Cellular Automata and Kan Extensions
aa r X i v : . [ c s . D C ] M a r Cellular Automata and Kan Extensions
Alexandre Fernandez, Luidnel Maignan, and Antoine Spicher
Univ Paris Est Creteil, LACL, 94000, Creteil, France
Abstract
In this paper, we formalize precisely the sense in which the applicationof a cellular automaton to partial configurations is a natural extension ofits local transition function through the categorical notion of Kan ex-tension. In fact, the two possible ways to do such an extension and theingredients involved in their definition are related through Kan extensionsin many ways. These relations provide additional links between computerscience and category theory, and also give a new point of view on thefamous Curtis-Hedlung theorem of cellular automata from the extendedtopological point of view provided by category theory. These links also al-low to relatively easily generalize concepts pioneered by cellular automatato arbitrary kind of possibly evolving spaces. No prior knowledge of cat-egory theory is assumed.
Cellular automata are usually presented either as a local behavior extended toa global and uniform one or as a continuous uniform global behavior for theappropriate topology [1]. We offer here a third, fruitful, point of view easingmany generalizations of the concepts pioneered by cellular automata, e.g. viaso-called global transformation [4, 2]. The goal of this paper is not to elabo-rate on these generalizations but to focus on some simple foundational bridgesallowing these generalizations. In particular, we focus on Kan extensions, acategorical notion allowing, as we show here, to capture local/global descrip-tions [3]. While categories are generalizations of monoids and posets, the caseof cellular automata can be fully treated in terms of posets only. Once the in-volved structures made clear via posets, the transition to category is preciselywhat enables the generalizations in a surprisingly smooth way as discussed inthe final section.In this paper, we recall the direct definitions of cellular automata on groups,local transition function, global transition function, shift action, and also con-sider the counterparts of these functions on arbitrary partial configurations.This bigger picture allows to show that the various local/global relations be-tween these objects are all captured by left and right Kan extensions, the latterproviding a alternative definition of these objects. The proofs are provided indetail to show how the concept can be easily manipulated once understood. Wealso introduce slightly more generality that one would typically need in orderto enrich the presentation of Kan extensions in a hopefully useful way. In the1nal section, we comment on the link with Curtis-Hedlung theorem and discussbriefly the smooth transition to more general systems where the space itself hasto evolve.
Let us give some basic definitions to fix the notations. We also note smallcaveats early on, to avoid having to deal with many unrelated details at thesame time in a single proof or construction latter on.
Definition 1. A group is a set G with a binary operation − · − : G × G → G which is associative, which has a neutral element and for which any g ∈ G has inverse g − . A right action of the group on a set X is a binary operation − ◭ − : X × G → X such that x ◭ x and ( x ◭ g ) ◭ h = x ◭ ( g · h ) . In cellular automata, the group G represents the space, each element g ∈ G being at the same time an absolute and a relative position. This space isdecorated with states that evolve through local interactions only. The classicalformal definitions go as follows and work with the entire, often infinite, space. Definition 2. A cellular automaton on a group G is given by a neighborhood N ⊆ G , a finite set of states Q , and a local transition function δ : Q N → Q .The elements of the set Q N are called local configurations . The elements of theset Q G are called global configurations and a right action − ◭ − : Q G × G → Q G is defined on Q G by ( c ◭ g )( h ) = c ( g · h ) . The global transition function ∆ : Q G → Q G of such a cellular automaton is defined as ∆( c )( g ) = δ (( c ◭ g ) ↾ N ) . Proposition 3.
The latter right action is indeed a right action.Proof.
For any g, h ∈ G , we have (( c ◭ g ) ◭ h )( i ) = ( c ◭ g )( h.i ) = c ( g · h · i ) =( c ◭ ( g · h ))( i ) for any i ∈ G , so (( c ◭ g ) ◭ h ) = ( c ◭ ( g · h )) and also( c ◭ i ) = c (1 · i ) = c ( i ) as required by Definition 1 of right actions.This choice of definition and right notation for the so called shift action hastwo advantages. Firstly, the definition of the action is a simple associativity.Secondly, when instantiated with G = Z with sum, the content of c ◭ c shifted to the left, as the symbols indicates. Indeed, for c ′ = c ◭ c ′ ( −
5) = c (0) and c ′ (0) = c (5). Proposition 4.
For all c ∈ Q G and g ∈ G , ∆( c )( g ) is only function of c ↾ g · N .Proof. Indeed, ∆( c )( g ) = δ (( c ◭ g ) ↾ N ) so the value is determined by ( c ◭ g ) ↾ N . But for any n ∈ N , ( c ◭ g )( n ) = c ( g · n ) by definition of ◭ .In common cellular automata terms, this proposition means that the neigh-borhood of g is g · N , in this order. Let us informally call objects of the form c ↾ g · N ∈ S g ∈ G Q g · N a shifted local configuration . Note that, at our level ofgenerality, two different positions g = g ′ ∈ G might have the same neighborhood g · N = g ′ · N . Although the injectivity of the function ( − · N ) could be a usefulconstraints to add, which is often verified in practice, we do not impose it so2he reader should keep this in mind. A second thing to keep in mind is that wedo not require here that the neighborhood should be finite. This is done onlybecause this property is not used in the formal development below. Proposition 5.
