# Cellular Automata: Tutorial

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Cellular Automata: Tutorial Jarkko Kari Department of Mathematics, University of Turku, Finland TUCS(Turku Centre for Computer Science), Turku, Finland

2 Cellular Automata: examples A Cellular Automaton (CA) is an infinite, regular lattice of simple finite state machines that change their states synchronously, according to a local update rule that specifies the new state of each cell based on the old states of its neighbors.

3 Cellular Automata: examples A Cellular Automaton (CA) is an infinite, regular lattice of simple finite state machines that change their states synchronously, according to a local update rule that specifies the new state of each cell based on the old states of its neighbors. The most widely known example is the Game-of-Life by John Conway. It is a two-dimensional CA which means that the space is an infinite checker-board. Each cell (=square of the checker-board) has two possible states, Alive and Dead, represented by drawing the corresponding square black or white, respectively.

4 The local update rule asks each cell to check the present states of the eight surrounding cells. If the cell is alive then it stays alive (survives) iff it has two or three live neighbors. Otherwise it dies of loneliness or overcrowding.

5 The local update rule asks each cell to check the present states of the eight surrounding cells. If the cell is alive then it stays alive (survives) iff it has two or three live neighbors. Otherwise it dies of loneliness or overcrowding. If the cell is dead then it becomes alive iff it has exactly three living neighbors.

6 The local update rule asks each cell to check the present states of the eight surrounding cells. If the cell is alive then it stays alive (survives) iff it has two or three live neighbors. Otherwise it dies of loneliness or overcrowding. If the cell is dead then it becomes alive iff it has exactly three living neighbors. All cells apply this rule simultaneously. As the process is repeated over and over again, a dynamical system is obtained that exhibits surprisingly complex behavior.

7 The local update rule asks each cell to check the present states of the eight surrounding cells. If the cell is alive then it stays alive (survives) iff it has two or three live neighbors. Otherwise it dies of loneliness or overcrowding. If the cell is dead then it becomes alive iff it has exactly three living neighbors

8 The local update rule asks each cell to check the present states of the eight surrounding cells. If the cell is alive then it stays alive (survives) iff it has two or three live neighbors. Otherwise it dies of loneliness or overcrowding. If the cell is dead then it becomes alive iff it has exactly three living neighbors

9 The local update rule asks each cell to check the present states of the eight surrounding cells. If the cell is alive then it stays alive (survives) iff it has two or three live neighbors. Otherwise it dies of loneliness or overcrowding. If the cell is dead then it becomes alive iff it has exactly three living neighbors

10 The local update rule asks each cell to check the present states of the eight surrounding cells. If the cell is alive then it stays alive (survives) iff it has two or three live neighbors. Otherwise it dies of loneliness or overcrowding. If the cell is dead then it becomes alive iff it has exactly three living neighbors

11 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life)

12 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators)

13 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators)

14 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships)

15 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships)

16 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships)

17 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships)

18 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships)

19 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships)

20 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships)

21 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships)

22 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships)

23 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships), emit spaceships (guns), etc.

24 Game of Life was invented in 1970 by John Conway. Since then many interesting creatures living in this universe have been identified. These include patterns that remain unchanged (still life), oscillate periodically (oscillators), glide through space as they oscillate (spaceships), emit spaceships (guns), etc. Using such behaviors it is possible to implement information carrying signals and logical operations on these signals. It was soon discovered that Game-of-Life is a universal computer: for any given Turing machine M and input word w one can effectively construct a start configuration c to the Game-of-Life such that all cells eventually die if and only if M accepts w.

25 Another famous computationally universal CA is rule 110 by S.Wolfram. This is a one-dimensional CA with binary state set {0,1}, i.e. a two-way infinite sequence of 0 s and 1 s. Each cell is updated based on its old state and the states of its left and right neighbors as follows:

26 Rule 110 was proved universal by M.Cook while working for Wolfram. Analogously to Game-of-Life, it is possible to identify signals that transmit information, and collisions that represent logic operations, but building a computer in the one-dimensional space involves solving difficult issues such as designing ways for the signals to cross each other.

27 Rule 110 was proved universal by M.Cook while working for Wolfram. Analogously to Game-of-Life, it is possible to identify signals that transmit information, and collisions that represent logic operations, but building a computer in the one-dimensional space involves solving difficult issues such as designing ways for the signals to cross each other. One-dimensional, binary state CA that use the nearest neighbors to determine their next state are called elementary cellular automata. There are only 2 8 = 256 elementary CA, and it is quite remarkable that one of them is computationally universal.

28 Elementary CA were experimentally investigated in the 1980 s by S.Wolfram. He designed a simple naming scheme that is still used today. The local update rule gets fully specified when one gives the next state for all 8 different contexts: 111 b b b b b b b b 0 The Wolfram number of this CA is the integer whose binary expansion is b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0. Elementary CA are known by their Wolfram numbers.

29 For example, elementary CA number 102 has local update rule In this rule a b c b + c (mod 2) and it will be referred to as the xor-ca.

30 A space-time diagram is a pictorial representation of the time evolution of the CA where consecutive configurations are drawn under each other. Horizontal lines represent space, and time increases downwards. For example, the space-time diagram of the xor-ca starting from a single cell in state 1 is the self-similar Sierpinski-triangle: Time

31 Wolfram classified elementary CA into four classes based on experiments on random initial configurations. (W1) Almost all initial configurations lead to the same uniform fixed point configuration. Rule 160

32 (W2) Almost all initial configurations lead to a periodically repeating configuration. Rule 108

33 (W3) Almost all initial configurations lead to chaotic, random looking behavior. Rule 126

34 (W4) Localized structures with complex interactions emerge. Rule 110

35 Class (W4) has the most interesting behavior. In addition to rule 110, also rule 54 is in class (W4). It is conjectured but not proved yet that also rule 54 is computationally universal.

36 Outline of the talk(s) (1) Definitions (configurations, neighborhoods, cellular automata, reversibility, injectivity, surjectivity)

37 Outline of the talk(s) (1) Definitions (configurations, neighborhoods, cellular automata, reversibility, injectivity, surjectivity) (2) Classical results (Hedlund s theorem, balance in surjective CA, Garden-Of-Eden theorem)

38 Outline of the talk(s) (1) Definitions (configurations, neighborhoods, cellular automata, reversibility, injectivity, surjectivity) (2) Classical results (Hedlund s theorem, balance in surjective CA, Garden-Of-Eden theorem) (3) Reversible CA (universality, billiard ball CA)

39 Outline of the talk(s) (1) Definitions (configurations, neighborhoods, cellular automata, reversibility, injectivity, surjectivity) (2) Classical results (Hedlund s theorem, balance in surjective CA, Garden-Of-Eden theorem) (3) Reversible CA (universality, billiard ball CA) (4) Decidability questions (Wang tiles, domino problem, snake tiles, undecidability of nilpotency, reversibility, surjectivity, periodicity)

40 Outline of the talk(s) (1) Definitions (configurations, neighborhoods, cellular automata, reversibility, injectivity, surjectivity) (2) Classical results (Hedlund s theorem, balance in surjective CA, Garden-Of-Eden theorem) (3) Reversible CA (universality, billiard ball CA) (4) Decidability questions (Wang tiles, domino problem, snake tiles, undecidability of nilpotency, reversibility, surjectivity, periodicity) (5) Dynamical systems aspects (equicontinuity classification, limit sets)

41 Cellular Automata: definitions A Cellular Automaton (CA) is an infinite lattice of finite state machines, called cells. The cells of a d-dimensional CA are positioned at the integer lattice points of the d-dimensional Euclidean space, and they are addressed by the elements of Z d. Let S be the finite state set. A configuration of the CA is a function c : Z d S where c( x) is the current state of the cell x. The set S Zd of all configurations is uncountably infinite.

