Uniform continuity of relations and nondeterministic cellular automata

Theoretical Computer Science 673 (2017) 19–29

Contents lists available at ScienceDirect

Theoretical Computer Science www.elsevier.com/locate/tcs

Uniform continuity of relations and nondeterministic cellular automata Hitoshi Furusawa Department of Mathematics and Computer Science, Kagoshima University, Kagoshima, 890-0065, Japan

a r t i c l e

i n f o

Article history: Received 15 September 2016 Received in revised form 30 January 2017 Accepted 2 February 2017 Available online 16 February 2017 Communicated by J. Kari

a b s t r a c t This paper provides a definition of uniformly continuous relations between uniform spaces and studies relationships with continuous relations between topological spaces associated with the uniform spaces. Using the notion of uniform continuity, we give a global characterisation of nondeterministic cellular automata on a group and a possibly infinite alphabet. © 2017 Elsevier B.V. All rights reserved.

Keywords: Uniform spaces Uniformly continuous relations Nondeterministic cellular automata Global characterisation

1. Introduction

A cellular automaton is a transition system on a given configuration space whose behavior is prescribed by a local rule. Such a transition system on the configuration space consisting of all maps from a fixed group to a fixed alphabet (which is just a set) is called cellular automaton over the group and the alphabet. When the local rule is a map, the cellular automaton is called deterministic and when it is a binary relation, the cellular automaton is called nondeterministic. There are some characterisations of cellular automata not mentioning their local rules. We call such characterisations global. In [6], Hedlund gave a global characterisation of deterministic cellular automata over a finite alphabet by using the notion of continuous functions with respect to a certain topological space. It is known that this result may be extended:

• deterministic cellular automata over a group and a possibly infinite alphabet are characterised globally by using the notion of uniformly continuous functions rather than continuous functions;

• nondeterministic cellular automata over a group and a finite alphabet are characterised globally by using the notion of continuous relations rather than continuous functions. The first extension was given by Ceccherini-Silberstein and Coornaert [2,3]. The second one was given by Richardson [10] and [5] proved it in a relation-algebraic style. The aim of this paper is to provide a global characterisation of nondeterministic cellular automata over a group and a possibly infinite alphabet by introducing a notion of uniformly continuous relations. This extends the characterisation in [10,5].

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.tcs.2017.02.004 0304-3975/© 2017 Elsevier B.V. All rights reserved.

20

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

In order to reach our objective, a notion of uniformly continuous relations is needed. In [9], Pauly and Ziegler provided some notions of continuity for relations with respect to metric spaces. We generalise one of them to a notion for general uniform spaces and study some basic properties. This paper is organised as follows. Section 2 provides basic notations and notions of relations and uniform structures. In section 3 we introduce the notion of uniformly continuous relations and show properties about relationship between continuity and uniform continuity. In Section 4 we define nondeterministic cellular automata over a group and a possibly infinite alphabet, and give a (not yet global) characterisation of them. Then in Section 5 we give a global characterisation and show that it extends the characterisation in [10,5] by using the general properties given in Section 3. Section 6 summarises this paper. 2. Preliminary Let R ⊆ X × Y be a relation from a set X to a set Y . The converse of R, the composition of relations R ⊆ X × Y and R  ⊆ Y × Z , and the identity relation on X will be denoted by R −1 , R ; R  , and  X , respectively. For x ∈ X , A ⊆ X , y ∈ Y and B ⊆ Y we define

[x] R = { y ∈ Y | (x, y ) ∈ R } R [ y ] = {x ∈ X | (x, y ) ∈ R }

 [ A ] R = a∈ A [a] R R [ B ] = b∈ B R [b] .

Of course, [x] R = R −1 [x], [ A ] R = R −1 [ A ], R [ y ] = [ y ] R −1 and R [ B ] = [ B ] R −1 . For a map f : X → Y , we also write f (x) and f ( A ) for the function value at x and image of A, respectively. d( R ) and r ( R ) denote domain and range of R, namely,

d( R ) = R [ Y ]

r(R ) = [ X ]R ,

respectively. So, d( R ) = r ( R −1 ) and r ( R ) = d( R −1 ). A uniform structure on a set X is a nonempty set U of relations on X satisfying the following conditions:

• • • • •

if if if if if

V V V V V

∈ U , then  X ⊆ V ; ∈ U and V ⊆ V  ⊆ X × X , then V  ∈ U ; ∈ U and W ∈ U , then V ∩ W ∈ U ; ∈ U , then V −1 ∈ U ; ∈ U , then there exists W ∈ U such that W ; W ⊆ V .

A pair of a set and a uniform structure on it is called a uniform space. It is obvious that the set of relations on X containing

 X is a uniform structure on X and is called discrete. If d is a metric on X , the uniform structure on X associated with d is the set of all subsets W ⊆ X × X such that V ε ⊆ W for some ε > 0, where V ε = {(x, y ) ∈ X × X | d(x, y ) < ε }. A subset B ⊆ U satisfying

∀ W ∈ U . ∃ V ∈ B. V ⊆ W is called a base of U . Thus the singleton set { X } is a base for the discrete uniform structure on X and so is the set { V ε | ε > 0} for the uniform structure associated with a metric. Let ( X , U X ) and (Y , UY ) be uniform spaces. A map f : X → Y satisfying

∀ W ∈ UY . ∃ V ∈ U X . ( f × f )( V ) ⊆ W is called uniformly continuous, where f × f is a map from X × X to Y × Y defined by ( f × f )(x, x ) = ( f (x), f (x )) for all (x, x ) ∈ X × X . If B ⊆ U X and B  ⊆ UY are bases of uniform structures U X and UY , respectively, a map f is uniformly continuous iff

∀ W ∈ B . ∃ V ∈ B. ( f × f )( V ) ⊆ W . Note that this condition is equivalent to ( f × f )−1 ( W ) ∈ U X for each entourage W ∈ UY . Let ( X , U ) be a uniform space. We define the family OU of subsets of X as follows.