The function − · N : G → G is not necessarily injective.Proof. Considering any group G and N = G , we have g · N = G for any g ∈ G . Considering the group G = Z / Z × Z and N = { (0 , , (1 , } , we have(0 ,
0) + N = (1 ,
0) + N = { (0 , , (1 , } because of the torsion.Because of this, it is useful to replace the shifted local configurations, i.e. theunion S g ∈ G Q g · N , by the disjoint union S g ∈ G ( { g } × Q g · N ). The elements of thelatter are of the form ( g ∈ G, c ∈ Q g · N ) and keep track of the considered “center”of the neighborhood. More explicitly, two elements ( g , c ↾ g · N ) , ( g , c ↾ g · N ) ∈ S g ∈ G ( { g }× Q g · N ) are different as soon as g = g even if g · N = g · N .This encodes things according to the intuition of a centered neighborhood. In the previous formal statements, one sees different kinds of configurations,explicitly or implicitly: global configurations c ∈ Q G , local configurations ( c ◭ g ) ↾ N ∈ Q N , shifted local configurations c ↾ g · N ∈ Q g · N , and their resulting“placed states” ( g ∆( c )( g )) ∈ Q { g } . In the cellular automata literature, oneoften considers configurations defined on other subsets of the space, e.g. finiteconnected subsets. More generally, we are interested in all partial configurations Q S for arbitrary subsets S ⊆ G . The restriction operation ( − ↾ − ) used manytimes above gives a partial ordering of these partial configurations. Definition 6. A (partial) configuration c is a partial function from G to Q .Its domain of definition is denoted | c | and called its support . The set of allconfigurations is denoted Conf = S S ⊆ G Q S . We extend the previous right action ◭ and define it to map each c ∈ Conf to c ◭ g having support | c ◭ g | = { h ∈ G | g · h ∈ | c |} and states ( c ◭ g )( h ) = c ( g · h ) . Proposition 7.
The latter right action is well-defined and is a right action.Proof.
The configuration c ◭ g is well-defined on all of its support. Indeed forany h ∈ | c ◭ g | , ( c ◭ g )( h ) = c ( g · h ) and g · h ∈ | c | by definition of h . The rightaction property is verified as in the proof of Proposition 3.Let us restate Proposition 4 more precisely using Definition 6. Proposition 8.
For all c ∈ Q G and g ∈ G , ( c ◭ g ) ↾ N = ( c ↾ g · N ) ◭ g .Proof. Indeed, | ( c ↾ g · N ) ◭ g | = { h ∈ G | g · h ∈ | ( c ↾ g · N ) |} = { h ∈ G | g · h ∈ g · N } = N = | ( c ◭ g ) ↾ N | . Also for any n ∈ N , (( c ◭ g ) ↾ N )( n ) = ( c ◭ g )( n ) = c ( g · n ) and (( c ↾ g · N ) ◭ g )( n ) = ( c ↾ g · N )( g · n ) = c ( g · n ). Definition 9. A partial order on a set X is a binary relation (cid:22) ⊆ X × X whichis reflexive, transitive, and antisymmetric. A set endowed with a partial orderis called a partially ordered set , or poset for short. Definition 10.
Given any two configurations c, c ′ ∈ Conf , we set c (cid:22) c ′ if andonly if ∀ g ∈ | c | , g ∈ | c ′ | ∧ c ( g ) = c ′ ( g ) . This is read “ c is a subconfiguration of c ′ ” or “ c ′ is a superconfiguration of c ”. roposition 11. The set
Conf with this binary relation is a poset. In thisposet, the shifted local configurations c ∈ S g ∈ G Q g · N are subconfigurations ofthe (appropriate) global configurations c ′ ∈ Q G . Shifted local configurationsform an antichain. Global configurations form an antichain.Proof. As can be readily seen, since each global configuration restricts to manyshifted local configurations, and recalling that an antichain is a subset S of theposet such that neither x (cid:22) x ′ nor x ′ (cid:22) x hold for any two different x, x ′ ∈ S . Given three sets A , B and C such that A ⊆ B , we say that a function g : B → C extends a function f : A → C if g ↾ A = f , or equivalently if g ◦ i = f where i is the obvious injective function from A to B . For a given f : A → C ,there are typically many possible extensions. Roughly speaking, Kan extensionsformalizes, among many things, the mathematical practice where extensions arerarely arbitrary. One usually chooses the “best” or “most natural” extensions.There is therefore an implicit comparison considered between the extensions.This is the reason why Kan extensions are formally defined at the level of2-categories: A , B , C are objects, f , g , i , and all (not necessarily “most nat-ural”) extensions are 1-morphisms between these objects, and the “naturality”comparison between 1-morphisms are 2-morphisms. However, we do not needto discuss things at this level of generality here. For our particular case, theobjects are posets, the 1-morphisms are monotonic functions and the monotonicfunctions are compared pointwise. Definition 12.