42 The cells change their states synchronously at discrete time steps. The next state of each cell depends on the current states of the neighboring cells according to an update rule. All cells use the same rule, and the rule is applied to all cells at the same time.

43 The cells change their states synchronously at discrete time steps. The next state of each cell depends on the current states of the neighboring cells according to an update rule. All cells use the same rule, and the rule is applied to all cells at the same time. The neighboring cells may be the nearest cells surrounding each cell, but more general neighborhoods can be specified by giving the relative offsets of the neighbors. Let N = ( x 1, x 2,..., x n ) be a vector of n distinct elements of Z d. Then the neighbors of a cell at location x Z d are the n cells at locations x + x i, for i = 1,2,...,n.

44 Typical two-dimensional neighborhoods: c c Von Neumann Moore neighborhood neighborhood {(0,0),(±1,0),(0, ±1)} { 1,0,1} { 1,0,1}

45 The smallest non-trivial one-dimensional neighborhood is the size two neighborhood (0,1). This is the neighborhood of the xor-ca. This case is one-way as information can not flow to the right.

46 If the configurations are shifted by half a cell at each step we obtain the radius- 1 2 neighborhood. The space-time diagram is then symmetric:

47 If the configurations are shifted by half a cell at each step we obtain the radius- 1 2 neighborhood. The space-time diagram is then symmetric:

48 The local rule is a function f : S n S where n is the size of the neighborhood. State f(a 1,a 2,...,a n ) is the new state of a cell whose n neighbors were at states a 1,a 2,...,a n one time step before.

49 The local rule is a function f : S n S where n is the size of the neighborhood. State f(a 1,a 2,...,a n ) is the new state of a cell whose n neighbors were at states a 1,a 2,...,a n one time step before. This update rule then determines the global dynamics of the CA: Configuration c becomes in one time step the configuration e where, for all x Z d, e( x) = f(c( x + x 1 ),c( x + x 2 ),...,c( x + x n )). We say that e = G(c), and call G : S Zd S Zd the global transition function of the CA.

50 We typically identify the CA with its global transition function G and talk about cellular automaton function G, or simply cellular automaton G. In algorithmic questions, however, the CA is always specified by the finite items d (the dimension of the space), S (the state set), N (the neighborhood) and f (the local rule).

51 Cellular Automata can be viewed as one of the first models of natural computing. They have many properties of the physical world: They consist of large numbers of simple objects, operate in parallel, evolve based on local interactions, are homogeneous in time and space. CA have traditionally been used in simulations of physical systems. Good examples are lattice gases. These are CA simulations of fluid or gas dynamics based on storing individual molecules in the cells and implementing particle interactions by the CA local rule.

52 Consider, for example, the simplest lattice gas model, called HPP (due to Hardy, Pomeau and de Pazzis). It is a two-dimensional CA where each cell can store up to four moving particles. Each particle has a direction of movement which can be up, down, left or right. There can be no more than one particle of each direction in any individual cell. So there are 2 4 = 16 possible states.

53 At each time step each particle moves to the neighboring cell as indicated by the direction of the particle. If a cell receives exactly two particles moving in opposite directions then the directions of the particles are changed into the two orthogonal directions.

54 At each time step each particle moves to the neighboring cell as indicated by the direction of the particle. If a cell receives exactly two particles moving in opposite directions then the directions of the particles are changed into the two orthogonal directions.

55 At each time step each particle moves to the neighboring cell as indicated by the direction of the particle. If a cell receives exactly two particles moving in opposite directions then the directions of the particles are changed into the two orthogonal directions.

56 At each time step each particle moves to the neighboring cell as indicated by the direction of the particle. If a cell receives exactly two particles moving in opposite directions then the directions of the particles are changed into the two orthogonal directions.

57 At each time step each particle moves to the neighboring cell as indicated by the direction of the particle. If a cell receives exactly two particles moving in opposite directions then the directions of the particles are changed into the two orthogonal directions.

58 At each time step each particle moves to the neighboring cell as indicated by the direction of the particle. If a cell receives exactly two particles moving in opposite directions then the directions of the particles are changed into the two orthogonal directions.

59 The local rule of HPP preserves the total number of particles and their total momentum. These are conservation laws that hold in the HPP automaton. (Note, however, that HPP is not a realistic lattice gas model as it satisfies incorrect conservation laws. For example, the horizontal momentum is preserved on each horizontal line individually.)

60 The local rule of HPP preserves the total number of particles and their total momentum. These are conservation laws that hold in the HPP automaton. (Note, however, that HPP is not a realistic lattice gas model as it satisfies incorrect conservation laws. For example, the horizontal momentum is preserved on each horizontal line individually.) Even more interestingly, the HPP local rule fully preserves information. There is another CA that traces back the configurations in the reverse direction. This inverse CA simply moves the particles to the opposite direction, and applies the same collision rule as HPP.

61 HPP is an example of reversible CA (RCA), also called invertible CA. These are CA functions G such that there is another CA function F that is its inverse, i.e. G F = F G = identity function. RCA G and F are called the inverse automata of each other. Clearly, in order to be reversible G has to be a bijective function S Zd S Zd.

62 HPP is an example of reversible CA (RCA), also called invertible CA. These are CA functions G such that there is another CA function F that is its inverse, i.e. G F = F G = identity function. RCA G and F are called the inverse automata of each other. Clearly, in order to be reversible G has to be a bijective function S Zd S Zd. A CA is called injective if G is one-to-one, surjective if G is onto, bijective if G is both one-to-one and onto.

63 Outline of the talk(s) (1) Definitions (configurations, neighborhoods, cellular automata, reversibility, injectivity, surjectivity) (2) Classical results (Hedlund s theorem, balance in surjective CA, Garden-Of-Eden theorem) (3) Reversible CA (universality, billiard ball CA) (4) Decidability questions (Wang tiles, domino problem, snake tiles, undecidability of nilpotency, reversibility, surjectivity, periodicity) (5) Dynamical systems aspects (equicontinuity classification, limit sets)

64 Classical results (1960 s and 70 s) It is useful to endow the set S Zd of configurations with the Cantor topology, i.e. the topology generated by the cylinder sets Cyl(c,M) = {e S Zd e( x) = c( x) for all x M} for c S Zd and finite M Z d. The topology is compact, and it is induced by a metric.

65 Classical results (1960 s and 70 s) It is useful to endow the set S Zd of configurations with the Cantor topology, i.e. the topology generated by the cylinder sets Cyl(c,M) = {e S Zd e( x) = c( x) for all x M} for c S Zd and finite M Z d. The topology is compact, and it is induced by a metric. All cylinder sets are clopen, i.e. closed and open. Cylinders for fixed finite M Z d form a finite partitioning of S Zd.

66 Under this topology, a sequence c 1,c 2,... of configurations converges to c S Zd if and only if for all cells x Z d and for all sufficiently large i holds c i ( x) = c( x). Compactness of the topology means that all infinite sequences c 1,c 2,... of configurations have converging subsequences.

67 All cellular automata are continuous transformations S Zd S Zd under the topology. Indeed, locality of the update rule means that if c 1,c 2,... is a converging sequence of configurations then converges as well, and G(c 1 ),G(c 2 ),... lim i G(c i) = G( lim i c i ).

68 The translation τ determined by vector r Z d is the transformation that maps c e where S Zd S Zd e( x) = c( x r) for all x Z d. (It is the CA whose local rule is the identity function and whose neighborhood consists of r alone.) Since all cells of a CA use the same local rule, the CA commutes with all translations: G τ = τ G.