O ∈ OU ⇐⇒ ∀x ∈ O . ∃ W ∈ U . W [x] = O Then ( X , OU ) is a topological space and is called topological space associated with the uniform structure U . A subset N ⊆ X is a neighborhood of x ∈ X for this topology iff there exists an entourage W such that N = W [x]. Note that the topological space associated with the discrete uniform structure is the discrete space. Also note that the topology defined by the uniform structure associated with a metric d is the topology defined by d. It is known that every uniformly continuous map is continuous with respect to the topologies associated with the uniform structure on its domain and codomain. Conversely, every continuous map whose domain is compact is uniformly continuous. [3, Appendix A and B] explains topologies and uniform structures related to deterministic cellular automata on Groups to the point. We refer more details about topologies and uniform structures to [7,8].

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

21

3. Continuity and uniform continuity of relations According to Brattka and Hertling [1], we give a definition of continuous relations. Let ( X , O X ) and (Y , OY ) be topological spaces and let R ⊆ X × Y . R is continuous at (x, y ) ∈ R iff for all neighborhoods N of y there exists a neighborhood N  of x such that

∀x ∈ N  ∩ d( R ). N ∩ [x ] R = ∅ . R is continuous iff R is continuous at every point (x, y ) ∈ R. [1, Lemma 2.2] shows that the condition is equivalent to

∀ O ∈ OY . ∃ O  ∈ O X . R [ O ] = O  ∩ d( R ) . Hence a total relation R is continuous iff R [ O ] ∈ O X for all O ∈ OY . Therefore all continuous functions are continuous relations. [1, Example 2.5] contains a few continuous relations. Next, generalising definition of uniformly (strongly) continuous relations between metric spaces given by Pauly and Ziegler [9], we give a definition of uniformly continuous relations between uniform spaces in general. Let ( X , U X ) and (Y , UY ) be uniform spaces. R ⊆ X × Y is uniformly continuous iff for all W ∈ UY there exists W  ∈ U X such that

∀x ∈ d( R ). ∀ y ∈ [x] R . ∀x ∈ W  [x] ∩ d( R ). W [ y ] ∩ [x ] R = ∅ . Noting that W [ y ] ∩ [x ] R = ∅ iff (x , y ) ∈ R ; W , we have the following equivalent condition:

∀ W ∈ UY . ∃ W  ∈ U X . W  ; R ∩ (d( R ) × Y ) ⊆ R ; W . Hence a total relation is uniformly continuous iff for all W ∈ UY there exists W  ∈ U X such that W  ; R ⊆ R ; W . Moreover, uniformly continuous functions are uniformly continuous relations since

W  ; f ⊆ f ; W =⇒ ( f × f )( W  ) = f −1 ; W  ; f ⊆ f −1 ; f ; W ⊆ W ( f × f )( W  ) ⊆ W =⇒ W  ; f ⊆ f ; f −1 ; W  ; f = f ;(( f × f )( W  )) ⊆ f ; W hold for a map f : X → Y by f −1 ; f ⊆ Y and  X ⊆ f ; f −1 . Let B and B  be bases of the uniform structures of X and Y , respectively. Then a relation R ⊆ X × Y is uniformly continuous iff

∀ V  ∈ B . ∃ V ∈ B. V ; R ∩ (d( R ) × Y ) ⊆ R ; V  . [9, Example 2.3] contains a uniformly continuous relation. The following property is a generalisation of [9, Lemma 2.2.f]. Proposition 3.1. Let X = ( X , U X ) and Y = (Y , UY ) be uniform spaces. If a relation R ⊆ X × Y is uniformly continuous, it is continuous with respect to the topologies on X and Y associated with the uniform structures. Proof. Suppose that R ⊆ X × Y is uniformly continuous and let (x, y ) ∈ R and N be a neighborhood of y. Then, there exists W ∈ UY such that N = W [ y ]. Since R is uniformly continuous, there exists W  ∈ U X such that W  ; R ∩ (d( R ) × Y ) ⊆ R ; W . Let N  = W  [x] then N  is a neighborhood of x. For x ∈ N  , (x , y ) ∈ W  ; R since (x , x) ∈ W  and (x, y ) ∈ R. Thus, if x ∈ d( R ), (x , y ) ∈ R ; W . This implies N ∩ [x ] R = W [ y ] ∩ [x ] R = ∅ for each x ∈ N  . 2 A counterexample of the converse of Proposition 3.1 is given by Pauly and Ziegler [9, Example 2.3]. Compactness of continuous relations implies uniform continuity. The following property is a generalisation of [9, Lemma 2.2.j]. Proposition 3.2. Let X = ( X , U X ) and Y = (Y , UY ) be uniform spaces. If a relation R ⊆ X × Y is continuous with respect to the topologies on X and Y associated with the uniform structures and compact in the product space X × Y , it is uniformly continuous. Proof. Let R ⊆ X × Y be continuous and compact and let W ∈ UY . By the definition of uniform structure, we may find a symmetric entourage S such that S ; S ⊆ W . Since R is continuous, there exists, for each (x, y ) ∈ R, V x ∈ U X such that

∀x ∈ V x [x] ∩ d( R ). S [ y ] ∩ [x ] R = ∅ . By the definition of uniform structure, we may find W x ∈ U X such that W x ; W x ⊆ V x , and then, the set W x [x] × S [ y ] is a neighborhood of (x, y ) ∈ R. By compactness of R there exists a finite subset F ⊆ R such that R ⊆ (x, y )∈ F W x [x] × S [ y ].  Note that (x, y ) ∈ (x, y )∈ F W x [x] × S [ y ] iff there exists (a, b) ∈ F such that (x, a) ∈ W a and (b, y ) ∈ S since S is symmetric. Let

W =

 a∈d( F )

W a .

