Automata 2005, Gda´ nsk 3-5 September 2005 Revised version, 7 September 2005 Conjugacy Results for Cellular Automata Silvio Capobianco, Universit` a degli Studi di Roma “La Sapienza”, capobian@mat.uniroma1.it 1
Introduction A cellular automaton is basically a “short” encoding of a transformation between colorings of the nodes of a regular graph. Transformations having such encodings, in turn, define dynamical systems. This leads to some interesting problems: 1. given a dynamical system, determine whether it can be described by a CA; 2. in this case, find which classes of CA are fit to describe it. 2
References • S. Capobianco, Structure and invertibility in cellular automata, Tesi di Dottorato • S. Capobianco, Cellular automata over semi-direct group products: reduction and invertibility results, submitted for publication • F . Fiorenzi, Cellular automata and strongly irreducible shifts of finite type, Theor. Comp. Sci. 299 (2003) 477–493 • D. Lind, B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press (1995) • A. Mach` ı, F . Mignosi, Garden of Eden configurations for cellular automata on Cayley graphs on groups, SIAM J. Disc. Math. 6 (1993) 44–56 • T. Toffoli, N. Margolus, Invertible cellular automata: A review, Physica D 45 (1990) 229–253 3
Evolution of the concept • von Neumann, 1950s: uniform local rule on the plane • Richardson, 1972: d -dimensional cellular automata • Hardy, de Pazzis, Pomeau 1976: lattice gas cellular automata • Mach` ı and Mignosi, 1993: cellular automata over groups of polynomial growth • Lind and Marcus, 1995: “sliding block codes” between shift subspaces of A Z • Fiorenzi, 2000: cellular automata over arbitrary finitely generated groups 4
The original form • Infinite square grid, identifiable with Z 2 . • A finite number of states for each point of the grid. • A finite set { n 1 , . . . , n k } ⊆ Z 2 . • A law of the form ( F ( c )) x = f ( c x + n 1 , . . . , c x + n k ) . Hypercubic grid of arbitrary dimension d ⇒ “classical” CA of the form: � d, Q, N , f � 5
Cellular automata as presentations Cellular automata are not dynamical systems per se . Rather, they are “short” ways to encode dynamics which can be very complex. Thus, it is more correct to say that CA are presentations of dynamical systems. 6
A special property A = � d, Q, N , f � ; φ : Q Z d × Z d → Q Z d : ( φ ( c, x )) y = c x + y Two properties of φ : 1. A ’s global evolution function commutes with φ ; 2. the map: π : Q Z d → Q ; π ( c ) = c 0 is continuous and satisfies: ∀ c 1 � = c 2 ∈ Q Z d ∃ g ∈ G : π ( φ ( c 1 , g )) � = π ( φ ( c 2 , g )) ( X, F ) “isomorphic” to the one described by A ⇒ ∃ an action of Z d on X with similar properties. 7
A special property (continued) Conjecture 1: (Toffoli, 1980s) Every dynamical system admitting of an action of Z d on its phase space having properties similar to 1 and 2, also admits of a presentation as a d -dimensional cellular automaton. Status: plausible, but missing an important point. 8
Terminology Alphabet : 1 < | A | < ∞ . G group, S ⊆ G : g ∈ G : ∃ n ∈ N , s 1 , . . . , s n ∈ S ∪ S − 1 : g = s 1 . . . s n � � � S � = G = � S � for | S | < ∞ : G finitely generated (f.g.). Cayley graph of G w.r.t. S : • V = G , ( g, gs ) , g ∈ G, s ∈ ( S ∪ S − 1 ) \ { 1 G } � � • E = . Distance of g and h w.r.t. S : minimum length of a path from g to h . 9
Examples of Cayley graphs The square grid is the Cayley graph of G = Z 2 with respect to S = { (1 , 0) , (0 , 1) } . 10
Examples of Cayley graphs (continued) The hexagonal grid is (isomorphic to) the Cayley graph of G = Z 2 with respect to S = { (1 , 0) , (1 , 1) , (0 , 1) } . 11
Terminology (continued) A alphabet, G f.g. group ⇒ c ∈ A G configuration . c ∈ A G : g ∈ G �→ c g ∈ A A G is given the product topology: • G finite ⇒ A G discrete; • G countable ⇒ A G Cantor. c 1 , c 2 “near” ⇔ “equal on a large disk centered in the origin”. 12
Terminology (continued) Action of G on X : 1. φ : X × G → X ; 2. φ ( φ ( x, g 1 ) , g 2 ) = φ ( x, g 1 g 2 ) for all x, g 1 , g 2 ; 3. φ ( x, 1 G ) = x for all x . φ continuous : x �→ φ ( x, g ) continuous ∀ g . ( σ G ( c, g )) h = c gh natural action of G on A G . Translations: the maps c �→ σ G ( c, g ) . 13