69 We have seen that all CA are continuous, translation commuting maps S Zd S Zd. The Curtis-Hedlund- Lyndon theorem from 1969 states also the converse: Theorem: A function G : S Zd S Zd is a CA function if and only if (i) G is continuous, and (ii) G commutes with translations.

70 Corollary: A cellular automaton G is reversible if and only if it is bijective. Proof: If G is a reversible CA function then G is by definition bijective. Conversely, suppose that G is a bijective CA function. Then G has an inverse function G 1 that clearly commutes with the shifts. The inverse function G 1 is also continuous because the space S Zd is compact. It now follows from the Curtis-Hedlund- Lyndon theorem that G 1 is a cellular automaton.

71 Corollary: A cellular automaton G is reversible if and only if it is bijective. Proof: If G is a reversible CA function then G is by definition bijective. Conversely, suppose that G is a bijective CA function. Then G has an inverse function G 1 that clearly commutes with the shifts. The inverse function G 1 is also continuous because the space S Zd is compact. It now follows from the Curtis-Hedlund- Lyndon theorem that G 1 is a cellular automaton. The point of the corollary is that in bijective CA each cell can determine its previous state by looking at the current states in some bounded neighborhood around them.

72 Configurations that do not have a pre-image are called Garden-Of-Eden -configurations. A CA has GOE configurations if and only if it is non-surjective. A finite pattern consists of a finite domain M Z d and an assignment p : M S of states. Configuration c contains the pattern if for some translation τ. τ(c) M = p Finite pattern is called an orphan for CA G if every configuration containing the pattern is a GOE.

73 Let us show that every GOE contains an orphan: Let c be a GOE, and let M 1, M 2, M 3,... be such that each M i Z d is finite, M 1 M 2 M 3..., and M 1 M 2 M 3... = Z d.

74 Let us show that every GOE contains an orphan: Let c be a GOE, and let M 1, M 2, M 3,... be such that each M i Z d is finite, M 1 M 2 M 3..., and M 1 M 2 M 3... = Z d. Since G is continuous also G(S Zd ) is compact. Hence sets C i = Cyl(c, M i ) G(S Zd ) are compact and form a decreasing chain C 1 C 2 C 3...

75 Let us show that every GOE contains an orphan: Let c be a GOE, and let M 1, M 2, M 3,... be such that each M i Z d is finite, M 1 M 2 M 3..., and M 1 M 2 M 3... = Z d. Since G is continuous also G(S Zd ) is compact. Hence sets C i = Cyl(c, M i ) G(S Zd ) are compact and form a decreasing chain C 1 C 2 C 3... Because C 1 C 2 C 3... = it follows from compactness of that C i = for some i. Then is an orphan. c Mi

76 All surjective CA have balanced local rules: for every a S f 1 (a) = S n 1.

77 All surjective CA have balanced local rules: for every a S f 1 (a) = S n 1. Indeed, consider a non-balanced local rule such as rule 110 where five contexts give new state 1 while only three contexts give state 0:

78 Consider finite patterns where state 0 appears in every third position. There are 2 2(k 1) = 4 k 1 such patterns where k is the number of 0 s

79 Consider finite patterns where state 0 appears in every third position. There are 2 2(k 1) = 4 k 1 such patterns where k is the number of 0 s A pre-image of such a pattern must consist of k segments of length three, each of which is mapped to 0 by the local rule. There are 3 k choices. As for large values of k we have 3 k < 4 k 1, there are fewer choices for the red cells than for the blue ones. Hence some pattern has no pre-image and it must be an orphan.

80 One can also verify directly that pattern is an orphan of rule 110. It is the shortest orphan.

81 One can also verify directly that pattern is an orphan of rule 110. It is the shortest orphan. Balance of the local rule is not sufficient for surjectivity. For example, the majority CA (Wolfram number 232) is a counter example. The local rule f(a,b,c) = 1 if and only if a + b + c 2 is clearly balanced, but is an orphan.

82 Sometimes one state q S satisfying f(q,q,...,q) = q is specified as a quiescent state. A configuration is called finite if only a finite number of cells are non-quiescent: q q q q q q q q q q q q q q If c is finite and G is a CA function then also G(c) is finite. Let us denote by G F the restriction of G on the finite configurations.

83 The Garden-Of-Eden -theorem by Moore (1962) and Myhill (1963) states that G is surjective if and only if G F is injective. Let us show this using rule 110 as a running example.

84 1) G not surjective = G F not injective: Since rule 110 is not surjective it has an orphan of width five. Consider a segment of length 5k 2, for some k, and finite configurations c that are quiescent outside this segment. There are 2 5k 2 = 32 k /4 such configurations. 5k

85 1) G not surjective = G F not injective: The non-quiescent part of G(c) is within a segment of length 5k. Partition this segment into k parts of length 5. Pattern cannot appear in any part, so only = 31 different patterns show up in the subsegments. There are at most 31 k possible configurations G(c). 5k k

86 1) G not surjective = G F not injective: The non-quiescent part of G(c) is within a segment of length 5k. Partition this segment into k parts of length 5. Pattern cannot appear in any part, so only = 31 different patterns show up in the subsegments. There are at most 31 k possible configurations G(c). 5k k As 32 k /4 > 31 k for large k, there are more choices for red than blue segments. So there must exist two different red segments with the same image.

87 2) G F not injective = G not surjective: In rule 110 p q so patterns p and q of length 8 can be exchanged to each other in any configuration without affecting its image. There exist just = 255 essentially different blocks of length 8.

88 2) G F not injective = G not surjective: Consider a segment of 8k cells, consisting of k parts of length 8. Patterns p and q are exchangeable, so the segment has at most 255 k different images. 8k 8k-2

89 2) G F not injective = G not surjective: Consider a segment of 8k cells, consisting of k parts of length 8. Patterns p and q are exchangeable, so the segment has at most 255 k different images. 8k 8k-2 There are, however, 2 8k 2 = 256 k /4 different patterns of size 8k 2. Because 255 k < 256 k /4 for large k, there are blue patterns without any pre-image.

90 Theorem: CA G is surjective if and only if its restriction G F to finite configurations is injective.

91 Theorem: CA G is surjective if and only if its restriction G F to finite configurations is injective. Corollary: Every injective CA is also surjective. Injectivity, bijectivity and reversibility are equivalent concepts. Proof: If G is injective then also G F is injective. The claim follows from the Garden-Of-Eden -theorem.

92 Theorem: CA G is surjective if and only if its restriction G F to finite configurations is injective. Corollary: Every injective CA is also surjective. Injectivity, bijectivity and reversibility are equivalent concepts. Proof: If G is injective then also G F is injective. The claim follows from the Garden-Of-Eden -theorem. Corollary: If G is injective then G F is surjective. Proof: If G is injective then it is reversible. The quiescent state of G is also quiescent in G 1, so G 1 maps finite configurations to finite configurations. So every finite configuration has a finite pre-image.

93 G injective G bijective G reversible G surjective G bijective F F G surjective G injective F

94 G injective G bijective G reversible G surjective G bijective F F XOR rule G surjective G injective F

95 In the xor-ca every configuration has exactly two pre-images, so G is surjective: One can freely choose one value in the pre-image, after which all remaining states are uniquely determined by the left-permutativity and the right-permutativity of xor.

96 In the xor-ca every configuration has exactly two pre-images, so G is surjective: One can freely choose one value in the pre-image, after which all remaining states are uniquely determined by the left-permutativity and the right-permutativity of xor.

97 In the xor-ca every configuration has exactly two pre-images, so G is surjective: One can freely choose one value in the pre-image, after which all remaining states are uniquely determined by the left-permutativity and the right-permutativity of xor.

98 In the xor-ca every configuration has exactly two pre-images, so G is surjective: One can freely choose one value in the pre-image, after which all remaining states are uniquely determined by the left-permutativity and the right-permutativity of xor.