22

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

Then, by the definition of uniform structure, W  ∈ U X . Suppose that x ∈ d( R ) and (x, y ) ∈ W  ; R. Then, there exists x ∈ X such that (x, x ) ∈ W  and (x , y ) ∈ R. So, we have

∃(a, b) ∈ F . (x , a) ∈ W a and (b, y ) ∈ S , and for this a

(x, x ) ∈ W 

=⇒ =⇒ =⇒

(x, x ) ∈ W a ( W  ⊆ W a )   (x, a) ∈ W a ; W a ((x , a) ∈ W a ) ( W a ; W a ⊆ V a ) (x, a) ∈ V a

holds. Hence, by (a, b) ∈ R and x ∈ V a [a] ∩ d( R ), we have S [b] ∩ [x] R = ∅. So, (x, b) ∈ R ; S and this implies (x, y ) ∈ R ; S ; S. Therefore (x, y ) ∈ R ; W . 2 The following property shows that the domain restriction preserves uniform continuity. Proposition 3.3. Let X = ( X , U X ) and Y = (Y , UY ) be uniform spaces. If a relation R ⊆ X × Y is uniformly continuous, so is R ∩ ( A × Y ) for each subset A ⊆ X . Proof. Suppose that R ⊆ X × Y is uniformly continuous and let W ∈ UY . Then, there exists W  ∈ U X such that

W  ; R ∩ (d( R ) × Y ) ⊆ R ; W . Clearly, such W  also satisfies

W  ;( R ∩ ( A × Y )) ∩ ((d( R ) ∩ A ) × Y ) ⊆ R ; W ∩ ( A × Y ) for each A ⊆ X . Thus, by d( R ) ∩ A = d( R ∩ ( A × Y )) and R ; W ∩ ( A × Y ) = ( R ∩ ( A × Y )); W , R ∩ ( A × Y ) is uniformly continuous. 2 4. Nondeterministic cellular automata A nondeterministic cellular automaton roughly is a relation on the set of configurations determined by a local rule which presents options of the state of the transited configuration at a given cell in accordance with the states of the original configuration on a finite neighborhood of the given cell. This section provides a formal definition of nondeterministic cellular automata and a characterisation of them. The characterisation given here is a nondeterministic variant of [3, Proposition 1.4.6], that is to say, not yet global. 4.1. Configurations Let G be a group with unit 1G and A a set. We write A G for the set of all maps from G to A. The set A is called the alphabet and the group G is called the universe. The elements of A are called the states or the symbols, the elements of G are called the cells, and the elements of A G are called the configurations. For a configuration x and a cell g, the value x( g ) stands for the state of x at g. The projection from A G to A which maps x to x( g ) is denoted by π g . For a subset H ⊆ G and a configuration x ∈ A G the restriction of x to H is denoted by x| H and the relation E H on A G is defined by

(x, y ) ∈ E H ⇐⇒ x| H = y | H . Clearly, E H is a equivalence relation on A G . Note that E { g } = π g ;π g−1 for each g ∈ G. So, E H = hold. For a cell g ∈ G and a configuration x ∈ A G , we define the configuration gx ∈ A G by



h∈ H

E {h} =



h∈ H

πh ;πh−1

gx( g  ) = x( g −1 g  ) for all g  ∈ G and the map assigning gx ∈ A G to x ∈ A G is called the shift. Clearly, a configuration x ∈ A G satisfies

x(1G ) = gx( g ) and x( g ) = g −1 x(1G ) for all g ∈ G. Also we have

1G x = x and g 1 ( g 2 x) = ( g 1 g 2 )x for all g 1 , g 2 ∈ G by 1G x( g ) = x(1G g ) = x( g ) and ( g 1 ( g 2 x))( g ) = x( g 2−1 ( g 1−1 g )) = x(( g 1 g 2 )−1 g ) = (( g 1 g 2 )x)( g ). Moreover g −1 x = y implies x = g y by x(h) = x( g g −1 h) = g −1 x( g −1 h) = y ( g −1 h) = g y (h) and conversely x = g y implies g −1 x = y by g −1 x(h) = x( gh) = g y ( gh) = g −1 g y (h) = y (h). Thus, for all x, y ∈ A G and for all g ∈ G the following equivalence holds.

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

23

g −1 x = y ⇐⇒ x = g y For subsets H , K ⊆ G we write H K for the subset {hk ∈ G | h ∈ H , k ∈ K } of G. Then the equivalence

(x, y ) ∈ E H K ⇐⇒ ∀h ∈ H . (h−1 x, h−1 y ) ∈ E K holds for all H , K ⊆ G by

(x, y ) ∈ E H K

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

x| H K = y | H K ∀h ∈ H . ∀k ∈ K . x(hk) = y (hk) ∀h ∈ H . ∀k ∈ K . h−1 x(k) = h−1 y (k) ∀h ∈ H . (h−1 x)| K = (h−1 y )| K ∀h ∈ H . (h−1 x, h−1 y ) ∈ E K .

