An order-preserving property of additive invariant for Takesue-type reversible cellular automata
aa r X i v : . [ n li n . C G ] J u l An order-preserving property of additive invariant forTakesue-type reversible cellular automata
Gianluca Caterina ∗ and Bruce M. Boghosian Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02420, USA
October 25, 2018
Abstract
We show that, for a fairly large class of reversible, one-dimensional cellular au-tomata, the set of additive invariants exhibits an algebraic structure. More precisely,if f and g are one-dimensional, reversible cellular automata of the kind considered byTakesue [1], we show that there is a binary operation on these automata ∨ such that ψ ( f ) ⊆ ψ ( f ∨ g ), where ψ ( f ) denotes the set of additive invariants of f and ⊆ denotesthe inclusion relation between real subspaces. Additive invariants for cellular automata (CA) have been widely studied over the lasttwo decades, especially in connection with the fundamental role they play in physicalmodeling. The problem of finding explicit conditions for a one-dimensional CA to haveconserved densities of a given size (in a sense that will be described in this article) hasbeen solved by Hattori and Takesue in their remarkable 1991 paper [2]. Interestingly,they expressed those conditions in terms of discrete current laws, thereby recovering afundamental property of continuous systems in a discrete context.This striking analogy with continuous models suggested that a discrete versionof Noether’s theorem – an elegant and fundamental connection between conservedquantities and symmetries of physical systems – might be formulable. On the otherhand, the proof of Noether’s theorem is based on the properties of differential operatorswhich are absent in a discrete context.For these reasons we decided to approach this problem first from a computationalpoint of view. We considered a class F I of second-order, reversible CA based ona binary alphabet defined over a fixed “neighbor window” I and, for each f ∈ F I ,computed all its additive conserved quantities, the set of which may be identified witha finite-dimensional vector space denoted by ψ ( f ). Then we looked at the equivalenceclasses of CA rules given by the relation ∗ Present address: Department of Mathematics, Northeastern University, 360 Huntington Avenue, Boston,MA 02115, USA ∼ g ⇐⇒ ψ ( f ) = ψ ( g )and observed that, if we choose two representatives f and g from the same equivalenceclass ψ ( f ), then there is a pointwise binary operation ∨ on F I such that ψ ( f ) ⊆ ψ ( f ∨ g ) . (1)This relation is interesting because it relates a dynamical system to its invariants. SinceNoether’s theorem endeavors to relate the latter with the symmetries of the system,we thought it was worth investigating this problem further. In this work we present aproof of Eq.(1) for a large class of reversible, second-order dynamical systems definedon an arbitrary finite set. In a seminal paper [1], Takesue argues that a family of fully-discrete models calledElementary Reversible Cellular Automata (ERCA) can be used to study the thermo-dynamic behavior of large dynamical systems. ERCA are a family of one-dimensionalreversible CA with two Boolean independent variables at each site. It is possible toassociate to these systems certain additive conserved quantities which can be regardedas a form of energy. Also, discreteness of the variables implies that the phace spacevolume is preserved under the dynamics, and therefore the statistical mechanics of themodel can be constructed.Examples of physical applications of these quantities are given, for instance, byPomeau [6] and Boccara [3, 4] who have used them successfully to model traffic. Inmost of these works, special symmetries of the rules are considered and conjectures aredrawn on how they relate to the associated set of additive invariants.In our approach, we give primacy to the equivalence relation which identifies tworules if they have the same set of invariants. The main result is a theorem which showsthat symmetries of the rules are reflected in an algebraic structure on their respectiveequivalence classes. Since “energies” are hydrodynamic quantities which one hopesto see emerging from the dynamics, our result can be seen as a contribution towardunderstanding how certain qualities of the dynamics reflect on the symmetries of theassociated physically relevant quantities. In the next section we start the discussionby generalizing ERCA to an arbitrary, finite alphabet.
