULES

• Synchronized updating: at time 𝑢 the state value of the cells is frozen,

and all cells update their state based on their own state and that of their neighbors, then time steps up to 𝑢 + 1 and process is repeated

• States are updated in sequence or in parallel, depending on the

available hardware, but it doesn’t matter for the final result

• Asynchronous updating: at time 𝑢 the state of one of more cells is

updated based on their own state and that of their neighbors at 𝑢, at 𝑢 + 1 the state of possibly different cells is updated and process is repeated

• States are selected according to some criterion, or self-trigger the

update, the updating sequence matters for the final result

𝑏𝑗 𝑏𝑗−1 𝑏𝑗+1 𝑏𝑗+2 𝑏𝑗−2 𝑏𝑗 𝑏𝑗−1 𝑏𝑗+1 𝑏𝑗+2 𝑏𝑗−2

𝑢 𝑢 + 1

𝐺(𝑏𝑗, 𝑏𝑗−1, 𝑏𝑗+1)

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DESIGN CHOICES

• In principle a great freedom choosing:
• number and type of states,
• state transition functions (for each cell),
• topology and neighborhood mapping (for each cell),
• cells updating scheme,
• number of cells,
• boundary conditions
• …
• In an homogeneous CA, neighborhoods, state transition functions, topology, are

the same for all cells, in a non homogenous CA there’s some heterogeneity, in space and/or time, in terms of transitions, topology / neighborhood

• Freedom in the design space has been exploited in a number of interesting

applications, that precisely might require a diversity of local behaviors, problem- specific interconnection topologies that reflect complex realities such as ecosystems, immune systems, car traffic flows, bio-chemical reactions,…

• CAs are discrete time and space models of partial differential equations

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DESIGN CHOICES TO STUDY CAS

• Being multidimensional iterated maps, CAs are very complex entities, therefore, to

study them, let’s make a few reasonably simplifying assumptions:

• Homogeneous CAs:
• Lattice is a regular grid, in 1D or 2D
• All cell functions 𝑏 have the same (relatively simple) neighborhood

mapping 𝒪(𝑏)  they all have the same number of neighbors defined according to the lattice

• All cell functions have the same state transition function, 𝐺(𝑏, 𝒪 𝑏 )
• States are encoded in a few bits, typically, 2 or 3
• Synchronous updating

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1D CA

• Simplest case: State variables / Cells are Boolean units, 𝑇 = {0,1}
• The neighborhood of a cell 𝑏𝑗, 𝒪 𝑏𝑗 corresponds to the one or two closest

neighbors in both left and right directions

•  Transition function 𝐺 is a Boolean function of 𝑜 = 3 or 𝑜 = 5 arguments

𝐺(𝑏𝑗, 𝑏𝑗−1, 𝑏𝑗+1, 𝑏𝑗−2, 𝑏𝑗+2) = ቊ1 if 𝑏𝑗 + 𝑏𝑗−1 + 𝑏𝑗+1 + 𝑏𝑗−2 + 𝑏𝑗+2 > 2

• therwise
•  A 1D Boolean CA with 𝑜 cells is an 𝑜-dimensional binary vector

𝒃(t), the state vector of the CA, that evolves over time by the iterated application of the map 𝐺 : 𝒃 t + 1 = 𝐺(𝒃 t )

• State space of the CA: All possible configurations of the vector 𝒃

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1D BOOLEAN CA, SOME NUMBERS

• 𝑙 = 𝑇 = number of (cell) states
• 𝑇 = {0,1}  𝑙 = 2
• 𝑁 = number of cells  2𝑁 possible configurations of CA’s state vector,
• 𝑁 = 100, 𝑙 = 2  2100 ≈ 1030 !!!!

One specific function 𝐺

𝒪 = 8

• 𝑠 = range = 𝒪 𝑏 /2 (assuming a symmetric neighborhood)
• 𝑙2𝑠+1 = 𝑙|𝒪|+1 possible configurations of neighbor set
• If 𝑠 = 1, 𝑙 = 2  8 possible neighbor configurations
• If 𝑠 = 2, 𝑙 = 2  32 possible neighbor configurations
• 𝑙𝑙2𝑠+1 = 𝑙𝑙|𝒪|+1 = possible evolution functions for the CA
• If 𝑠 = 1, 𝑙 = 2  256 possible Boolean evolution functions
• If 𝑠 = 2, 𝑙 = 2  4 ∙ 109 possible Boolean functions!

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ELEMENTARY CA: WOLFRAM CODE

• 𝑇 = 0,1 , 𝑠 = 1  𝑙 = 2, 𝒪 + 1 = 8, 256 possible Boolean functions

Transition function 𝑮 (rule of the CA) Example: Rule 30 This is a bit string  Decimal number Rule 30: (00011110)  30 Wolfram code

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SOME RULES …

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STUDYING CAS: NON-LINEAR BUSINESS AS USUAL

Direct problem (Prediction): Given the function, what’s the behavior?

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RULE 30

Class 3 cellular automata: overall the evolution presents regularities, however, the state sequence generated by the central cell is used as random generator in Mathematica! (randomness deriving from a purely deterministic process with no external ’noisy’ inputs)

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A ZOO OF BEHAVIORS