Here is a tiny example of notions defined above. Example 4.1. Let the alphabet A be {0, 1} and the universe G the quotient group Z/2Z = {0, 1}, where, as usual, Z and 2Z denote additive groups consisting of all integers and even integers and 0, 1 are equivalence classes of 0 ∈ Z and 1 ∈ Z, respectively. In this case, the unit 1G is 0. The set of configurations A G consists of the following maps.

x0 : 0 → 0; 1 → 0

x1 : 0 → 0; 1 → 1

x2 : 0 → 1; 1 → 0

x3 : 0 → 1; 1 → 1

These configurations satisfy the following equations. −1

−1

−1

−1

1x0 = x0 = 1 x0 1x2 = x1 = 1 x2 1x1 = x2 = 1 x1 1x3 = x3 = 1 x3 x0 |{0} = (0 → 0) = x1 |{0} x2 |{0} = (0 → 1) = x3 |{0} x0 |{1} = (1 → 0) = x2 |{1}

x1 |{1} = (1 → 1) = x3 |{1}

So, we have

E {0} = ({x0 , x1 } × {x0 , x1 }) ∪ ({x2 , x3 } × {x2 , x3 }) and E {1} = ({x0 , x2 } × {x0 , x2 }) ∪ ({x1 , x3 } × {x1 , x3 }) . 4.2. Relations on the set of configurations We introduce three notions on relations on the set of configurations. These notions will be used to characterise cellular automata. Let T be a relation on A G . We call T G-independent if



T ; E {g} = T .

g ∈G

If T is univalent (or a partial map in other words), it is G-independent since =  AG .



g ∈G



T ; E { g } = T ;(

g ∈G

E { g } ) and



g ∈G

E {g}

Example 4.2. Consider the same situation as Example 4.1. Then, by

E {0} = ({x0 , x1 } × {x0 , x1 }) ∪ ({x2 , x3 } × {x2 , x3 }) and E {1} = ({x0 , x2 } × {x0 , x2 }) ∪ ({x1 , x3 } × {x1 , x3 }) , the relation {(x3 , x2 ), (x3 , x3 )} is G-independent. Contrastingly, the relation {(x3 , x0 ), (x3 , x3 )} is not since



{(x3 , x0 ), (x3 , x3 )}; E { g } = {x3 } × A G .

g∈ AG

A relation T ⊆ A G × A G satisfying, for all x, y ∈ A G and for all g ∈ G

( gx, y ) ∈ T ⇐⇒ (x, g −1 y ) ∈ T is called G-equivariant. Since g g −1 x = x and g −1 g y = y, any G-equivariant relation T satisfies

( g −1 x, g −1 y ) ∈ T ⇐⇒ (x, y ) ∈ T ⇐⇒ ( gx, g y ) ∈ T and g −1 x ∈ d( T ) ⇐⇒ x ∈ d( T ) ⇐⇒ gx ∈ d( T ) for all x, y ∈ A G and for all g ∈ G. Note that a map T : A G → A G is G-equivariant iff T ( gx) = g T (x) for all x ∈ A G and all g ∈ G since y = T (x) iff (x, y ) ∈ T .

24

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

−1

Example 4.3. Consider the same situation as Example 4.1. Then, by 1x2 = x1 = 1

−1

x2 and 1x1 = x2 = 1

x1 the relation −1

{(x1 , x2 ), (x2 , x1 )} is G-equivariant. Contrastingly, the relation {(x0 , x1 ), (x1 , x0 )} is not since it does not contain (x2 , 1 (x2 , x0 ) although it contains (1x2 , x0 ) = (x1 , x0 ).

x0 ) =

Here is a more general example of G-equivariant relations. Example 4.4. Let G be a group and A a set. For a subset S ⊆ G and a relation M ⊆ A S × A, define a relation T M ⊆ A G × A G by

(x, y ) ∈ T M ⇐⇒ ∀ g ∈ G . (( g −1 x)| S , y ( g )) ∈ M . Then T M is G-equivariant since

( gx, y ) ∈ T M

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

∀h ∈ G . ((h−1 gx)| S , y (h)) ∈ M ∀h ∈ G . ((( g −1 h)−1 x)| S , g −1 y ( g −1 h)) ∈ M ∀k ∈ G . ((k−1 x)| S , g −1 y (k)) ∈ M (x, g −1 y ) ∈ T M

by G = { g −1 }G. The following lemma will be used to prove Proposition 4.6, 4.9, and 4.12. Lemma 4.5. If a relation T is G-equivariant, the following equivalence holds for any subset H ⊆ G.

(x, y ) ∈



T ; E {h} ⇐⇒ ∀h ∈ H . (h−1 x, h−1 y ) ∈ T ; E {1G }

h∈ H

Proof. This equivalence holds by

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

∀h ∈ H . (h−1 x, h−1 y ) ∈ T ; E {1G } ∀h ∈ H . ∃ yh ∈ A G . (h−1 x, yh ) ∈ T and yh (1G ) = h−1 y (1G ) ∀h ∈ H . ∃ yh ∈ A G . (x, hyh ) ∈ T and yh (1G ) = h−1 y (1G ) ∀h ∈ H . ∃ yh ∈ A G . (x, hyh ) ∈ T and hyh (h) = y (h)    G ∀h ∈ H . ∃ yh ∈ A . (x, yh ) ∈ T and yh (h) = y (h) (x, y ) ∈ h∈ H T ; E {h}

where the second equivalence follows from G-equivariance of T .