The instantaneous state of many discrete dynamical systems is most conveniently de-scribed by n dependent variables, each of which takes its values in a set S , with | S | = N < ∞ . We denote the j ’th dependent variable at time step t by x j ( t ) ∈ S , for0 ≤ j < n and t ≥
0, so that the state of the system at time tx ( t ) = { x ( t ) , x ( t ) , . . . , x n − ( t ) } may be thought of as an n -vector. CA dynamical systems of this kind endowed with ashift-invariant map [5]. ore precisely, consider the set I β = {− β, − β + 1 , . . . , , . . . , β − , β } ( β ∈ Z ) anda function f : S I β −→ S. Then a one-dimensional cellular automaton is the dynamical system defined by thetriple (
S, β, f ) , with the dynamics being described by the following map: x i ( t + 1) = f ( x i − β ( t ) , . . . , x i ( t ) , . . . , x i + β ( t )) . By defining ˆ x βi ( t ) := { x i − β ( t ) , . . . , x i ( t ) , . . . , x i + β ( t ) } , the above can be rewritten as x i ( t + 1) = f (ˆ x βi ( t )) . We refer to the associate global map F : x ( t ) x ( t + 1) as the global map induced by f . In what follows, we will also assume periodic boundary conditions , which meansthat x ( i + j ) = x ( i + j ) mod n and we refer to the system so defined as a cellular automaton(CA) of first order and neighborhood size β .We now define the concept of second-order reversible CA, mentioned in the Intro-duction. Let γ : S −→ S be a second-order reversible map; that is γ satisfies ∀ x,y ∈ S : γ ( y, γ ( y, x )) = x. Then we can define ˜ f : S I β × S −→ S (ˆ x i ( t + 1) , x i ( t )) γ ( f (ˆ x i ( t + 1)) , x i ( t )) = x i ( t + 2) . (2)Consistent with the notation introduced above, we call ˜ f a second-order CA of neigh-borhood size β and reversible combiner γ . Since the binary version of these objectswas first introduced in [2], we will also refer to them as Takesue-type cellular automata.When it will be clear from the context, we will drop the tilde from the notation.We notice that:˜ f ( x i ( t + 1) , x i ( t + 2)) = γ ( f (ˆ x i ( t + 1)) , x i ( t + 2)) == γ ( f (ˆ x i ( t + 1)) , γ ( f (ˆ x i ( t + 1)) , x i ( t ))) = x i ( t ) . (3)Hence, by Eqs.(2) and (3), we deduce that the second-order dynamical system ˜ F in-duced by ˜ f , ˜ F ( x ( t + 1) , x ( t )) = x ( t + 2)implies ˜ F ( x ( t + 1) , x ( t + 2)) = x ( t )demonstrating that ˜ F is indeed a reversible dynamical system. Additive conserved quantities
Consider a second-order reversible CA f of neighborhood size β on a periodic latticeof size N . We define an “energy density function” ǫ : S α +1 × S −→ R (ˆ x αi ( t ) , x i ( t − ǫ (ˆ x αi ( t ) , x i ( t − . Remark 1
In what follows, two distinct neighborhoods will appear: One is relativeto the automaton (neighborhood size β ), and one is relative to the conserved density(neighborhood size α ). However, the α -neighborhood will appear exclusively as an argu-ment of ǫ , whereas the β -neighborhood will appear only as a property of the automaton.Hence, in order to simplify the notation, we can drop the superscripts α and β with noambiguity. We define the total energy E N of the system at time t associated to ǫ as: E N ( x ( t ) , x ( t − , ǫ ) = N − X i =0 ǫ (ˆ x i ( t ) , x i ( t − . (4)When the density ǫ is given, or clear from the context, it will be dropped from thearguments for E .We say that ǫ is a additive conserved density for ˜ f if, for any t > N > E N ( x ( t ) , x ( t − , ǫ ) = E N ( ˜ F ( x ( t ) , x ( t − , x ( t ) , ǫ ) . (5)According to this definition, in order for ǫ to be a conserved quantity, an infinite numberof conditions must hold: namely, conserved quantities are solutions to the system ofequations defined by Eq. (5) for any N ∈ N + . In their seminal paper [2] Hattori andTakesue proved that only a finite number of linear conditions are sufficient to find allthe solutions to Eq. (5).These conditions can be rephrased in a slightly different, though equivalent, fashion.Indeed, let us denote by ψ ( ˜ f , N, α ) the set of all the additive conserved densities ofneighborhood size α for ˜ f on a periodic lattice of size N . Then we have: Lemma 1
If there is N such that ψ ( ˜ f , N , α ) = ψ ( ˜ f , N + 1 , α ) then ψ ( ˜ f , M, α ) = ( ˜ f , N , α ) , ∀ M > N Lemma 2
For any ˜ f and any α there exists N ≤ ∞ such that ψ ( ˜ f , N , α ) = ψ ( ˜ f , N + 1 , α ) . For the proof of these lemmas we refer to [2]. Let us notice that the above resulttells us that the set of invariants for a given automaton ˜ f is, for N sufficiently large,independent of the size of the periodic lattice on which the system evolves. Thereforewe can drop the dependence on N and define the set of additive conserved quantitiesfor ˜ f to be ψ ( ˜ f , α ) = ψ ( ˜ f , N , α ). An algebraic property of additive invariants
In this section we prove that additive invariants for the second-order, reversible CAconsidered above possess a natural equivalence class structure.
