1. 1 Stefano Redaelli LIntAr - Department of Computer Science - Unversity of Milano-Bicocca redaelli@disco.unimib.it Space and Cellular Automata

2. 2 Continuous or discrete?

3. 3 Space and discrete representations Euclid(x + y) 2 = x 2 + 2xy + y 2 Kepler Newton Einstein Onda luminosa Fotoni o quanti di luce 0 1 0 1 numbers

4. 4 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

5. 5 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

6. 6 Formal definition A Cellular Automata is a tuple –L: a uniform lattice –Q: finite state set –q 0 : initial state –u: the local connection template, or automaton’s neighborhood u : L  L k k is a positive integer –f: the automaton transition rule f : Q k  Q

7. 7 Space A grid n×n - Square lattice Each cell has different states The world is represented through space

8. 8 Just an idea The model of a classroom Free place Occupied place

9. 9 Interactions Distance Adjacency –Only two near cells can interact each other –When two cells are near? d = 1d = 2

10. 10 Interactions The concept of neighborhood –Each cell has the set of cells adjacent to it in its neighborhood Local

11. 11 Neighborhood A grid n×n Neighborhood: - Moore - Von Neumann - Square lattice

12. 12 Neigborhood radius r = 1r = 2r = 3

13. 13 Not-square lattice A grid n×n - Square lattice - Triangular - Hexagonal

14. 14 Not-square lattice A grid n×n - Square lattice - Triangular - Hexagonal

15. 15 Just an idea Application of the rule: “to have a lot of space it is more confortable” The model of a classroom

16. 16 Border condition ? Time: step12

17. 17 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

18. 18 Border condition

19. 19 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…

20. 20 Why to use a CA approach Local perception and partial knowledge of the environment Complexity of global dynamic –a bottom-up approach is easier ?

21. 21 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

22. 22 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.

23. 23 Life: example 1.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.

24. 24 Emergent patterns in Life Static patterns (the most famous) –Still life object: Block Beehive Boat Ship Loaf

25. 25 Emergent patterns in Life Dynamic patterns (the most famous) –Oscillators: Blinker Toad Gliders –Moving patterns:

26. 26 The problem of CA approach The problem of Action at a distance: –How to make local a long-ranged interaction Long-ranged interaction Local interaction A trace in the space Local interaction!!!

27. 27 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?

28. 28 Example: shadowing must follow

29. 29 Example: shadowing must follow

30. 30 Example: shadowing must follow

31. 31 Example: shadowing must follow

32. 32 Example: shadowing must follow

33. 33 Example: shadowing must follow ?

34. 34 Example: shadowing must follow if in N(s) if

35. 35 Example: shadowing must follow if in N(s) if

36. 36 Example: shadowing must follow if in N(s) if

37. 37 Example: shadowing must follow if in N(s) if

38. 38 Example: shadowing if and in N(*)or if and in N(*) if and in N(*)

39. 39 Action at a distance problem Application of the rule: “to have a lot of space it is more confortable” The model of a classroom

40. 40 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!

41. 41 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

42. 42 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

43. 43 SCA (Situated Cellular Agent) –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

44. 44 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

45. 45 Fields Mean for agent asynchronous communication Fields are generated by agents –W f : set of field values –Diffusion f : P ×W f × P → (W f )+: field distribution function –Compose f : (W f )+ → W f : field composition function –Compare f : W f ×W f → {True, False} field comparison function

46. 46 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…

47. 47 Importance of spatial interactions in crowd context Example: a group getting through a crowded area –Weak bonds: keeping sight –Strong bonds: keeping by hand

48. 48 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

49. 49 Example: cohesion and movement Crowd phenomenon Physical interpretation Computational SCA-model

50. 50 Example: cohesion and movement Crowd phenomenon Physical interpretation Computational SCA-model

51. 51 Example: cohesion and movement Crowd phenomenon Physical interpretation Computational SCA-model

52. 52 Example: cohesion and movement Crowd phenomenon Physical interpretation Computational SCA-model

53. 53 Example: cohesion and movement Crowd phenomenon Physical interpretation Computational SCA-model