99 In the xor-ca every configuration has exactly two pre-images, so G is surjective: One can freely choose one value in the pre-image, after which all remaining states are uniquely determined by the left-permutativity and the right-permutativity of xor.

100 In the xor-ca every configuration has exactly two pre-images, so G is surjective: One can freely choose one value in the pre-image, after which all remaining states are uniquely determined by the left-permutativity and the right-permutativity of xor.

101 The two pre-images of the finite configuration 0 0 are both infinite: So G F is not surjective.

102 G injective G bijective G reversible G surjective G bijective F F XOR rule G surjective G injective F

103 G injective G bijective G reversible controlled XOR rule G surjective G bijective F F XOR rule G surjective G injective F

104 The controlled-xor CA has two binary tracks. The lower track does not change, but it indicates which cells are active. Non-active cells do not change at all, while active cells apply the xor rule on the first track: A A A A 0 A A

105 The controlled-xor CA has two binary tracks. The lower track does not change, but it indicates which cells are active. Non-active cells do not change at all, while active cells apply the xor rule on the first track: A A A A 0 A A

106 The controlled-xor CA has two binary tracks. The lower track does not change, but it indicates which cells are active. Non-active cells do not change at all, while active cells apply the xor rule on the first track: A A A A 0 A A The quiescent state is 0

107 The controlled-xor CA has two binary tracks. The lower track does not change, but it indicates which cells are active. Non-active cells do not change at all, while active cells apply the xor rule on the first track: A A A A 0 A A The quiescent state is 0 The CA is not injective as all active 0 s and all active 1 s have the same image, but every finite configuration has a finite pre-image, so G F is surjective.

108 G injective G bijective G reversible controlled XOR rule G surjective G bijective F F XOR rule G surjective G injective F

109 Examples: The majority rule is not surjective: finite configurations and have the same image, so G F is not injective. Pattern is an orphan.

110 Examples: In Game-Of-Life a lonely living cell dies immediately, so G F is not injective. So GOL is not surjective. Interestingly, no small orphans are known for Game-Of-Life. Currently, the smallest known orphan consists of 113 cells:

111 Examples: The Traffic CA is the elementary CA number The local rule replaces pattern 01 by pattern 10.

112

113

114

115

116 The local rule is balanced. However, there are two finite configurations with the same successor: and hence traffic CA is not surjective.

117 There is an orphan of size four:

118 Outline of the talk(s) (1) Definitions (configurations, neighborhoods, cellular automata, reversibility, injectivity, surjectivity) (2) Classical results (Hedlund s theorem, balance in surjective CA, Garden-Of-Eden theorem) (3) Reversible CA (universality, billiard ball CA) (4) Decidability questions (Wang tiles, domino problem, snake tiles, undecidability of nilpotency, reversibility, surjectivity, periodicity) (5) Dynamical systems aspects (equicontinuity classification, limit sets)

119 Reversible cellular automata (RCA) Reversibility is an important aspect of physics at the microscopic scale, so it is natural that simulations are done using reversible CA. Despite being a very restrictive condition, reversibility does not prevent universal computation. It is clear that any Turing machine can be simulated by a one-dimensional CA. In 1977 T.Toffoli demonstrated how any d-dimensional CA can be simulated by a d + 1-dimensional RCA. Hence two-dimensional universal RCA exist.

120 In 1989 K.Morita and M.Harao showed how to simulate any reversible Turing machine by a one-dimensional RCA. Since reversible Turing machines can be computation universal (Bennett 1973), the following result follows: Theorem: One-dimensional reversible cellular automata exist that are computationally universal.

121 In 1989 K.Morita and M.Harao showed how to simulate any reversible Turing machine by a one-dimensional RCA. Since reversible Turing machines can be computation universal (Bennett 1973), the following result follows: Theorem: One-dimensional reversible cellular automata exist that are computationally universal. In the following I give a simple construction of a one-dimensional reversible CA that simulates in real time any Turing machine, reversible or not.

122 Partitioned CA (PCA) are particular types of reversible cellular automata. The state set of a PCA is a cartesian product of finite sets: S = S 1 S 2...S k. The k components are called tracks. The local update rule is specified by providing a translation τ i for each track i = 1,2,...,k, and a bijective transformation of the entire state set. π : S S The update is done in two steps: First π is applied at all cells, after which all tracks are translated according to τ i.

123 Example: A three track PCA where each track is binary. Translations of tracks are (1) shift left, (2) no move and (3) shift right. Permutation π is a b c a a + b (mod 2) b + c (mod 2) PCA are clearly reversible as both elementary steps are reversible.

124 Example: A three track PCA where each track is binary. Translations of tracks are (1) shift left, (2) no move and (3) shift right. Permutation π is a b c a a + b (mod 2) b + c (mod 2) PCA are clearly reversible as both elementary steps are reversible.

125 Example: A three track PCA where each track is binary. Translations of tracks are (1) shift left, (2) no move and (3) shift right. Permutation π is a b c a a + b (mod 2) b + c (mod 2) PCA are clearly reversible as both elementary steps are reversible.

126 Example: A three track PCA where each track is binary. Translations of tracks are (1) shift left, (2) no move and (3) shift right. Permutation π is a b c a a + b (mod 2) b + c (mod 2) PCA are clearly reversible as both elementary steps are reversible.

127 Example: A three track PCA where each track is binary. Translations of tracks are (1) shift left, (2) no move and (3) shift right. Permutation π is a b c a a + b (mod 2) b + c (mod 2) PCA are clearly reversible as both elementary steps are reversible.

128 Example: A three track PCA where each track is binary. Translations of tracks are (1) shift left, (2) no move and (3) shift right. Permutation π is a b c a a + b (mod 2) b + c (mod 2) PCA are clearly reversible as both elementary steps are reversible.

129 Example: A three track PCA where each track is binary. Translations of tracks are (1) shift left, (2) no move and (3) shift right. Permutation π is a b c a a + b (mod 2) b + c (mod 2) PCA are clearly reversible as both elementary steps are reversible.

130 To simulate a Turing machine we use a PCA with four tracks: Track (1) is identical to the tape of the Turing machine. It is not translated.

131 To simulate a Turing machine we use a PCA with four tracks: Track (1) is identical to the tape of the Turing machine. It is not translated. Track (2) or (3) stores the Turing machine state. They are shifted one cell left and right respectively. The state is stored on the track which moves to the direction indicated by the TM instruction being executed.

132 To simulate a Turing machine we use a PCA with four tracks: Track (1) is identical to the tape of the Turing machine. It is not translated. Track (2) or (3) stores the Turing machine state. They are shifted one cell left and right respectively. The state is stored on the track which moves to the direction indicated by the TM instruction being executed. Track (4) is a garbage track. It is translated by two cells so that a new empty trash bin always appears at the position of the Turing machine.