Shift subspaces Origin: Symbolic dynamics. σ : A Z → A Z the shift map : ( σ ( c )) x = c x +1 X ⊆ A Z : ( X, σ ) subsystem of ( A Z , σ ) . Generalization: X ⊆ A G such that ( X, σ G ( · , g )) subsystem of ( A G , σ G ( · , g )) ∀ g ∈ G . A G is called the full shift . σ G satisfies properties 1 and 2 when restricted to a shift subspace ⇒ it cannot tell the full shift from an arbitrary shift subspace. This is precisely the point missed by Conjecture 1. 14
Characterization of shift subspaces Pattern: p : E → A , E ⊆ G finite. p ∈ A E occurs in c ∈ A G : ( σ G ( c, g )) | E = p for some g ∈ G . X ⊆ A G . The following are equivalent: 1. X is a shift subspace; 2. X is closed and translation invariant; 3. ∃ a set F of patterns such that: X = X F = { c ∈ A G : ∀ p ∈ F , g ∈ G : p ∈ A E , ( σ G ( c, g )) | E � = p } X = X F with F finite: X is of finite type . 15
UL-definable functions F : A G → A G UL-definable (i.e., uniformly locally definable) ⇔ ∃ N ⊆ G , |N| < ∞ , f : A N → A s.t. ∀ c ∈ A G , g ∈ G � � � � ( F ( c )) g = f ( σ G ( c, g )) |N = f c gn 1 , . . . , c gn |N | G f.g. ⇒ N can be a disk . Hedlund’s Theorem: X ⊆ A G shift subspace. F : X → A G : 1. continuous 2. σ G ( F ( c ) , g ) = F ( σ G ( c, g )) ∀ g ∈ G, c ∈ X ⇒ F is the restriction to X of a UL-definable function. 16
The reasons for a broader definition CA are uniform local presentations of global dynamics. Locality is linked to invariance by translation. Uniformity is linked to compactness. Thus: Shift subspaces still possess the requirements for the definition of a dynamics in uniform, local terms. 17
Generalized cellular automata A = � X, N , f � with: • X ⊆ A G shift subspace for some G f.g., 1 < | A | < ∞ ; • N ⊆ G, |N| < ∞ ; • f : A N → A : F ( X ) ⊆ X , where: � � ( F ( c )) g = f ( σ G ( c, g ) |N ) G : tessellation group of A . X : support of A . ( X, F ) : associate dynamical system of A . A is full if X = A G . 18
Presenting dynamical systems by generalized CA Topological conjugacy from ( X, F ) to ( X ′ , F ′ ) : homeomorphism t : X → X ′ such that t ◦ F = F ′ ◦ t : F X X t t F’ X’ X’ � X, N , f � presentation of ( X ′ , F ′ ) ⇔ ( X, F ) topologically conjugate to ( X ′ , F ′ ) . Shift subspaces with a countable number of elements exist ⇒ the definition is actually broader. 19
A not-so-trivial example of presentation by CA Let A = { 0 , 1 } , G = Z , N = { +1 } , f (1 �→ a ) = a . Let σ : { 0 , 1 } Z → { 0 , 1 } Z be the shift map. { 0 , 1 } Z , σ − 1 � � { 0 , 1 } Z , N , f � � Then, is a presentation of . The topological conjugacy is given by the reversing map: ( r ( c )) x = c − x 20
Disambiguation Lind and Marcus 1995: conjugacy between shift subspaces X, Y ⊆ A Z : 1. UL-definable F : A Z → A Z 2. F injective over X 3. F ( X ) = Y . Equivalently: F topological conjugacy between ( X, σ ) and ( Y, σ ) ( shift conjugacy ). The reversing map is not UL-definable, thus it is not a shift conjugacy. 21
Discernibility A alphabet, φ : X × G → X action. X discernible over A by φ ⇔ ∃ π : X → A : • π is continuous • ∀ x 1 � = x 2 ∈ X ∃ g ∈ G : π ( φ ( x 1 , g )) � = π ( φ ( x 2 , g )) . Conjecture 1 can then be restated as such: Commutation with and discernibility by the same group action characterize dynamical systems presentable as CA. 22
Theorem 1 ( X, F ) compact dynamical system. The following are equivalent: 1. ( X, F ) has a presentation as a generalized cellular automaton; 2. ∃ A , G f.g., φ : X × G → X continuous s.t.: (a) F commutes with φ ; (b) X is discernible over A by φ . In this case, ( X, F ) has a presentation as a generalized CA with alphabet A and tessellation group G . In other words: Conjecture 1 is true for generalized CA. 23
Discussion Condition 2 only implies that X is homeomorphic to a shift subspace of A G . Reason: • φ and σ G have the same role • ⇒ neither can tell an arbitrary shift subspace from A G ⇒ Dynamical systems presentable as full CA must possess properties in addition to those in condition 2. 24
Discussion (continued) No special properties of G used in proof of Theorem 1. (Aside from being finitely generated.) ⇒ Further restrictions on the tessellation group can be done, yielding corresponding variants of Theorem 1. Important if one wants all translations to be global evolution functions of CA, possible iff G is Abelian. (The global map induced by f ( g �→ a ) = a obeys ( F ( c )) h = c hg , which in general is not c gh unless g is central.) 25
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