The good side of running away

Transcription

1 The good side of running away Introducing signalling into Conways Game of Life Simon Schulz 20. Januar 2013

2 Overview Introduction How to improve the game The GOLS Game Implementation Results

3 Introduction

4 Conways Game of Life Is a cellular automaton, which follows four rules:

5 Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population).

6 Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population). 2 Any live cell with two or three live neighbours lives on to the next generation.

7 Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population). 2 Any live cell with two or three live neighbours lives on to the next generation. 3 Any live cell with more than three live neighbours dies(overcrowding).

8 Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population). 2 Any live cell with two or three live neighbours lives on to the next generation. 3 Any live cell with more than three live neighbours dies(overcrowding). 4 Any dead cell with exactly three live neighbours becomes a live cell (reproduction)

9 Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population). 2 Any live cell with two or three live neighbours lives on to the next generation. 3 Any live cell with more than three live neighbours dies(overcrowding). 4 Any dead cell with exactly three live neighbours becomes a live cell (reproduction)

10 Conways Game of Life Rules applied simultaneously Zero player game, seed at beginning Deterministic

11 Conways Game of Life Rules applied simultaneously Zero player game, seed at beginning Deterministic Gardner, Martin (1970/10) Because of Life s analogies with the rise, fall and alterations of a society of living organisms, it belongs to a growing class of what are called simulation games

12 Point of view Commonly on cellular level

13 Point of view Commonly on cellular level But macro view:

14 Point of view Commonly on cellular level But macro view: Cells as cities in a given area

15 Point of view Commonly on cellular level But macro view: Cells as cities in a given area Relation of available resources...

16 Point of view Commonly on cellular level But macro view: Cells as cities in a given area Relation of available resources and e.g. trading.

17 Point of view Commonly on cellular level But macro view: Cells as cities in a given area Relation of available resources and e.g. trading. Communication

18 Communication Requires some intelligence Intelligence gives ability to make decisions

19 Communication Requires some intelligence Intelligence gives ability to make decisions Is there any decision that increases amount of surviving cities?

20 How to improve the game

21 Case studies Allow action between generations What are reasonable actions?

22 Case studies Allow action between generations What are reasonable actions? Influence on the rules Influence on the global survivability

23 Case studies Allow action between generations What are reasonable actions? Influence on the rules Influence on the global survivability What s about ABANDONMENT?

24 4-die-modification vs. normal game 2 4die/basic game 1.5 ratio 4die/basic runs

25 4-die-modification vs. normal game 1600 Long term behaviour 4die Number living cells Number of Generations

26 Use that knowledge to give the cells the ability to abandon Depending on signalling Preplay communication Step before basic GoL rule appliance

27 Use that knowledge to give the cells the ability to abandon Depending on signalling Preplay communication Step before basic GoL rule appliance Example: citizens can decide to abandon the area e.g. when resources run short

28 The signalling system

29 Definitions Definition: State For a given cell c i the state is t n, where n = 0,..., 8, number of living cells around c i. The set of states is therefore defined as T = {t 0,..., t 8 }.

30 Definitions Definition: State For a given cell c i the state is t n, where n = 0,..., 8, number of living cells around c i. The set of states is therefore defined as T = {t 0,..., t 8 }. Definition: Sender & Receiver The sender is a cell c s. The receiver is a cell c r, c s c r, S, R C = {c i : c i Living cells}

31 Definitions Definition: State For a given cell c i the state is t n, where n = 0,..., 8, number of living cells around c i. The set of states is therefore defined as T = {t 0,..., t 8 }. Definition: Sender & Receiver The sender is a cell c s. The receiver is a cell c r, c s c r, S, R C = {c i : c i Living cells} Definition: Action An action is the decision to die (a 2 ) or not to die (a 1 ), before the next game-based selection process begins, A = {a 1, a 2 }.

32 Definitions Definition: State For a given cell c i the state is t n, where n = 0,..., 8, number of living cells around c i. The set of states is therefore defined as T = {t 0,..., t 8 }. Definition: Sender & Receiver The sender is a cell c s. The receiver is a cell c r, c s c r, S, R C = {c i : c i Living cells} Definition: Action An action is the decision to die (a 2 ) or not to die (a 1 ), before the next game-based selection process begins, A = {a 1, a 2 }. Definition: Messages A message is a signal that can be choosen from := {M 0,..., M 8 }.