133 The permutation π copies first the contents of tracks (1),(2) and (3) to the garbage track. This copying is done only at the cell that contains the Turing machine head. Then the first three tracks are updated according to the Turing machine instruction. b a b a a b b q TM(q,b) = (s,c, )

134 The permutation π copies first the contents of tracks (1),(2) and (3) to the garbage track. This copying is done only at the cell that contains the Turing machine head. Then the first three tracks are updated according to the Turing machine instruction. c a b a a b b s b q TM(q,b) = (s,c, )

135 The permutation π copies first the contents of tracks (1),(2) and (3) to the garbage track. This copying is done only at the cell that contains the Turing machine head. Then the first three tracks are updated according to the Turing machine instruction. c a b a a b b b q s TM(s,a) = (r,d, )

136 The permutation π copies first the contents of tracks (1),(2) and (3) to the garbage track. This copying is done only at the cell that contains the Turing machine head. Then the first three tracks are updated according to the Turing machine instruction. c d b a a b b r b q a s TM(s,a) = (r,d, )

137 The permutation π copies first the contents of tracks (1),(2) and (3) to the garbage track. This copying is done only at the cell that contains the Turing machine head. Then the first three tracks are updated according to the Turing machine instruction. c d b a a b b r b q a s TM(r,c) = (q,a, )

138 The permutation π copies first the contents of tracks (1),(2) and (3) to the garbage track. This copying is done only at the cell that contains the Turing machine head. Then the first three tracks are updated according to the Turing machine instruction. a d b a a b b q b q a s c r TM(r,c) = (q,a, )

139 The permutation π copies first the contents of tracks (1),(2) and (3) to the garbage track. This copying is done only at the cell that contains the Turing machine head. Then the first three tracks are updated according to the Turing machine instruction. q a d b a a b b b q a s c r TM(q, ) = (r,a, )

140 The permutation π copies first the contents of tracks (1),(2) and (3) to the garbage track. This copying is done only at the cell that contains the Turing machine head. Then the first three tracks are updated according to the Turing machine instruction. a a d b a a b b b q a s r c r q TM(q, ) = (r,a, )

141 The permutation π copies first the contents of tracks (1),(2) and (3) to the garbage track. This copying is done only at the cell that contains the Turing machine head. Then the first three tracks are updated according to the Turing machine instruction. a a d b a a b b b q a s c r q r TM(r, a) = (s,d, )

142 It is clear that the partially defined π is one-to-one. Any partially defined injective map S S can be completed into a bijection by matching the missing elements in the domain and range arbitrarily. Note that the missing elements correspond to situations that never occur during valid simulations of the Turing machine (for example, when the incoming new garbage bin is not empty).

143 N.Margolus introduced a particularly simple two-dimensional RCA that is computationally universal, called the Billiard Ball RCA. This CA also introduced the technique of block permutations to guarantee reversibility. In this technique the update is done in two steps:

144 N.Margolus introduced a particularly simple two-dimensional RCA that is computationally universal, called the Billiard Ball RCA. This CA also introduced the technique of block permutations to guarantee reversibility. In this technique the update is done in two steps: Partition the plane into 2 2 blocks and apply some permutation π 1 of S 4 inside each block.

145 N.Margolus introduced a particularly simple two-dimensional RCA that is computationally universal, called the Billiard Ball RCA. This CA also introduced the technique of block permutations to guarantee reversibility. In this technique the update is done in two steps: Partition the plane into 2 2 blocks and apply some permutation π 1 of S 4 inside each block.

146 N.Margolus introduced a particularly simple two-dimensional RCA that is computationally universal, called the Billiard Ball RCA. This CA also introduced the technique of block permutations to guarantee reversibility. In this technique the update is done in two steps: Shift the partitioning horizontally and vertically, and apply another permutation π 2 of S 4 on the new blocks.

147 N.Margolus introduced a particularly simple two-dimensional RCA that is computationally universal, called the Billiard Ball RCA. This CA also introduced the technique of block permutations to guarantee reversibility. In this technique the update is done in two steps: Shift the partitioning horizontally and vertically, and apply another permutation π 2 of S 4 on the new blocks.

148 N.Margolus introduced a particularly simple two-dimensional RCA that is computationally universal, called the Billiard Ball RCA. This CA also introduced the technique of block permutations to guarantee reversibility. In this technique the update is done in two steps: The composition of the two block permutation is one iteration of the CA. It is trivially reversible.

149 In the Billiard Ball RCA S is a binary set, and both permutations π 1 and π 2 are identical. They only make the following exchanges:

150 Because of alternating partitioning, a single black state propagates in one of the four diagonal directions:

151 Because of alternating partitioning, a single black state propagates in one of the four diagonal directions:

152 Because of alternating partitioning, a single black state propagates in one of the four diagonal directions:

153 Because of alternating partitioning, a single black state propagates in one of the four diagonal directions:

154 Because of alternating partitioning, a single black state propagates in one of the four diagonal directions:

155 Because of alternating partitioning, a single black state propagates in one of the four diagonal directions:

156 Two particles that collide sideways pause for a moment and reverse their direction:

157 Two particles that collide sideways pause for a moment and reverse their direction:

158 Two particles that collide sideways pause for a moment and reverse their direction:

159 Two particles that collide sideways pause for a moment and reverse their direction:

160 Two particles that collide sideways pause for a moment and reverse their direction:

161 Two particles that collide sideways pause for a moment and reverse their direction:

162 In the Billiard Ball RCA one can simulate the motion and collisions of balls of positive diameter. Walls from which the balls bounce can also be created. Amazingly, these constructs are sufficient to perform arbitrary computation, proving the computational universality of the Billiard Ball RCA.

163 In the Billiard Ball RCA one can simulate the motion and collisions of balls of positive diameter. Walls from which the balls bounce can also be created. Amazingly, these constructs are sufficient to perform arbitrary computation, proving the computational universality of the Billiard Ball RCA. Reversibility is trivially obtained using the block permutation technique. Also conservation laws are easy to enforce. For example, if the permutations π 1 and π 2 are such that they preserve the numbers of black states (e.g. as in the Billiard Ball RCA) then the whole CA conserves the number of black states.

164 Reversible CA come up naturally when simulating reversible physical systems.

165 Reversible CA come up naturally when simulating reversible physical systems. Conversely, physically building a cellular automata device in the reversible physical universe is most energy efficient if the CA is reversible. In fact, whenever a physical device erases information (by performing, say, a logical AND operation) it must dissipate energy from the system, usually in the form of heat. It is more energy efficient to use only reversible logical operations during computation.

166 The problem with reversible CA is that reversibility is global: the transformation G : S Zd S Zd is bijective on infinite configurations. In order to exploit the reversibility of RCA they should be implemented using finite reversible logical gates. However, CA are given in terms of the local update rule f : S n S that is not reversible: f

167 Rather than using the non-reversible local rule f we would need to use reversible local functions π Partitioned CA, and the block partitioning technique of the Billiard Ball RCA are examples where the local rule was given in terms of reversible local rules. A natural question: Can every RCA be expressed using reversible local rules?

168 In other words, we would like to replace a traditional CA f f f f f f f with something like that computes the same global function G : S Zd S Zd.

169 The answer is positive in one- and two-dimensional spaces: Every one-dimensional RCA is a composition of two block permutations and a translation of a track.

170 The answer is positive in one- and two-dimensional spaces: Every one-dimensional RCA is a composition of two block permutations and a translation of a track. Terminology used: A block permutation consists of partitioning the space into congruent finite segments (rectangles, hypercubes in higher dimensions), and applying a permutation π inside every block:

171 The answer is positive in one- and two-dimensional spaces: Every one-dimensional RCA is a composition of two block permutations and a translation of a track. Translating a track means expressing the state set as a cartesian product S = S 1 S 2 and translating the S 1 -components.

172 The answer is positive in one- and two-dimensional spaces: Every one-dimensional RCA is a composition of two block permutations and a translation of a track. Every two-dimensional RCA is a composition of three block permutations and a translation of a track.

173 The answer is positive in one- and two-dimensional spaces: Every one-dimensional RCA is a composition of two block permutations and a translation of a track. Every two-dimensional RCA is a composition of three block permutations and a translation of a track. The question is open for three-dimensional CA. Conjecture: Every d-dimensional RCA can be expressed as a composition of block permutations and a translation of a track. (We know that if the conjecture holds then d + 1 block permutations suffice.)