Given two posets ( X, (cid:22) X ) and ( Y, (cid:22) Y ) , a function f : X → Y is said to be monotonic if for all x, x ′ ∈ X , x (cid:22) X x ′ implies f ( x ) (cid:22) Y f ( x ′ ) . Proposition 13.
For any g ∈ G , the function ( − ◭ g ) : Conf → Conf ismonotonic.Proof.
Given any c, c ′ ∈ Conf such that c (cid:22) c ′ , this claim is equivalent to:( c ◭ g ) (cid:22) ( c ′ ◭ g ) (by Def 12) ⇐⇒ ∀ h ∈ | c ◭ g | ; h ∈ | c ′ ◭ g | ∧ ( c ◭ g )( h ) = ( c ′ ◭ g )( h ) (Def 10) ⇐⇒ ∀ h ∈ G s.t. g · h ∈ | c | ; g · h ∈ | c ′ | ∧ c ( g · h ) = c ′ ( g · h ) (Def 6)which is true by the application of Definition 10 of c (cid:22) c ′ on g · h . Definition 14.
Given two posets ( X, (cid:22) X ) and ( Y, (cid:22) Y ) , we define the binaryrelation − ⇒ − on the set of all monotonic functions from X to Y by f ⇒ f ′ ⇐⇒ ∀ c ∈ Conf , f ( c ) (cid:22) Y f ′ ( c ) . Proposition 15.
Given two posets ( X, (cid:22) X ) and ( Y, (cid:22) Y ) , the set of monotonicfunctions between them together with this binary relation forms a poset.Proof. As one can easily check.
Definition 16.
In this setting, given three posets A , B and C , and three mono-tonic functions i : A → B , f : A → C and g : B → C , g is said to be the left (resp. right) Kan extension of f along i if g is the ⇒ -minimum (resp. ⇒ -maximum) element in the set of monotonic functions { h : B → C | f ⇒ h ◦ i } (resp. { h : B → C | h ◦ i ⇒ f } ). Proposition 17.
The left Kan extension g is unique when it exists.Proof. It is defined as the minimum of a set, and a minimum is always uniquewhen it exists.Another suggestive way to read the concept of Kan extensions with respectto this paper is to say that a function g on a poset can be summarized into, orgenerated by, a part of its behavior f on just a small part of the poset. Notehowever that i does not need to be injective in this definition. The first, intuitive, approach is to take a configuration c , look for all places g where the whole neighborhood g · N is defined and to take the local transitionresult of these places only. We first give a direct formal definition, and thenshow that this is a left Kan extension. This shows in particular that the globaltransition function is the left Kan extension of the “fully shifted” local transition.The sense of “fully shifted” is described below and is only necessary because werestrict ourselves to posets, as discussed in the final section of this paper. The interior of a subset S ⊆ G is int( S ) = { g ∈ G | g · N ⊆ S } . Definition 19.
The coarse transition function Φ : Conf → Conf is defined forall c ∈ Conf as | Φ( c ) | = int( | c | ) and Φ( c )( g ) = δ (( c ◭ g ) ↾ N ) . Proposition 20.
For any c ∈ Conf and g ∈ G , the statements g ∈ int( | c | ) , g · N ⊆ | c | , and N ⊆ | c ◭ g | are equivalent. (So Φ is well-defined in Definition 19.)Proof. The first and second statements are equivalent by Definition 18 of int.The second and third statements are equivalent by Definition 6 of ◭ .Remember Proposition 5. If we do have injectivity of neighborhoods, we haveint( g · N ) = { g } . But since we do not assume it, we only have the following. Proposition 21.
Let S ⊆ G . In general, S ⊆ int( S · N ) but we do not neces-sarily have equality, even when S = { g } for some g ∈ G .Proof. Consider the examples in the proof of Proposition 5.Another useful remark on which we come back below is the following.
Proposition 22.
For any g ∈ G , any M ( N , and any c ∈ Q g · M , | Φ( c ) | = ∅ .Also, for any c ∈ Q G , | Φ( c ) | = ∆( c ) .Proof. By Definition 19 of Φ. 5 .1.2 Characterization as a Left Kan Extension
The coarse transition function Φ is defined on the set of all configurations Confand we claim that it is generated, in the Kan extension sense, by the local tran-sition function δ shifted everywhere. We define the latter, with Proposition 5in mind. Definition 23.
We define
Loc to be the poset
Loc = S g ∈ G ( { g } × Q g · N ) withtrivial partial order ( g, c ) (cid:22) ( g ′ , c ′ ) ⇐⇒ ( g, c ) = ( g ′ , c ′ ) . The “fully shiftedlocal transition function” δ : Loc → Conf is defined, for any ( g, c ) ∈ Loc as | δ ( g, c ) | = { g } and δ ( g, c )( g ) = δ ( c ◭ g ) . The second projection of Loc is themonotonic function π : Loc → Conf defined as π ( g, c ) = c . Proposition 24.