33 Utility-function The utility is calculated by u(c r, a i ) = N(c r, t + 1, a i ) N(c r, t) where c r is a receiver cell a i is a specific action t is the time N is the neighbourhood function

34 The signalling game Definition: Signalling game for Conways GoL G = (S, R, T, M, A, u)

35 GOLS Game Definition: GOLS Game SS = (G, ϕ, Pr m, Pr a, µ)

36 GOLS Game Definition: GOLS Game SS = (G, ϕ, Pr m, Pr a, µ) Where G is the game

37 GOLS Game Definition: GOLS Game SS = (G, ϕ, Pr m, Pr a, µ) Where G is the game ϕ selects the sender & receiver

38 GOLS Game Definition: GOLS Game SS = (G, ϕ, Pr m, Pr a, µ) Where G is the game ϕ selects the sender & receiver µ is the update dynamic

39 Selection functions

40 Selection functions Definition of Pr m Pr m : T M

41 Selection functions Definition of Pr m Pr m : T M Definition of Pr a Pr a : T M M

42 Agent selection function Definition: Agent selection function C = (Cell 1,..., Cell n ) ϕ(c) = {(Cell i, Cell j )} ϕ(c \ {Cell i, Cell j }) where i, j N, i, j < n, i random, j such that Cell j random neighbour of Cell i

43 Update dynamics Definition: Update function Bush-Mosteller-Reinforcement µ(x) = pr old (x) + α (1 pr old (x)) α = lp u, where lp is the learning parameter u is the utility gotten

44 Paste it together (C s, C r ) ϕ(c)

45 Paste it together (C s, C r ) ϕ(c) T s R m M Tr R a A

46 Paste it together (C s, C r ) ϕ(c) T s R m M Tr R a A ut = u(c r, A)

47 Paste it together (C s, C r ) ϕ(c) T s R m M Tr R a A ut = u(c r, A) µ(t s ), µ(m, T r )

48 Conclusion Global knowledge of all cells

49 Conclusion Global knowledge of all cells Cells only know the neighbours

50 Conclusion Global knowledge of all cells Cells only know the neighbours... of a random neighbour

51 Conclusion Global knowledge of all cells Cells only know the neighbours... of a random neighbour... their selves

52 Conclusion Global knowledge of all cells Cells only know the neighbours... of a random neighbour... their selves Action based on local knowledge

53 Conclusion Global knowledge of all cells Cells only know the neighbours... of a random neighbour... their selves Action based on local knowledge Communication on a random base

54 Implementation

55 Problems that occured Testing showed... Poor signalling gets punished hard and fast (Populations die out fast)

56 Problems that occured Testing showed... Poor signalling gets punished hard and fast (Populations die out fast) Time depended strategies: High density vs few left

57 Problems that occured Testing showed... Poor signalling gets punished hard and fast (Populations die out fast) Time depended strategies: High density vs few left Chaos - Difficult to measure

58 Problems that occured Testing showed... Poor signalling gets punished hard and fast (Populations die out fast) Time depended strategies: High density vs few left Chaos - Difficult to measure Utility choice of utility function

59 Solutions Solution for hard punishment Repeat update progress n-times Large initial populations

60 Solutions Solution for hard punishment Repeat update progress n-times Large initial populations Solution for time-depended strategies Bush-Mosteller reinforcement for faster learning Roth-Erev reinforcement too slow Repeat update progress n-times

61 Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average

62 Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average c = 1 t n CellsNum(t) t=1

63 Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average c = 1 t n CellsNum(t) t=1 And then build the ratio r = c signalling c basic

64 Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average c = 1 t n CellsNum(t) t=1 And then build the ratio r = c signalling c basic This ratio is used as a measurement for the survivability.

65 Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average c = 1 t n CellsNum(t) t=1 And then build the ratio r = c signalling c basic This ratio is used as a measurement for the survivability.

66 Solutions Solution for finding a fairly good utility function Number of global cells not representative Again... chaos Commonly number of cells shrinking Examining all constellations too complex Solution already introduced function

67 Demonstration

68 Results

69 Quantitative evaluation Grid size = Random cells p C =.25 modification vs basic 5 repetitions of signalling before each round actions in 5-th round

70 With a given meaning 2 signalling nm/basic game ratio signalling nm/basic runs

71 With a no given meaning 2 signalling wm/basic game ratio signalling wm/basic runs

72 Results compared Runs Successful Average 4-die % 136% given meaning % 119% no giv. meaning % 114%

73 Long term with a given meaning 2500 Long term behaviour signaling with given meaning 2000 Number living cells Number of generations

74 Long term without a given meaning 2500 Long term behaviour signaling with no given meaning 2000 Number living cells Number of generations

75 Qualitative evaluation Signalling strong when huge population Big grid size Important to get a good strategy fast Weak when only a few left Wrong strategy hard punishment

76 Final statement Taking Conways Game of Life as a metaphor for life, one could say Signalling can achieve higher survivability Signalling can successfully evolve in a high-punishing environment Poor signalling is deadly in a high-punishing environment