174 Outline of the talk(s) (1) Definitions (configurations, neighborhoods, cellular automata, reversibility, injectivity, surjectivity) (2) Classical results (Hedlund s theorem, balance in surjective CA, Garden-Of-Eden theorem) (3) Reversible CA (universality, billiard ball CA) (4) Decidability questions (Wang tiles, domino problem, snake tiles, undecidability of nilpotency, reversibility, surjectivity, periodicity) (5) Dynamical systems aspects (equicontinuity classification, limit sets)

### Cellular Automata and Tilings

Cellular Automata and Tilings Jarkko Kari Department of Mathematics, University of Turku, Finland TUCS(Turku Centre for Computer Science), Turku, Finland Outline of the talk (1) Cellular automata (CA)

XX Eesti Arvutiteaduse Talvekool Cellular automata, tilings and (un)computability Jarkko Kari Department of Mathematics and Statistics University of Turku Lecture 1: Tutorial on Cellular automata Introduction

### Cellular Automata. Jason Frank Mathematical Institute

Cellular Automata Jason Frank Mathematical Institute WISM484 Introduction to Complex Systems, Utrecht University, 2015 Cellular Automata Game of Life: Simulator: http://www.bitstorm.org/gameoflife/ Hawking:

### Modelling with cellular automata

Modelling with cellular automata Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Outline Outline of Topics Concepts about cellular

### Outline 1 Introduction Tiling definitions 2 Conway s Game of Life 3 The Projection Method

A Game of Life on Penrose Tilings Kathryn Lindsey Department of Mathematics Cornell University Olivetti Club, Sept. 1, 2009 Outline 1 Introduction Tiling definitions 2 Conway s Game of Life 3 The Projection

### Motivation. Evolution has rediscovered several times multicellularity as a way to build complex living systems

Cellular Systems 1 Motivation Evolution has rediscovered several times multicellularity as a way to build complex living systems Multicellular systems are composed by many copies of a unique fundamental

### biologically-inspired computing lecture 12 Informatics luis rocha 2015 INDIANA UNIVERSITY biologically Inspired computing

lecture 12 -inspired Sections I485/H400 course outlook Assignments: 35% Students will complete 4/5 assignments based on algorithms presented in class Lab meets in I1 (West) 109 on Lab Wednesdays Lab 0

### Dimension sensitive properties of cellular automata and subshifts of finite type

Charalampos Zinoviadis Dimension sensitive properties of cellular automata and subshifts of finite type TUCS Technical Report No 977, May 2010 Dimension sensitive properties of cellular automata and subshifts

### Bio-inspired Models of Computation Seminar. Daniele Sgandurra. 16 October 2009

Bio-inspired Models of Computation Seminar Università di Pisa 16 October 2009 Outline Introduction Motivation History Cellular Systems Wolfram Classes Variants and Extensions Extended Topics Garden of

### Universality in Elementary Cellular Automata

Universality in Elementary Cellular Automata Matthew Cook Department of Computation and Neural Systems, Caltech, Mail Stop 136-93, Pasadena, California 91125, USA The purpose of this paper is to prove

### Classification of two-dimensional binary cellular automata with respect to surjectivity

Classification of two-dimensional binary cellular automata with respect to surjectivity Henryk Fukś and Andrew Skelton epartment of Mathematics Brock University St. Catharines, ON, Canada Abstract While

### Introduction to Topology

Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

### Structure and Invertibility in Cellular Automata

Università degli Studi di Roma La Sapienza Dipartimento di Matematica G. Castelnuovo Tesi di Dottorato in Matematica Structure and Invertibility in Cellular Automata Silvio Capobianco December 2004 2 Acknowledgements

### A Topological Classification of D-Dimensional Cellular Automata

A Topological Classification of D-Dimensional Cellular Automata by Emily Gamber A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the

### Complexity Classes in the Two-dimensional Life Cellular Automata Subspace

Complexity Classes in the Two-dimensional Life Cellular Automata Subspace Michael Magnier Claude Lattaud Laboratoire d Intelligence Artificielle de Paris V, Université René Descartes, 45 rue des Saints

### Periodicity and Immortality in Reversible Computing

Periodicity and Immortality in Reversible Computing Jarkko Kari, Nicolas Ollinger To cite this version: Jarkko Kari, Nicolas Ollinger. Periodicity and Immortality in Reversible Computing. Additional material

### Lecture 23 : Nondeterministic Finite Automata DRAFT Connection between Regular Expressions and Finite Automata

CS/Math 24: Introduction to Discrete Mathematics 4/2/2 Lecture 23 : Nondeterministic Finite Automata Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last time we designed finite state automata

### DRAFT. Diagonalization. Chapter 4

Chapter 4 Diagonalization..the relativized P =?NP question has a positive answer for some oracles and a negative answer for other oracles. We feel that this is further evidence of the difficulty of the

### Logic Programming for Cellular Automata

Technical Communications of ICLP 2015. Copyright with the Authors. 1 Logic Programming for Cellular Automata Marcus Völker RWTH Aachen University Thomashofstraße 5, 52070 Aachen, Germany (e-mail: marcus.voelker@rwth-aachen.de)

### Elementary cellular automata

Cellular Automata Cellular automata (CA) models epitomize the idea that simple rules can generate complex pa8erns. A CA consists of an array of cells each with an integer state. On each?me step a local

### Some Function Problems SOLUTIONS Isabel Vogt Last Edited: May 24, 2013

Some Function Problems SOLUTIONS Isabel Vogt Last Edited: May 24, 23 Most of these problems were written for my students in Math 23a/b at Harvard in 2/22 and 22/23. They cover basic function theory, countability,

### From Glider to Chaos: A Transitive Subsystem Derived From Glider B of CA Rule 110

From Glider to Chaos: A Transitive Subsystem Derived From Glider B of CA Rule 110 Pingping Liu, Fangyue Chen, Lingxiao Si, and Fang Wang School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang,

### Chaotic Subsystem Come From Glider E 3 of CA Rule 110

Chaotic Subsystem Come From Glider E 3 of CA Rule 110 Lingxiao Si, Fangyue Chen, Fang Wang, and Pingping Liu School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, P. R. China Abstract The

### CS 21 Decidability and Tractability Winter Solution Set 3

CS 21 Decidability and Tractability Winter 2018 Posted: January 31 Solution Set 3 If you have not yet turned in the Problem Set, you should not consult these solutions. 1. (a) A 2-NPDA is a 7-tuple (Q,,

### Decision Problems with TM s. Lecture 31: Halting Problem. Universe of discourse. Semi-decidable. Look at following sets: CSCI 81 Spring, 2012

Decision Problems with TM s Look at following sets: Lecture 31: Halting Problem CSCI 81 Spring, 2012 Kim Bruce A TM = { M,w M is a TM and w L(M)} H TM = { M,w M is a TM which halts on input w} TOTAL TM

### CSCE 551: Chin-Tser Huang. University of South Carolina

CSCE 551: Theory of Computation Chin-Tser Huang huangct@cse.sc.edu University of South Carolina Church-Turing Thesis The definition of the algorithm came in the 1936 papers of Alonzo Church h and Alan

### Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009

Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009 Our main goal is here is to do counting using functions. For that, we

### Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,

### State Space Search Problems

State Space Search Problems Lecture Heuristic Search Stefan Edelkamp 1 Overview Different state space formalisms, including labelled, implicit and explicit weighted graph representations Alternative formalisms:

### * 8 Groups, with Appendix containing Rings and Fields.

* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that

### Computability Theory. CS215, Lecture 6,

Computability Theory CS215, Lecture 6, 2000 1 The Birth of Turing Machines At the end of the 19th century, Gottlob Frege conjectured that mathematics could be built from fundamental logic In 1900 David

### LIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon

LIP Laboratoire de l Informatique du Parallélisme Ecole Normale Supérieure de Lyon Institut IMAG Unité de recherche associée au CNRS n 1398 Inversion of 2D cellular automata: some complexity results runo

### Turing machines Finite automaton no storage Pushdown automaton storage is a stack What if we give the automaton a more flexible storage?