Loc is a poset and δ and π are monotonic functions.Proof. Indeed, the identity relation is an order relation and any function respectsthe identity relation.
Proposition 25.
The coarse transition function Φ is monotonic.Proof. Indeed, take c, c ′ ∈ Conf such that c (cid:22) c ′ . We want to prove thatΦ( c ) (cid:22) Φ( c ′ ) and this is equivalent to: ∀ g ∈ | Φ( c ) | , g ∈ | Φ( c ′ ) | ∧ Φ( c )( g ) = Φ( c ′ )( g ) ⇐⇒ ∀ g ∈ int( | c | ) , g ∈ int( | c ′ | ) ∧ δ ( c ◭ g ↾ N ) = δ ( c ′ ◭ g ↾ N ) ⇐⇒ ∀ g ∈ G s.t. g · N ⊆ | c | , g · N ⊆ | c ′ | ∧ δ ( c ◭ g ↾ N ) = δ ( c ′ ◭ g ↾ N ) , by Definition 6 of (cid:22) , Definition 19 of Φ, and Definition 18 of int. The finalstatement is implied by: ∀ g ∈ G s.t. g · N ⊆ | c | , g · N ⊆ | c ′ | ∧ c ◭ g ↾ N = c ′ ◭ g ↾ N ⇐⇒ ∀ g ∈ G s.t. g · N ⊆ | c | , g · N ⊆ | c ′ | ∧ ∀ n ∈ N, ( c ◭ g )( n ) = ( c ′ ◭ g )( n ) ⇐⇒ ∀ g ∈ G s.t. g · N ⊆ | c | , g · N ⊆ | c ′ | ∧ ∀ n ∈ N, c ( g · n ) = c ′ ( g · n ) , the last equivalence being by Definition 6. To prove it, take g ∈ G such that g · N ⊆ | c | , and n ∈ N . Since c (cid:22) c ′ and g · n ∈ | c | , we have g · n ∈ | c ′ | , and c ( g · n ) = c ′ ( g · n ) as wanted. Proposition 26. Φ is the left Kan extension of δ along π : Loc → Conf .Proof.
By Definition 16 of left Kan extensions, we need to prove firstly that Φis such that δ ⇒ Φ ◦ π , and secondly that it is smaller than any other suchmonotonic functions.For the first part, δ ⇒ Φ ◦ π is equivalent to: ∀ ( g, c ) ∈ Loc , δ ( g, c ) (cid:22) Φ( c ) (Defs. 14 and 23 of ⇒ and π ) ⇐⇒ ∀ ( g, c ) ∈ Loc , ∀ h ∈ | δ ( g, c ) | , h ∈ | Φ( c ) | ∧ δ ( g, c )( h ) = Φ( c )( h ) (D. 10 of (cid:22) ) ⇐⇒ ∀ ( g, c ) ∈ Loc , g ∈ | Φ( c ) | ∧ δ ( c ◭ g ) = Φ( c )( g ) (Def. 23 of δ ) ⇐⇒ ∀ ( g, c ) ∈ Loc , g · N ∈ | c | ∧ δ ( c ◭ g ) = δ (( c ◭ g ) ↾ N ) (Defs 18, 19 of Φ).This last statement is true by Definition 23 of Loc, since c ∈ Q g · N .6or the second part, let F : Conf → Conf be a monotonic function such that δ ⇒ F ◦ π . We want to show that Φ ⇒ F , which is equivalent to: ∀ c ∈ Conf , Φ( c ) (cid:22) F ( c ) (Def. 14 of ⇒ ) ⇐⇒ ∀ c ∈ Conf , ∀ g ∈ | Φ( c ) | , g ∈ | F ( c ) | ∧ Φ( c )( g ) = F ( c )( g ) (Def. 10 of (cid:22) ) ⇐⇒ ∀ c ∈ Conf , ∀ g ∈ int( | c | ) , g ∈ | F ( c ) | ∧ F ( c )( g ) = δ (( c ◭ g ) ↾ N ) (Def. 19)So take c ∈ Conf and g ∈ int( | c | ), and consider d g = c ↾ g · N . Since d g (cid:22) c and F is monotonic, we have F ( d g ) (cid:22) F ( c ). Moreover δ ⇒ F ◦ π and ( g, d g ) ∈{ g } × Q g · N ⊆ Loc, so δ ( g, d g ) (cid:22) F ( d g ) by Definitions 14 and 23 of ⇒ and π .By transitivity δ ( g, d g ) (cid:22) F ( c ). By Definition 23 of δ and Definition 10 of (cid:22) , weobtain g ∈ | F ( c ) | , and F ( c )( g ) = δ ( g, d g )( g ) = δ (( c ◭ g ) ↾ N ), as wanted.As a sidenote, remark that in order to have the equality δ = Φ ◦ π , one needsto have the injectivity of neighborhood function. Indeed, without injectivity,we have two different g, g ′ ∈ G having the same neighborhood M , i.e. M = g · N = g ′ · N . This means that, given any local configuration c ∈ Q M on thisneighborhood, each pair ( g, c ) , ( g ′ , c ) ∈ Loc have different results δ ( g, c ) ∈ Q { g } and δ ( g ′ , c ) ∈ Q { g ′ } with different support { g } and { g ′ } . However, their commonprojection π ( g, c ) = π ( g ′ , c ) = c have a unique result Φ( c ) with a support suchthat { g, g ′ } ⊆ | Φ( c ) | . So we have a strict comparison δ ⇒ Φ ◦ π . When theneighborhood function is injective, π is also injective and the previous situationcan not occur so we have equality δ = Φ ◦ π . For some applications, the