2

The following property will be used to show uniform continuity of cellular automata via a characterisation given as Proposition 4.12. Proposition 4.6. Let G be a group whose unit is 1G and let A be a set. If a relation T ⊆ A G × A G is G-equivariant, then, for a subset K ⊆ G the following conditions are equivalent. 1. E K ; T ∩ (d( T ) × A G ) ⊆ T  ; E {1G } . 2. E H K ; T ∩ (d( T ) × A G ) ⊆ h∈ H T ; E {h} for each subset H ⊆ G. Proof. By {1G } K = K , 2 implies 1. Conversely, let H ⊆ G. Then we have

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ =⇒

(x, y ) ∈ E H K ; T ∩ (d( T ) × A G ) x ∈ d( T ) and ∃x ∈ A G . x| H K = x | H K and (x , y ) ∈ T x ∈ d( T ) and ∃x ∈ A G . ∀h ∈ H . (h−1 x)| K = (h−1 x )| K and (x , y ) ∈ T ∃x ∈ A G . ∀h ∈ H . h−1 x ∈ d( T ) and (h−1 x)| K = (h−1 x )| K and (h−1 x , h−1 y ) ∈ T ∀h ∈ H . h−1 x ∈ d( T ) and (h−1 x, h−1 y ) ∈ E K ; T ∀h ∈ H . (h−1 x, h−1 y ) ∈ T ; E {1G }

where the third step  follows from G-equivariance of T and the implication follows from the inclusion 1. Then, by Lemma 4.5, we have (x, y ) ∈ h∈ H T ; E {h} . 2

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

25

Let H be a subset of G and M a relation from A H to A. We call T semi-compatible with M if

x ∈ d( T ) and (x| H , y (1G )) ∈ M ⇐⇒ (x, y ) ∈ T ; E {1G } . Note that the condition is equivalent to T (x)(1G ) = M (x| H ) if T and M are maps since (x| H , y (1G )) ∈ M iff M (x| H ) = y (1G ) and (x, y ) ∈ T ; E {1G } iff T (x)(1G ) = y (1G ). Example 4.7. Consider the same situation as Example 4.1. Also, let S = {0} and define M ⊆ A S × A by M = {(0 → 1, 0)}. Then, by

E {1G } = E {0} = ({x0 , x1 } × {x0 , x1 }) ∪ ({x2 , x3 } × {x2 , x3 }) (0 → 1, 0) = (x2 | S , x0 (1G )) = (x2 | S , x1 (1G )) = (x3 | S , x0 (1G )) = (x3 | S , x1 (1G )) , the relation {(x3 , x0 ), (x3 , x1 )} is semi-compatible with M. Contrastingly, the relation {(x3 , x2 ), (x3 , x3 )} is not since (x3 | S , x2 (1G )) ∈ / M although (x3 , x2 ) ∈ {(x3 , x2 ), (x3 , x3 )}; E {1G } . The following property will be used together with Proposition 4.6 to show uniform continuity of cellular automata. Proposition 4.8. Let G be a group with unit 1G and A a set. Consider a relation T ⊆ A G × A G . Let H be a subset of G and let M ⊆ A H × A. Then the inclusion

E H ; T ∩ (d( T ) × A G ) ⊆ T ; E {1G } holds if T is semi-compatible with M. Proof. Suppose that (x, y ) ∈ E H ; T ∩ (d( T ) × A G ). By  A G ⊆ E {1G } , (x, y ) ∈ E H ; T implies (x, y ) ∈ E H ; T ; E {1G } . So, there exists x ∈ A G such that (x, x ) ∈ E H and (x , y ) ∈ T ; E {1G } . By the definition of E H and semi-compatibility of T and M, we have x| H = x | H and (x | H , y (1G )) ∈ M. Hence (x| H , y (1G )) ∈ M. Also, x ∈ d( T ) since (x, y ) ∈ d( T ) × A G . Thus (x, y ) ∈ T ; E {1G } by semi-compatibility of T and M. 2 G-independent, G-equivariant and semi-compatible relations satisfy the following generalised form of G-independence. It will also be used to show uniform continuity of cellular automata. Proposition 4.9. Let G be a group with unit 1G and A a set. Consider a relation T ⊆ A G × A G . Let H be a subset of G and let M ⊆ A H × A. Then the equation



T ; E {h} = T ; E H

h∈ H

holds if T is G-independent, G-equivariant and semi-compatible with M.





G G Proof. Since E H = h∈ H E {h} , the inclusion  T ; E H ⊆ h∈ H T ; E {h} holds for any relation T ⊆ A × A . So, it is sufficient to show the converse inclusion. Let (x, y ) ∈ h∈ H T ; E {h} . Then, since T is G-equivariant, we have

∀h ∈ H . (h−1 x, h−1 y ) ∈ T ; E {1G } by Lemma 4.5. So, by semi-compatibility of T and M we have

∀h ∈ H . ((h−1 x)| H , h−1 y (1G )) ∈ M and h−1 x ∈ d( T ) . On the other hand, it is obvious that x ∈ d( T ). This implies

∃ y  ∈ A G . (x, y  ) ∈



T ; E {g}

g ∈G

since  A G ⊆ E { g } for any g ∈ G, and hence we have

∃ y  ∈ A G . ∀ g ∈ G . (( g −1 x)| H , g −1 y  (1G )) ∈ M and g −1 x ∈ d( T ) by, again, Lemma 4.5 and semi-compatibility of T and M. Then the configuration z ∈ A G defined by



z( g ) =

y( g ) y( g )

(g ∈ H ) (otherwise)

satisfies z(h) = y (h) for any h ∈ H , (( g −1 x)| H , g −1 z(1G )) ∈  M and g −1 x ∈ d( T ) for any g ∈ G. Thus, clearly ( z, y ) ∈ E H and, by semi-compatibility of T and M and Lemma 4.5 (x, z) ∈ g ∈G T ; E { g } . Moreover, by G-independence (x, z) ∈ T . Therefore (x, y ) ∈ T ; E H . 2