Turing machines Finite automaton no storage Pushdown automaton storage is a stack What if we give the automaton a more flexible storage? What is the most powerful of automata? In this lecture we will introduce

### Variations of the Turing Machine

Variations of the Turing Machine 1 The Standard Model Infinite Tape a a b a b b c a c a Read-Write Head (Left or Right) Control Unit Deterministic 2 Variations of the Standard Model Turing machines with:

### 1 Partitions and Equivalence Relations

Today we re going to talk about partitions of sets, equivalence relations and how they are equivalent. Then we are going to talk about the size of a set and will see our first example of a diagonalisation

### Variants of Turing Machine (intro)

CHAPTER 3 The Church-Turing Thesis Contents Turing Machines definitions, examples, Turing-recognizable and Turing-decidable languages Variants of Turing Machine Multi-tape Turing machines, non-deterministic

### arxiv: v2 [cs.fl] 21 Sep 2016

A A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications Kamalika Bhattacharjee, Indian Institute of Engineering Science and Technology, Shibpur Nazma Naskar, Seacom Engineering

### CpSc 421 Homework 9 Solution

CpSc 421 Homework 9 Solution Attempt any three of the six problems below. The homework is graded on a scale of 100 points, even though you can attempt fewer or more points than that. Your recorded grade

### Sphere Packings, Coverings and Lattices

Sphere Packings, Coverings and Lattices Anja Stein Supervised by: Prof Christopher Smyth September, 06 Abstract This article is the result of six weeks of research for a Summer Project undertaken at the

### b 0 + b 1 z b d z d

I. Introduction Definition 1. For z C, a rational function of degree d is any with a d, b d not both equal to 0. R(z) = P (z) Q(z) = a 0 + a 1 z +... + a d z d b 0 + b 1 z +... + b d z d It is exactly

### A Brief History of Cellular Automata

A Brief History of Cellular Automata PALASH SARKAR Indian Statistical Institute Cellular automata are simple models of computation which exhibit fascinatingly complex behavior. They have captured the attention

### acs-07: Decidability Decidability Andreas Karwath und Malte Helmert Informatik Theorie II (A) WS2009/10

Decidability Andreas Karwath und Malte Helmert 1 Overview An investigation into the solvable/decidable Decidable languages The halting problem (undecidable) 2 Decidable problems? Acceptance problem : decide

### MTAT Complexity Theory October 13th-14th, Lecture 6

MTAT.07.004 Complexity Theory October 13th-14th, 2011 Lecturer: Peeter Laud Lecture 6 Scribe(s): Riivo Talviste 1 Logarithmic memory Turing machines working in logarithmic space become interesting when

### Exact results for deterministic cellular automata traffic models

June 28, 2017 arxiv:comp-gas/9902001v2 2 Nov 1999 Exact results for deterministic cellular automata traffic models Henryk Fukś The Fields Institute for Research in Mathematical Sciences Toronto, Ontario

### Undecidable Problems and Reducibility

University of Georgia Fall 2014 Reducibility We show a problem decidable/undecidable by reducing it to another problem. One type of reduction: mapping reduction. Definition Let A, B be languages over Σ.

### Math 300: Final Exam Practice Solutions

Math 300: Final Exam Practice Solutions 1 Let A be the set of all real numbers which are zeros of polynomials with integer coefficients: A := {α R there exists p(x) = a n x n + + a 1 x + a 0 with all a

### Economics 204 Fall 2011 Problem Set 1 Suggested Solutions

Economics 204 Fall 2011 Problem Set 1 Suggested Solutions 1. Suppose k is a positive integer. Use induction to prove the following two statements. (a) For all n N 0, the inequality (k 2 + n)! k 2n holds.

### The Emptiness Problem for Valence Automata or: Another Decidable Extension of Petri Nets

The Emptiness Problem for Valence Automata or: Another Decidable Extension of Petri Nets Georg Zetzsche Technische Universität Kaiserslautern Reachability Problems 2015 Georg Zetzsche (TU KL) Emptiness

### cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 13 CHAPTER 4 TURING MACHINES 1. The definition of Turing machine 2. Computing with Turing machines 3. Extensions of Turing

### Busch Complexity Lectures: Turing s Thesis. Costas Busch - LSU 1

Busch Complexity Lectures: Turing s Thesis Costas Busch - LSU 1 Turing s thesis (1930): Any computation carried out by mechanical means can be performed by a Turing Machine Costas Busch - LSU 2 Algorithm:

### The Hurewicz Theorem

The Hurewicz Theorem April 5, 011 1 Introduction The fundamental group and homology groups both give extremely useful information, particularly about path-connected spaces. Both can be considered as functors,

### Chapter 2 Algorithms and Computation

Chapter 2 Algorithms and Computation In this chapter, we first discuss the principles of algorithm and computation in general framework, common both in classical and quantum computers, then we go to the

### Rings and Fields Theorems

Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)

### Verifying whether One-Tape Turing Machines Run in Linear Time

Electronic Colloquium on Computational Complexity, Report No. 36 (2015) Verifying whether One-Tape Turing Machines Run in Linear Time David Gajser IMFM, Jadranska 19, 1000 Ljubljana, Slovenija david.gajser@fmf.uni-lj.si

### A Complexity Theory for Parallel Computation

A Complexity Theory for Parallel Computation The questions: What is the computing power of reasonable parallel computation models? Is it possible to make technology independent statements? Is it possible

### Q = Set of states, IE661: Scheduling Theory (Fall 2003) Primer to Complexity Theory Satyaki Ghosh Dastidar

IE661: Scheduling Theory (Fall 2003) Primer to Complexity Theory Satyaki Ghosh Dastidar Turing Machine A Turing machine is an abstract representation of a computing device. It consists of a read/write

### Cellular Automata in the Cantor, Besicovitch, and Weyl Topological Spaces

Cellular Automata in the Cantor, Besicovitch, and Weyl Topological Spaces François Blanchard Institut de Mathématiques de Luminy - CNRS, Case 930, 163 avenue de Luminy, F-13288 Marseille cedex 09, France

### Computational Models Lecture 9, Spring 2009

Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Computational Models Lecture 9, Spring 2009 Reducibility among languages Mapping reductions More undecidable

### GOTO S CONSTRUCTION AND PASCAL S TRIANGLE: NEW INSIGHTS INTO CELLULAR AUTOMATA SYNCHRONIZATION JEAN-BAPTISTE YUNÈS 1

Journées Automates Cellulaires 2008 (Uzès), pp. 195-203 GOTO S CONSTRUCTION AND PASCAL S TRIANGLE: NEW INSIGHTS INTO CELLULAR AUTOMATA SYNCHRONIZATION JEAN-BAPTISTE YUNÈS 1 1 LIAFA, Université Paris Diderot/CNRS

### Computability Theory

Computability Theory Cristian S. Calude May 2012 Computability Theory 1 / 1 Bibliography M. Sipser. Introduction to the Theory of Computation, PWS 1997. (textbook) Computability Theory 2 / 1 Supplementary

### Learning Cellular Automaton Dynamics with Neural Networks

Learning Cellular Automaton Dynamics with Neural Networks N H Wulff* and J A Hertz t CONNECT, the Niels Bohr Institute and Nordita Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark Abstract We have trained

### Random Lifts of Graphs

27th Brazilian Math Colloquium, July 09 Plan of this talk A brief introduction to the probabilistic method. A quick review of expander graphs and their spectrum. Lifts, random lifts and their properties.