previous definitions are too naive. For example,it is common to consider two cellular automata to be essentially the same ifthey generate the same global transition functions. However, here, two suchcellular automata give different coarse transition function if they have a differentneighborhood.To refine the previous definitions, a second approach is to take a configura-tion c , and look at all places for which the result is already determined by thepartial data defined in c . So we consider all g ∈ G for which all completions ofthe data present on the defined neighborhood g · N ∩ | c | into a configuration onthe complete neighborhood g · N always lead to the same result by δ . For any c ∈ Conf and g ∈ G , let c g = c ↾ ( g · N ∩ | c | ) . Definition 28.
Given a configuration c ∈ Conf , its determined subset is det( c ) = { g ∈ G | ∃ q ∈ Q, ∀ c ′ ∈ Q g · N , c ′ ↾ | c g | = c g = ⇒ δ ( c ′ ◭ g ) = q } .For any g ∈ det( c ) , we denote q c,g ∈ Q the unique state q having the mentionedproperty. Note that this definition depends on the cellular automaton local transitionfunction δ and on the data of the configuration c , contrary to Definition 18 ofinterior that only depends on its neighborhood N and on the support of theconfiguration. 7 efinition 29. Given a cellular automaton, its fine transition function φ :Conf → Conf is defined as | φ ( c ) | = det( c ) and φ ( c )( g ) = q c,g , i.e. φ ( c )( g ) = δ ( c ′ ◭ g ) for any c ′ ∈ Q N such that c ′ ↾ | c g | = c g . Proposition 30.
The fine transition function φ is well defined.Proof. This is the case precisely because we restrict the support of φ ( c ) to thedetermined subset of the c . Proposition 31.
Consider the constant cellular automaton δ ( c ) = q ∀ c ∈ Q N for a specific q ∈ Q and regardless of the neighborhood N chosen to representit. We have | φ ( c ) | = G for any c ∈ Conf .Proof.
Indeed, even with no data at all, i.e. for c such that | c | = ∅ , the resultat all position is determined and is q . As for the coarse transition function, the fine transition function φ is definedon the set of all configurations Conf and we claim that it is generated, in theKan extension sense. We consider two ways to generate it and start by thesimplest one. The second one is considered in the following section using sub-local configurations in order to be closer to the direct definition and to be a“from local to global” characterization. Proposition 32.
For any g ∈ G , the function − g : Conf → Conf of Defini-tion 27 is monotonic.Proof.
As one can easily check.
Proposition 33.
The fine transition function φ is monotonic.Proof. Indeed, take c , c ∈ Conf such that c (cid:22) c . We want to prove that φ ( c ) (cid:22) φ ( c ) and this is equivalent to: ∀ g ∈ | φ ( c ) | , g ∈ | φ ( c ) | ∧ φ ( c )( g ) = φ ( c )( g ) (Def 10 of (cid:22) ) ⇐⇒ ∀ g ∈ det( c ) , g ∈ det( c ) ∧ q c ,g = q c ,g (Def 29 of φ )Take g ∈ det( c ). We want to prove that g ∈ det( c ), which means by Defini-tion 28 of det( c ): ∃ q ∈ Q, ∀ c ∈ Q g · N , c ↾ | ( c ) g | = ( c ) g = ⇒ δ ( c ◭ g ) = q We claim that the property is verified with q = q c ,g . Indeed, take any c ∈ Q g · N such that c ↾ | ( c ) g | = ( c ) g . We also have that c ↾ | ( c ) g | = ( c ) g since thehypothesis c (cid:22) c implies ( c ) g (cid:22) ( c ) g by Proposition 32. By Definition 28 ofdet( c ), we obtain that δ ( c ◭ g ) = q c ,g , so q = q c ,g has the wanted property,which implies that g ∈ det( c ) as wanted. But the above property of q set it tobe precisely what we denote by q c ,g (Def 28 of q c ,g ), so q c ,g = q c ,g . Proposition 34.
