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Stefano RedaelliLIntAr - Department of Computer Science - Unversity of Milano-Bicocca

redaelli@disco.unimib.it

Space and Cellular Automata

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Continuous or discrete?

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Space and discrete representationsEuclid (x + y)2 = x2 + 2xy + y2 Kepler

Newton

Einstein

Onda luminosa

Fotoni o quanti di luce

0 1

0 1

numbers

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Continuous vs Discrete

• Continuous:– More accurate– Computationally heavy– Space not explicitly

represented– Spatial equations– Suitable for analytical

approaches• Global dynamic• Top-down approach

• Discrete:– Less accurate– More simple– Structure represents

the space– Discrete systems– Suitable for Individual-

Oriented approaches• Local dynamic• Bottom-up approach

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Cellular Automata (CA):informal definition

• Cellular Automata are discrete dynamical systems– System: a set of interacting entities– Dynamic: temporal evolution on a set of steps– Discrete: space, time and properties of the

automaton can have only a finite, countable number of states

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Formal definition

• A Cellular Automata is a tuple < L,Q,q0,u,f >– L: a uniform lattice– Q: finite state set– q0: initial state– u: the local connection template, or automaton’s

neighborhood u : L Lk

• k is a positive integer – f: the automaton transition rule

f : Qk Q

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Space• A grid n×n

- Square lattice

• Each cell has different states

• The world is represented through space

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Just an idea

The model of a

classroom

Free place

Occupied place

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Interactions• Distance

• Adjacency– Only two near cells can interact each other– When two cells are near?

221 ),( Zxxx 2

21 ),( Zyyy 2

222

11 )()(),( yxyxyxd

d = 1d = 2

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Interactions• The concept of neighborhood

– Each cell has the set of cells adjacent to it in its neighborhood

• Local

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Neighborhood• A grid n×n

• Neighborhood:

- Moore

- Von Neumann

- Square lattice

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Neigborhood radiusr = 1 r = 2 r = 3

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Not-square lattice• A grid n×n

- Square lattice

- Triangular

- Hexagonal

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Not-square lattice• A grid n×n

- Square lattice

- Triangular

- Hexagonal

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Just an idea

Application of the rule: “to have a

lot of space it is more

confortable”

The model of a

classroom

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Border condition

?Time: step 12

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Border conditions: solutions

1. Opposite borders of the lattice are "sticked together". A one dimensional "line" becomes following that way a circle (a two dimensional lattice becomes a torus).

2. The border cells are mirrored: the consequence are symmetric border properties.

The more usual method is the possibility 1

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Border condition

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Example: the study of Pedestrian and Crowd Dynamics

• describing the behavior of crowd– Crowd (or group) formation– Crowd (or group) dispersion– Crowd (or group) movement– Crowd behavior in given spatial structures– Other…

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Why to use a CA approach

• Local perception and partial knowledge of the environment

• Complexity of global dynamic– a bottom-up approach is easier

?

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The strength of CA

• “CAs contain enough complexity to simulate surprising and novel change as reflected in emergent phenomena” (Mike Batty)

• Complex group behaviors can emerge from these simple individual behaviors

• Complexity emerges through spatial patterns

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Patterns

• A pattern is a form, template, or model • Patterns can be used to make or to generate things

or parts of a thing • The simplest patterns are based on

repetition/periodicity: several copies of a single template are combined without modification.

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Life: example1. Any live cell with

fewer than two neighbours dies of loneliness.

2. Any live cell with more than three neighbours dies of crowding.

3. Any dead cell with exactly three neighbours comes to life.

4. Any live cell with two or three neighbours lives, unchanged, to the next generation.

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Emergent patterns in Life

• Static patterns (the most famous)

– Still life object:• Block• Beehive

• Boat

• Ship

• Loaf

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Emergent patterns in Life

• Dynamic patterns (the most famous)

– Oscillators:• Blinker

• Toad

• Gliders

– Moving patterns:

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The problem of CA approach

• The problem of Action at a distance:– How to make local a long-ranged interaction

Long-ranged interaction

Local interactionA trace in the space

Local interaction!!!