26

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

4.3. Definition and characterisation using local rule Next we define nondeterministic cellular automata over a group and an alphabet. Definition 4.10. A (nondeterministic) cellular automaton over the group G and the alphabet A is a relation T ⊆ A G × A G satisfying the following property: There exists a finite subset S ⊆ G and a relation M ⊆ A S × A such that

(x, y ) ∈ T ⇐⇒ ∀ g ∈ G . (( g −1 x)| S , y ( g )) ∈ M and g −1 x ∈ d( T ) . Such a set S is called a memory set and the relation M is called a local defining relation for T . If M is a map, T is a partial map satisfying T (x)( g ) = M (( g −1 x)| S ) for all x ∈ d( T ), g ∈ G. Therefore the definition extends deterministic cellular automata given in [2, Section 1] and [3, Definition 1.4.1]. In this paper the alphabet of a cellular automaton may be infinite like deterministic cellular automata in [2,3] but unlike nondeterministic cellular automata in [5,10]. Example 4.11. The relation T M on A G from Example 4.4 is a cellular automaton admitting S as a memory set and M as a local defining relation if S is finite. Let T be a cellular automaton with a memory set S ⊆ G and a local defining relation M ⊆ A S × A. Then, T is G-independent since

 (x, y ) ∈ g ∈G T ; E { g } ∀ g ∈ G . ∃ y g ∈ A G . (x, y g ) ∈ T and y g ( g ) = y ( g ) ∀ g ∈ G . ∃ y g ∈ A G . y g ( g ) = y ( g ) and ∀h ∈ G . ((h−1 x)| S , y g (h)) ∈ M and h−1 x ∈ d( T ) ∀h ∈ G . ((h−1 x)| S , y (h)) ∈ M and h−1 x ∈ d( T ) (x, y ) ∈ T .

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

Also T is G-equivariant since

⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

( gx, y ) ∈ T ∀h ∈ G . ((h−1 gx)| S , y (h)) ∈ M and h−1 gx ∈ d( T ) ∀h ∈ G . ((( g −1 h)−1 x)| S , g −1 y ( g −1 h)) ∈ M and ( g −1 h)−1 x ∈ d( T ) ∀k ∈ G . ((k−1 x)| S , g −1 y (k)) ∈ M and k−1 x ∈ d( T ) (x, g −1 y ) ∈ T

by G = { g −1 }G. Moreover, (x, y ) ∈ T ; E {1G } implies x ∈ d( T ) and (x| S , y (1G )) ∈ M by

(x, y ) ∈ T ; E {1 g } ∃ y  ∈ A G . (x, y  ) ∈ T and y  (1G ) = y (1G ) ∃ y  ∈ A G . ∀ g ∈ G . (( g −1 x)| S , y  ( g )) ∈ M and g −1 x ∈ d( T ) and y  (1G ) = y (1G ) ∃ y  ∈ A G . (x| S , y  (1G )) ∈ M and x ∈ d( T ) and y  (1G ) = y (1G ) x ∈ d( T ) and (x| S , y (1G )) ∈ M .

⇐⇒ ⇐⇒ =⇒ ⇐⇒

Conversely, assume that x ∈ d( T ) and (x| S , y (1G )) ∈ M. Note that

x ∈ d( T ) ⇐⇒ ∃ y  ∈ A G . (x, y  ) ∈ T ⇐⇒ ∃ y  ∈ A G . ∀ g ∈ G . (( g −1 x)| S , y  ( g )) ∈ M and g −1 x ∈ d( T ) . Then the configuration z ∈ A G defined by



z( g ) =

y( g ) y ( g )

( g = 1G ) (otherwise)

satisfies z(1G ) = y (1G ), (( g −1 x)| S , z( g )) ∈ M and g −1 x ∈ d( T ) for any g ∈ G. Hence, ( z, y ) ∈ E {1G } and (x, z) ∈ T . So, we have (x, y ) ∈ T ; E {1G } . Therefore T is semi-compatible with M. The following property is a nondeterministic variant of [3, Proposition 1.4.6]. Proposition 4.12. Let G be a group and let A be a set. Consider a relation T ⊆ A G × A G . Let S be a finite subset of G and let M ⊆ A S × A. Then the following conditions are equivalent: 1. T is a cellular automaton admitting S as a memory set and M as a local defining relation; 2. T is G-independent, G-equivariant and semi-compatible with M.

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

27

Proof. We have already shown that 1 implies 2. Conversely, suppose 2. Then we have

(x, y ) ∈ T ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒

 (x, y ) ∈ g ∈G T ; E { g } ∀ g ∈ G . ( g −1 x, g −1 y ) ∈ T ; E {1G } ∀ g ∈ G . (( g −1 x)| S , g −1 y (1G )) ∈ M and g −1 x ∈ d( T ) ∀ g ∈ G . (( g −1 x)| S , y ( g )) ∈ M and g −1 x ∈ d( T )

where the first equivalence follows from G-independence, the second equivalence follows from G-equivariance (by Lemma 4.5), and the third equivalence follows from semi-compatibility of T and M. 2 By the last proposition, every cellular automaton is G-equivariant. Hence the following property is immediate from the definition of cellular automata. Corollary 4.13. Let T be a cellular automaton admitting S as a memory set and M as a local defining relation and define a relation T M by (x, y ) ∈ T M iff (( g −1 x)| S , y ( g )) ∈ M for each cell g (as in Example 4.4). Then, the equation T = T M ∩ (d( T ) × A G ) holds. As we have mentioned in Example 4.11, the relations T M and T M  on A G are cellular automata for given a finite subset S ⊆ G and relations M , M  ⊆ A S × A. Clearly T M ∪ M  is again a cellular automaton. However, T M ∪ T M  need not be a cellular automaton. The following example shows this fact with applying the last proposition. Example 4.14. Consider the same situation as Example 4.1. Also, let S = {0} and define M 1 , M 2 ⊆ A S × A by