The fine transition function φ is the right Kan extension ofthe global transition function ∆ along the inclusion i : Q G → Conf . roof. By Definition 16 of right Kan extensions, we need to prove firstly that φ is such that φ ◦ i ⇒ ∆, and secondly that it is greater than any other suchmonotonic functions.For the first part, we actually have φ ◦ i = ∆ since for any c ∈ Q G , | φ ( c ) | =det( c ) = G = | ∆( c ) | and for any g ∈ G , we have φ ( c )( g ) = q c,g = δ ( c g ◭ g ) = δ (( c ↾ g · N ) ◭ g ) = δ (( c ◭ g ) ↾ N ) = ∆( c )( g ) by Defs. 29, 28, 27, 2 of φ , det, c g and ∆ and Prop. 8.For the second part, let f : Conf → Conf be a monotonic function such that f ◦ i ⇒ ∆. We want to show that f ⇒ φ , which is equivalent to: ∀ c ∈ Conf , f ( c ) (cid:22) φ ( c ) (Def. 14 of ⇒ ) ⇐⇒ ∀ c ∈ Conf , ∀ g ∈ | f ( c ) | , g ∈ | φ ( c ) | ∧ f ( c )( g ) = φ ( c )( g ) (Def. 10 of (cid:22) ) ⇐⇒ ∀ c ∈ Conf , ∀ g ∈ | f ( c ) | , g ∈ det( c ) ∧ f ( c )( g ) = q c,g (Def. 35 of φ ) ⇐⇒ ∀ c ∈ Conf , ∀ g ∈ | f ( c ) | , ∀ c ′ ∈ Q g · N , c (cid:22) c ′ = ⇒ f ( c )( g ) = δ ( c ′ ◭ g ) (D. 28)So take c ∈ Conf and g ∈ | f ( c ) | and c ′ ∈ Q g · N such that c (cid:22) c ′ . Consider any c ′′ ∈ Q G such that c ′ (cid:22) c ′′ (or equivalently c ′′ ↾ g · N = c ′ ). Since f is monotonic,we have f ( c ) (cid:22) f ( c ′′ ), which means that f ( c )( g ) = f ( c ′′ )( g ) by Def. 10. Butsince f ◦ i ⇒ ∆, we have f ( c )( g ) = ∆( c ′′ )( g ) = δ (( c ′′ ◭ g ) ↾ N ) by Def. 14 of ⇒ and Def. 2 of ∆. But by Prop. 8, ( c ′′ ◭ g ) ↾ N = ( c ′′ ↾ g · N ) ◭ g = c ′ ◭ g . The direct definition of the fine transition function is explicitly about assigning aresult for a configuration c at a given g ∈ G even when the whole neighborhood g · N is not complete. By isolating these “shifted sub-local configuration” inthe poset of configurations, we can (right-)extend the local transition to themand show that, in the same way as the coarse transition function is the left Kanextension of the local transition function, the fine transition function is the leftKan extension of the sub-local transition function. We define
Sub = S g ∈ G,M ⊆ N ( { g } × Q g · M ) with partial orderdefined as ( g, c ) (cid:22) ( g ′ , c ′ ) if and only if g = g ′ and c (cid:22) c ′ . The “fully shifted sub-local transition function” δ : Sub → Conf is defined, for any g ∈ G , any M ⊆ N and any c ∈ Q g · M , as | δ ( g, c ) | = { g }∩ det( c ) and, if g ∈ det( c ) , δ ( g, c )( g ) = q c ′ ,g , i.e. δ ( g, c )( g ) = δ ( c ′ ◭ g ) for any c ′ ∈ Q g · N such that c = c ′ ↾ | c | . The secondprojection of Sub is the function π : Sub → Conf defined as π ( g, c ) = c . In this definition, a given sub-local configuration can result either in anempty configuration when the transition is not determined, or in a configurationwith only singleton support when the transition is determined.Note that for a given cellular automaton, it is possible to restrict the posetSub to an antichain. Indeed, any time a result is determined by a sub-local con-figuration ( g, c ), all bigger sub-local configuration ( g, c ′ ) with c (cid:22) c ′ does notcontribute anything new. We do not elaborate on this because this antichainwould be different for each cellular automaton, blurring the global picture pre-sented below. 9 .3.2 Characterization as a Right Kan ExtensionProposition 36. The fully shifted sub-local transition function δ is monotonicProof. As usual, take ( g, c ) , ( g ′ , c ′ ) ∈ Sub such that ( g, c ) (cid:22) ( g ′ , c ′ ). First notethat g = g ′ by Definition 35. We want to prove that δ ( g, c ) (cid:22) δ ( g, c ′ ) and thisis equivalent to: ∀ h ∈ | δ ( c ) | , h ∈ | δ ( c ′ ) | ∧ δ ( c )( h ) = δ ( c ′ )( h ) ⇐⇒ ∀ h ∈ { g } ∩ det( c ) , h ∈ { g } ∩ det( c ′ ) ∧ q c,g = q c ′ ,g ⇐⇒ g ∈ det( c ) = ⇒ g ∈ det( c ′ ) ∧ q c,g = q c ′ ,g ′ , by Definition 6 of (cid:22) and Definition 35 of δ . The end of this proof is similar tothe one of Proposition 33. Proposition 37.