### Computational Complexity: A Modern Approach. Draft of a book: Dated January 2007 Comments welcome!

i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University complexitybook@gmail.com Not to be reproduced or distributed

### PERCOLATION GAMES, PROBABILISTIC CELLULAR AUTOMATA, AND THE HARD-CORE MODEL

PERCOLATION GAMES, PROBABILISTIC CELLULAR AUTOMATA, AND THE HARD-CORE MODEL ALEXANDER E. HOLROYD, IRÈNE MARCOVICI, AND JAMES B. MARTIN Abstract. Let each site of the square lattice Z 2 be independently

### 47-831: Advanced Integer Programming Lecturer: Amitabh Basu Lecture 2 Date: 03/18/2010

47-831: Advanced Integer Programming Lecturer: Amitabh Basu Lecture Date: 03/18/010 We saw in the previous lecture that a lattice Λ can have many bases. In fact, if Λ is a lattice of a subspace L with

### 1 Computational Problems

Stanford University CS254: Computational Complexity Handout 2 Luca Trevisan March 31, 2010 Last revised 4/29/2010 In this lecture we define NP, we state the P versus NP problem, we prove that its formulation

### Week 4-5: Generating Permutations and Combinations

Week 4-5: Generating Permutations and Combinations February 27, 2017 1 Generating Permutations We have learned that there are n! permutations of {1, 2,...,n}. It is important in many instances to generate

### Uniquely Universal Sets

Uniquely Universal Sets 1 Uniquely Universal Sets Abstract 1 Arnold W. Miller We say that X Y satisfies the Uniquely Universal property (UU) iff there exists an open set U X Y such that for every open

### Topological dynamics of one-dimensional cellular automata

Topological dynamics of one-dimensional cellular automata Petr Kůrka Contents 1 Glossary 1 2 Definition 2 3 Introduction 2 4 Topological dynamics 3 5 Symbolic dynamics 4 6 Equicontinuity 6 7 Surjectivity

### One-to-one functions and onto functions

MA 3362 Lecture 7 - One-to-one and Onto Wednesday, October 22, 2008. Objectives: Formalize definitions of one-to-one and onto One-to-one functions and onto functions At the level of set theory, there are

### Cellular automata as emergent systems and models of physical behavior

Cellular automata as emergent systems and models of physical behavior Jason Merritt December 19, 2012 Abstract Cellular automata provide a basic model for complex systems generated by simplistic rulesets.

### Baltic Way 2003 Riga, November 2, 2003

altic Way 2003 Riga, November 2, 2003 Problems and solutions. Let Q + be the set of positive rational numbers. Find all functions f : Q + Q + which for all x Q + fulfil () f ( x ) = f (x) (2) ( + x ) f

### An algebraic view of topological -machines

An algebraic view of topological -machines Luke Grecki Graduate Group in Applied Mathematics lgrecki@math.ucdavis.edu June 8, 2010 1 Contents 1 Semigroups 3 2 Semigroups of automata 4 3 -machine words

### ON COMPUTAMBLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHENIDUGSPROBLEM. Turing 1936

ON COMPUTAMBLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHENIDUGSPROBLEM Turing 1936 Where are We? Ignoramus et ignorabimus Wir mussen wissen Wir werden wissen We do not know We shall not know We must know

### Homework 8. a b b a b a b. two-way, read/write

Homework 8 309 Homework 8 1. Describe a TM that accepts the set {a n n is a power of 2}. Your description should be at the level of the descriptions in Lecture 29 of the TM that accepts {ww w Σ } and the

### CS 208: Automata Theory and Logic

CS 208: Automata Theory and Logic b a a start A x(la(x) y.(x < y) L b (y)) B b Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 19 Logistics Course Web-page:

### @igorwhiletrue

Abstrakte Maschinen @igorwhiletrue Programming is hard Why? Link between our universe and computational universe Cellular automata are self-replicating abstract machines Humans are self-replicating biological

### Adequate set of connectives, logic gates and circuits

Adequate set of connectives, logic gates and circuits Lila Kari University of Waterloo Adequate set of connectives, logic gates and circuits CS245, Logic and Computation 1 / 59 Connectives We have mentioned

### Isometries. Chapter Transformations of the Plane

Chapter 1 Isometries The first three chapters of this book are dedicated to the study of isometries and their properties. Isometries, which are distance-preserving transformations from the plane to itself,

### CS 361 Meeting 26 11/10/17

CS 361 Meeting 26 11/10/17 1. Homework 8 due Announcements A Recognizable, but Undecidable Language 1. Last class, I presented a brief, somewhat inscrutable proof that the language A BT M = { M w M is

### Undecidability of the validity problem

Undecidability of the validity problem We prove the undecidability of the validity problem for formulas of predicate logic with equality. Recall: there is an algorithm that given a formula of predicate

### We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

More Books At www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com www.goalias.blogspot.com

### Lecture 14: State Tables, Diagrams, Latches, and Flip Flop

EE210: Switching Systems Lecture 14: State Tables, Diagrams, Latches, and Flip Flop Prof. YingLi Tian Nov. 6, 2017 Department of Electrical Engineering The City College of New York The City University

ADMISSIBILITY OF KNEADING SEQUENCES AND STRUCTURE OF HUBBARD TREES FOR QUADRATIC POLYNOMIALS HENK BRUIN AND DIERK SCHLEICHER Abstract. Hubbard trees are invariant trees connecting the points of the critical

### Cellular Automata. Introduction

Cellular Automata 1983 Introduction It appears that the basic laws of physics relevant to everyday phenomena are now known. Yet there are many everyday natural systems whose complex structure and behavior

### From Sequential Circuits to Real Computers

1 / 36 From Sequential Circuits to Real Computers Lecturer: Guillaume Beslon Original Author: Lionel Morel Computer Science and Information Technologies - INSA Lyon Fall 2017 2 / 36 Introduction What we

### Investigations of Game of Life Cellular Automata rules on Penrose Tilings: lifetime, ash, and oscillator statistics

Investigations of Game of Life Cellular Automata rules on Penrose Tilings: lifetime, ash, and oscillator statistics Nick Owens, Susan Stepney 2 Department of Electronics, University of York, UK 2 Department

### On the coinductive nature of centralizers

On the coinductive nature of centralizers Charles Grellois INRIA & University of Bologna Séminaire du LIFO Jan 16, 2017 Charles Grellois (INRIA & Bologna) On the coinductive nature of centralizers Jan

### The Quantum and Classical Complexity of Translationally Invariant Tiling and Hamiltonian Problems

The Quantum and Classical Complexity of Translationally Invariant Tiling and Hamiltonian Problems Daniel Gottesman Perimeter Institute for Theoretical Physics Waterloo, Ontario, Canada dgottesman@perimeterinstitute.ca

### Neural Turing Machine. Author: Alex Graves, Greg Wayne, Ivo Danihelka Presented By: Tinghui Wang (Steve)

Neural Turing Machine Author: Alex Graves, Greg Wayne, Ivo Danihelka Presented By: Tinghui Wang (Steve) Introduction Neural Turning Machine: Couple a Neural Network with external memory resources The combined

### FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM. Contents

FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM SAMUEL BLOOM Abstract. In this paper, we define the fundamental group of a topological space and explore its structure, and we proceed to prove Van-Kampen

### Turing Machines Part III

Turing Machines Part III Announcements Problem Set 6 due now. Problem Set 7 out, due Monday, March 4. Play around with Turing machines, their powers, and their limits. Some problems require Wednesday's

### MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

### GROUPS AND THEIR REPRESENTATIONS. 1. introduction

GROUPS AND THEIR REPRESENTATIONS KAREN E. SMITH 1. introduction Representation theory is the study of the concrete ways in which abstract groups can be realized as groups of rigid transformations of R

### CSCI3390-Assignment 2 Solutions

CSCI3390-Assignment 2 Solutions due February 3, 2016 1 TMs for Deciding Languages Write the specification of a Turing machine recognizing one of the following three languages. Do one of these problems.

### Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................