M 1 = {(0 → 1, 0)} and M 2 = {(0 → 1, 1)} . Then, defining T M 1 , T M 2 ⊆ A G × A G by

(x, y ) ∈ T M i ⇐⇒ ∀ g ∈ G . (( g −1 x)| S , y ( g )) ∈ M i

( i = 1, 2) ,

namely T M 1 = {(x3 , x0 )} and T M 2 = {(x3 , x3 )}, as we have seen in Example 4.2, T M 1 ∪ T M 2 is not G-independent. Therefore, by the last proposition, T M 1 ∪ T M 2 is not a cellular automaton. 5. Global characterisation In the last section we defined (nondeterministic) cellular automata and gave a characterisation of them. In both of them we needed to mention “local rules”. In this section we give another characterisation of cellular automata by using topological notions which enables us to characterise “globally”. 5.1. Prodiscrete uniform space The prodiscrete uniform structure on A G is, as in [2,3], the weakest uniform structure on A G such that each projection π g from A G to the discrete uniform space A is uniformly continuous. Denoting the family of all finite subsets of G by F , the family B = { E H | H ∈ F } is a base of entourages for the prodiscrete uniform structure on A G . Proposition 5.1. Every cellular automaton T ⊆ A G × A G is uniformly continuous on the prodiscrete uniform space A G . Proof. Suppose that T is a cellular automaton with a memory set S and a local defining relation M ⊆ A S × A. Let H be a finite subset of G. Then, clearly, H S is finite and E H , E H S ∈ B . So, it is sufficient to show the inclusion E H S ; T ∩ (d( T ) × A G ) ⊆ T ; E H . Let (x, y ) ∈ E H S ; T ∩ (d( T ) × A G ). By (x, y ) ∈ E H S ; T , there exists x ∈ A G such that

∀h ∈ H . (h−1 x )| S = (h−1 x)| S and ∀ g ∈ G . (( g −1 x )| S , y ( g )) ∈ M . By (x, y ) ∈ d( T ) × A G there exists y  ∈ A G such that (x, y  ) ∈ T . So the pair (x, y  ) satisfies

∀ g ∈ G . (( g −1 x)| S , y  ( g )) ∈ M and g −1 x ∈ d( T ) . Hence the configuration z ∈ A G defined by



z( g ) =

y( g ) y( g )

(g ∈ H ) (otherwise)

satisfies (( g −1 x)| S , z( g )) ∈ M for all g ∈ G, and z| H = y | H . So, (x, z) ∈ T by g −1 x ∈ d( T ), and ( z, y ) ∈ E H . Therefore (x, y ) ∈ T ;E H . 2

28

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

Though we have given a rather direct proof from the definition of cellular automata, the above proposition may be proved via Proposition 4.12. Since any cellular automata are semi-compatible with M, we have E S ; T ∩ (d( T ) × A G ) ⊆ T ; E {1G } by Proposition 4.8. Also, since any cellular automata are G-equivariant, the inclusion implies E H S ; T ∩(d( T ) × A G ) ⊆ h∈ H T ; E {h} by Proposition 4.6. Moreover, since any cellular automata are G-independent, G-equivariant and semi-compatible with M,  we have h∈ H T ; E {h} = T ; E H by Proposition 4.9. Therefore we have the inclusion E H S ; T ∩ (d( T ) × A G ) ⊆ T ; E H for any finite subset H ⊆ G. On the other hand, uniform continuity of T ⊆ A G × A G ensures existence of a finite subset S and a relation M ⊆ A S × A with which T is semi-compatible. Lemma 5.2. If T ⊆ A G × A G is uniformly continuous on the prodiscrete uniform space A G , there exists a finite subset S and a relation M ⊆ A S × A such that T is semi-compatible with M. Proof. Suppose that T ⊆ A G × A G is uniformly continuous on the prodiscrete uniform space A G . Then there exists a finite subset S ⊆ G such that E S ; T ∩ (d( T ) × A G ) ⊆ T ; E {1G } . Define M ⊆ A S × A by

( z, a) ∈ M ⇐⇒ ∃(x, y ) ∈ T . x| S = z and y (1G ) = a . Then we have

⇐⇒ ⇐⇒ =⇒ ⇐⇒

x ∈ d( T ) and (x| S , y (1G )) ∈ M x ∈ d( T ) and ∃(x0 , y 0 ) ∈ T . x0 | S = x| S and y 0 (1G ) = y (1G ) ∃ y 0 ∈ A G . (x, y 0 ) ∈ E S ; T ∩ (d( T ) × A G ) and y 0 (1G ) = y (1G ) ∃ y 0 ∈ A G . (x, y 0 ) ∈ T ; E {1G } and ( y 0 , y ) ∈ E {1G } (x, y ) ∈ T ; E {1G }

⇐⇒ ⇐⇒ =⇒ ⇐⇒

(x, y ) ∈ T ; E {1G } ∃ y 0 ∈ A G . (x, y 0 ) ∈ T and ( y 0 , y ) ∈ E {1G } x ∈ d( T ) and ∃ y 0 ∈ A G . (x, y 0 ) ∈ T and ( y 0 , y ) ∈ E {1G } x ∈ d( T ) and ∃(x0 , y 0 ) ∈ T . x0 | S = x| S and y 0 (1G ) = y (1G ) x ∈ d( T ) and (x| S , y (1G )) ∈ M .

and

Therefore T is semi-compatible with M.