The fully shifted sub-local transition function δ is the rightKan extension of the fully shifted local transition function δ along the inclusion i : Loc → Sub .Proof.
By Definition 16 of right Kan extensions, we need to prove firstly that δ is such that δ ◦ i ⇒ δ , and secondly that it is greater than any other suchmonotonic functions.For the first part, δ ◦ i ⇒ δ is equivalent to: ∀ ( g, c ) ∈ Loc , δ ( g, c ) (cid:22) δ ( g, c ) (Def. 14 of ⇒ ) ⇐⇒ ∀ ( g, c ) ∈ Loc , ∀ h ∈ | δ ( g, c ) | , h ∈ | δ ( g, c ) | ∧ δ ( g, c )( h ) = δ ( g, c )( h ) (D. 10 (cid:22) ) ⇐⇒ ∀ ( g, c ) ∈ Loc , g ∈ det( c ) = ⇒ g ∈ | δ ( g, c ) | ∧ q c,g = δ ( g, c )( g ) (Def. 35 of δ ) ⇐⇒ ∀ ( g, c ) ∈ Loc , g ∈ det( c ) = ⇒ g ∈ { g } ∧ q c,g = δ ( c ◭ g ) (Def 23 of δ ).This last statement is true by Def. 28 of q c,g .For the second part, let f : Sub → Conf be a monotonic function such that f ◦ i ⇒ δ . We want to show that f ⇒ δ , which is equivalent to: ∀ ( g, c ) ∈ Sub , f ( g, c ) (cid:22) δ ( g, c ) (Def. 14 of ⇒ ) ⇐⇒ ∀ ( g, c ) ∈ Sub , ∀ h ∈ | f ( g, c ) | , h ∈ | δ ( g, c ) | ∧ f ( g, c )( h ) = δ ( g, c )( h ) (Def. 10) ⇐⇒ ∀ ( g, c ) ∈ Sub , ∀ h ∈ | f ( g, c ) | , h ∈ { g } ∩ det( c ) ∧ f ( g, c )( h ) = q c,g (Def. 35)So take ( g, c ) ∈ Sub and h ∈ | f ( g, c ) | . Consider any c ′ ∈ Loc such that c (cid:22) c ′ .Since f is monotonic, we have f ( g, c ) (cid:22) f ( g, c ′ ), which means that h ∈ | f ( g, c ′ ) | and f ( c )( h ) = f ( c ′ )( h ) by Def. 10. But since f ◦ i ⇒ δ , we have h ∈ | δ ( g, c ′ ) | = { g } and f ( g, c ′ )( h ) = δ ( g, c ′ )( h ) = δ ( c ′ ◭ g ) by Def. 14 of ⇒ and Def. 23 of δ . Since this is true for any c ′ , this establishes exactly the defining property ofdet( c ) by Def. 28. The projection function π : Sub → Conf is monotonic.Proof.
As can be readily checked in Definition 35
Proposition 39. φ is the left Kan extension of δ along π : Sub → Conf . roof. By Definition 16 of left Kan extensions, we need to prove firstly that φ is such that δ ⇒ φ ◦ π , and secondly that it is smaller than any other suchmonotonic functions.For the first part, δ ⇒ φ ◦ π is equivalent to: ∀ ( g, c ) ∈ Sub , δ ( g, c ) (cid:22) φ ( c ) (Defs. 14 and 35 of ⇒ and π ) ⇐⇒ ∀ ( g, c ) ∈ Sub , ∀ h ∈ | δ ( g, c ) | , h ∈ | φ ( c ) | ∧ δ ( g, c )( h ) = φ ( c )( h ) (Def 10 of (cid:22) ) ⇐⇒ ∀ ( g, c ) ∈ Sub , g ∈ det( c ) = ⇒ g ∈ | φ ( c ) | ∧ q c,g = φ ( c )( g ) (Def. 35 of δ ) ⇐⇒ ∀ ( g, c ) ∈ Sub , g ∈ det( c ) = ⇒ g ∈ det( c ) ∧ q c,g = q c,g (Def 29 of φ ) , a most trivial statement.For the second part, let f : Conf → Conf be a monotonic function such that δ ⇒ f ◦ π . We want to show that φ ⇒ f , which is equivalent to: ∀ c ∈ Conf , φ ( c ) (cid:22) f ( c ) (Def. 14 of ⇒ ) ⇐⇒ ∀ c ∈ Conf , ∀ g ∈ | φ ( c ) | , g ∈ | f ( c ) | ∧ φ ( c )( g ) = f ( c )( g ) (Def. 10 of (cid:22) ) ⇐⇒ ∀ c ∈ Conf , ∀ g ∈ det( c ) , g ∈ | f ( c ) | ∧ q c,g = f ( c )( g ) (Def. 29 of φ )So take c ∈ Conf and g ∈ det( c ). Since c g (cid:22) c (Def 27) and f is monotonic, wehave f ( c g ) (cid:22) f ( c ). Moreover δ ⇒ f ◦ π and ( g, c g ) ∈ Sub so δ ( g, c g ) (cid:22) f ( c g )by Definitions 14 and 23 of ⇒ and π . By transitivity δ ( g, c g ) (cid:22) f ( c ). ByDefinition 35 of δ and Definition 10 of (cid:22) , we therefore have g ∈ | f ( c ) | , and f ( c )( g ) = δ ( g, d g )( g ) = q c,g as wanted. There are additional simple structural facts to note about the monotonic func-tions considered. The first one is that the shift action on partial configurations,as given in Definition 6, is the right Kan extension of the shift action on globalconfigurations, as given in Definition 2. Another one is that Φ ⇒ φ , hencethe names of these transition functions, coarse and fine. In fact, any mono-tonic function f : Conf → Conf such that δ ⇒ f ◦ π is necessarily such thatΦ ⇒ f ⇒ φ . This shows, in some sense, the efficiency of the simple constraintsof monotonicity and δ ⇒ f ◦ π .In the formal development presented here, we explicitly “copy” a single localbehavior δ on all g ∈ G to obtain δ and work with it. It is readily possible to puta different behavior on