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• A container? • A collection of

objects? • …or something more

Which cities are NEAR each other?

The space morphology influence the possibility of interaction between the objects!

Space is only a container?

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Example: shadowing

must follow

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Example: shadowing

must follow

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Example: shadowing

must follow

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Example: shadowing

must follow

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Example: shadowing

must follow

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Example: shadowing

must follow

?

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Example: shadowing

must follow

if in N(s)

if

if

if

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Example: shadowing

must follow

if in N(s)

if

if

if

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Example: shadowing

must follow

if in N(s)

if

if

if

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Example: shadowing

must follow

if in N(s)

if

if

if

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Example: shadowing

if

and in N(*) or

if

and in N(*)

if

and in N(*)

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Action at a distance problem

Application of the rule: “to have a

lot of space it is more

confortable”

The model of a

classroom

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Action At-a-Distance in CA

• Traditional CA– Local neighborhood

definition (e.g. Moore)

– Isotropic space• But... in real world In order to have interaction between

two cells far in space I have to extend the neighborhood

Space is anisotropic!

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Neighborhood and proximity matrices

• For example: in modeling geographical space, roads establish preferential directions.

• The neighborhood should consider this preferences

• but it should be different for each roads

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From Cells to Agents• Hybrid Automata

– The example of TerraML– TerraLib Modeling Language (TerraML) is a spatial dynamic

modeling language to simulate dynamic processes in environmental applications.

• Situated Cellular Agents (SCA)– The example of MMASS

– A model defining MAS whose entities are situated in an environment whose structure (i.e. space) is defined as an undirected graph of sites

– Agents in MMASS can emits fields that propagate signals through the space

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SCA (Situated Cellular Agent)• < Space, F,A >

– Space: models the spatial structure of the environment– A: set of situated agents– F: set of fields propagating throughout the Space

• Agent interaction– Asynchronous AAAD: field emission–propagation–perception

mechanism– Synchronous interaction: reaction among a set of agents of

given types and states and situated in adjacent sites

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Agent environment• Space: set P of sites arranged in a network • Each site p є P (containing at most one agent) is defined by the

3–tuple

where: agent situated in p: set of fields active in p: set of sites adjacent to p

• Then the Space is a not oriented graph of sites

ppp PFa ,,

Aap

FFp PPp

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Fields

• Mean for agent asynchronous communication• Fields are generated by agents

<Wf ,Diffusionf ,Comparef ,Composef >– Wf : set of field values

– Diffusionf : P ×Wf × P → (Wf )+: field distribution function

– Composef : (Wf )+ → Wf : field composition function

– Comparef : Wf ×Wf → {True, False} field comparison function

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The example of Crowd Dynamics

• describing the behavior of crowd– Crowd (or group) formation– Crowd (or group) dispersion– Crowd (or group) movement– Crowd behavior in given spatial structures– Other…

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Importance of spatial interactions in crowd context

• Example: a group getting through a crowded area– Weak bonds:

keeping sight– Strong bonds:

keeping by hand

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Importance of spatial interactions in crowd context• The force of

relationships influence the behavior:– Weak bonds: more

possibility to get through in few time but more possibility of members getting lost

– Strong bonds: few possibility to loose members but more difficulty to get through

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Example: cohesion and movement

Crowd phenomenon

Physical interpretation

Computational SCA-model

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Example: cohesion and movement

Crowd phenomenon

Physical interpretation

Computational SCA-model

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Example: cohesion and movement

Crowd phenomenon

Physical interpretation

Computational SCA-model

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Example: cohesion and movement

Crowd phenomenon

Physical interpretation

Computational SCA-model

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Example: cohesion and movement

Crowd phenomenon

Physical interpretation

Computational SCA-model