2

We have prepared to give a global characterisation of cellular automata. This is a nondeterministic variant of [2, Theorem 1.1] (Theorem 1.9.1 in [3]). Theorem 5.3. Let A be a set and let G be a group. For a relation T ⊆ A G × A G the following conditions are equivalent: 1. T is a cellular automaton; 2. T is G-independent, G-equivariant and uniformly continuous on the prodiscrete uniform space A G . Proof. By Proposition 4.12 and Proposition 5.1, 1 implies 2. The converse direction follows from Lemma 5.2 and Proposition 4.12. 2 5.2. Prodiscrete space The prodiscrete topology on A G is, as in [2,3], the weakest topology on A G such that each projection π g from A G to the discrete space A is continuous. In fact the prodiscrete topology on A G is the topology associated with the prodiscrete uniform structure on A G . Richardson has shown the following property in [10]. A modern relation-algebraic proof of it has been given in [5]. Theorem 5.4. Let A be a finite set and let G be a group. For a relation T ⊆ A G × A G the following conditions are equivalent: 1. T is a cellular automaton; 2. T is G-independent, G-equivariant, and there exists a relation T  ⊆ A G × A G that is continuous on the prodiscrete space A G and closed in the product space A G × A G , and satisfies T = T  ∩ (d( T ) × A G ). If A is finite, the second condition of Theorem 5.4 implies the second condition of Theorem 5.3 as follows. Suppose that, for a relation T ⊆ A G × A G there exists a continuous and closed relation T  ⊆ A G × A G satisfying T = T  ∩ (d( T ) × A G ). Then,

H. Furusawa / Theoretical Computer Science 673 (2017) 19–29

29

since A is finite, A G and A G × A G are compact by Tychonoff theorem. Hence the closed set T  is compact. Therefore, by Proposition 3.2, T  is uniformly continuous. Moreover, by Proposition 3.3, T is also uniformly continuous. For a cellular automaton T with a memory set S and a local defining relation M, by Example 4.11 and Corollary 4.13, the relation T M defined by

(x, y ) ∈ T M ⇐⇒ ∀ g ∈ G . (( g −1 x)| S , y ( g )) ∈ M is a cellular automaton satisfying T = T M ∩ (d( T ) × A G ). Since every cellular automaton is uniformly continuous, T M is continuous by Proposition 3.1. Also, since S is finite, A S × A is a discrete space. So, all the subsets of A S × A are open and closed. Moreover, since the shifts, the maps determined by the restriction to finite sets, and the projections are continuous (see [3, Section 1.2] for more detail), the map assigning (( g −1 x)| S , y ( g )) ∈ A S × A to (x, y ) ∈ A G × A G is continuous and the intersection of the inverse image of M under the map with respect to g ∈ G is equal to T M . Thus T M is closed. So, the second condition of Theorem 5.4 is derived from the second condition of Theorem 5.3. The deterministic cellular automata given in [2, Section 4] and [3, Example 1.8.2] show that, if A is infinite, then a relation on A G satisfying the second condition of Theorem 5.4 need not be a cellular automaton. The above discussion explains that Theorem 5.4 is a special case of Theorem 5.3. 6. Conclusion This paper has provided a notion of uniformly continuous relations with respect to general uniform spaces by generalising the notion in [9]. Then, by using it, we have globally characterised nondeterministic cellular automata over a group and a possibly infinite alphabet. This characterisation contains the global characterisations given in [6,2,3,10] as special cases. In [4] Di Lena and Margara investigated the class of nondeterministic codes that give rise to surjective and reversible nondeterministic cellular automata over the integers and a finite alphabet with at least two elements by using the notion of continuous relations between certain metric spaces. It might be interesting to generalise their results in our general setting. In addition, further study of uniformly continuous relations from a general-topological point of view is a future work since what we have done is minimally enough to give the global characterisation. Acknowledgements The author thanks Shuichi Inokuchi and Yasuo Kawahara for their valuable comments. I also thank Keita Suzuki since his master’s thesis research on deterministic cellular automata is a motivator of this work. The presentation of this article has benefited from the suggestions of the anonymous reviewers. This work was supported in part by JSPS KAKENHI grant number JP25330016. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Vasco Brattka, Peter Hertling, Continuity and Computability of Relations, Inform.-Ber., vol. 164, FernUniversität in Hagen, 1994. Tullio Ceccherini-Silberstein, Michel Coornaert, A generalization of the Curtis–Hedlund theorem, Theoret. Comput. Sci. 400 (2008) 225–229. Tullio Ceccherini-Silberstein, Michel Coornaert, Cellular Automata and Groups, Springer, 2010. Pietro Di Lena, Luciano Margara, Nondeterministic cellular automata, Inform. Sci. 287 (2014) 13–25. Hitoshi Furusawa, Toshikazu Ishida, Yasuo Kawahara, Continuous Relations and Richardson’s Theorem, Lecture Notes in Comput. Sci., vol. 7560, Springer, 2012, pp. 310–325. Gustav Hedlund, Endomorphisms and automorphisms of the shift dynamical systems, Math. Syst. Theory 3 (4) (1969) 320–375. Ioan James, Topological and Uniform Spaces, Springer, 1987. John L. Kelley, General Topology, Springer, 1975. Arno Pauly, Martin Ziegler, Relative computability and uniform continuity of relations, J. Log. Anal. 5 (7) (2013) 1–39. Daniel Richardson, Tessellations with local transformations, J. Comput. System Sci. 6 (1972) 373–388.