each g ∈ G , with no real modification to the proofs. Thestatements are therefore valid for non-uniform cellular automata and automatanetworks. As mentioned in the beginning, we did not even used the finitenessof the neighborhood either. At this point, the reader might have the feelingthat these results are not really about cellular automata, and there are at leastthree answers to that. The first answer is that one could easily impose the shiftand simultaneously prevent the use of a highly redundant “fully shifted localtransition function”, but this requires using a category of configuration insteadof a poset of configurations. The latter is very similar to the poset, except thatthe yes/no question “is this configuration a subconfiguration of this other one ?”is replaced by the open-ended question “where does this configuration appear inthis other one ?” [4]. The goal of this paper is indeed to introduce the conceptsneeded for this other point of view, among many others. The second answer is11hat the proofs are more about the decomposition/composition process involvedin the local/global definition of cellular automata. Because of the simplicity ofcellular spaces, groups, the description is very simple to make “directly”. Inother situations, a Kan extension presentation can be the most effective wayto describe the spatial extensions/restriction, for example when the space isan evolving graph [4, 2]. The third answer is that, with small modifications,this result is closely related to the Curtis-Hedlung theorem. Indeed, if onerestores the finite neighborhoods and finite states constraints, one can see thatthe poset of finite support configurations is a “generating” part of the poset ofopen subsets of the product topology. In this case, the fine transition function φ can be viewed as encoding an important part of the topological behavior ofthe global transition function ∆ [1].To finish, let us mention an important aspect of the Kan extensions con-sidered here and in other papers of the author [4, 2]. They have the propertyto be pointwise . Intuitively, this means that they can be computed “algorith-mically” using simple building blocks. This formulation in terms of buildingblocks is completely equivalent and is the one used in the other papers, firstlybecause it is via these building blocks that the authors discovered these linksbetween spatially-extended dynamical systems and category theory, and sec-ondly because this formulation is closer to the software implementations of theconsidered models. In fact, it is possible to have an implementation completelygeneric over the particular kind of space considered, e.g. evolving graphs of anysort, evolving higher-order structures such as abstract cell [4], evolving stringssuch as Lindenmayer systems [2], or Cayley graphs as considered here. References [1] Tullio Ceccherini-Silberstein and Michel Coornaert.
Cellular Automataand Groups , pages 778–791. Springer New York, New York, NY, 2009. doi:10.1007/978-0-387-30440-3_52 .[2] Alexandre Fernandez, Luidnel Maignan, and Antoine Spicher. Lindenmayersystems and global transformations. In Ian McQuillan and Shinnosuke Seki,editors,
Unconventional Computation and Natural Computation - 18th In-ternational Conference, UCNC 2019, Tokyo, Japan, June 3-7, 2019, Pro-ceedings , volume 11493 of
Lecture Notes in Computer Science , pages 65–78.Springer, 2019. doi:10.1007/978-3-030-19311-9\_7 .[3] S. MacLane.
Categories for the Working Mathematician . Graduate Texts inMathematics. Springer New York, 2013.[4] Luidnel Maignan and Antoine Spicher. Global graph transformations.In Detlef Plump, editor,
Proceedings of the 6th International Work-shop on Graph Computation Models co-located with the 8th Interna-tional Conference on Graph Transformation (ICGT 2015) part of theSoftware Technologies: Applications and Foundations (STAF 2015) fed-eration of conferences, L’Aquila, Italy, July 20, 2015 , volume 1403 of
CEUR Workshop Proceedings , pages 34–49. CEUR-WS.org, 2015. URL: http://ceur-ws.org/Vol-1403/paper4.pdfhttp://ceur-ws.org/Vol-1403/paper4.pdf