Remark 2
The choice for the space-time window neighborhood (in particular, thechoice of the “central” point of the neighborhood at time t − ) which defines the den-sity ǫ is crucial to our analysis. The idea is to force the window for ǫ to have thesame “shape” as the one for the second-order automaton ˜ f : In the next section theconsequences of this choice on our argument will appear more clearly. Using the notation introduced above, let us consider a second-order reversible CAwith S = { , } , β = 1 and α = 1. This corresponds to the following time evolution: x i ( t + 2) = f (ˆ x i ( t + 1)) ⊕ x i ( t )where x i ( t ) ∈ { , } for all i, s, and ⊕ is addition modulo 2. Since we set α = 1,the density ǫ has the form ǫ = ǫ ( x i − ( t ) , x i ( t ) , x i +1 ( t ) , x i ( t − F the set of all the functions { , } −→ { , } , and, for any f ∈ F , let φ ( f ) be the set of all second-order additive conserved densitiesfor ˜ f with α = 1. The equivalence relation f ∼ g ⇐⇒ ψ ( f ) = ψ ( g )then induces a partition of F into equivalence classes. We did an exhaustive computersearch and found that ∼ partitions F into 21 distinct equivalence classes. Example 1
The following independent row vectors, written in matrix form as: − − − − − − − − − − − − are all the possible conserved quantities of the CA represented, using Wolfram’s nota-tion [7], by 8,64 and 72. Therefore { , , } comprises a class in F / ∼ . In the above example, CA 8, 64 and 72 correspond, respectively, to the followingbinary sequences:8 (0 , , , , , , , , (0 , , , , , , , , (0 , , , , , , , f ∨ g to be the bitwise inclusive or of the binary representations of f and g , we can easily verify that the class formed by 8 , ,
72 is closed with respect tothis operation. Interestingly, this closure property holds for all the classes except two.However, the following slightly weaker property holds in this context: onjecture 1 If f, g belongs to the same equivalence class, then we have that ψ ( f ) ⊆ ψ ( f ∨ g ) . This is interesting, since it says that there is an “invariant-preserving” operation definedon the class of CA with a given window function. In the next section we will presenta lemma which will be used to prove, in Section 7, a generalization of this result to anarbitrary finite alphabet.
Following the notation introduced in the previous section, let ˜ f be a second orderreversible cellular automaton defined by˜ f : S β +1 × S −→ S ( x i ( t ) , x i ( t − γ ( f ( x i ( t )) , x i ( t − , where γ : S −→ S is such that ∀ x,y ∈ S : γ ( y, γ ( y, x )) = x. Notice that, on a binary alphabet, the only non-trivial function with this property isaddition modulo 2 (up to conjugation), which we denote by ⊕ . Notice also that, since ⊕ is associative and commutative, it satisfies( x ⊕ ( y ⊕ z )) = ( y ⊕ ( x ⊕ z )) . The next lemma shows that these two properties of the reversible combiner are sufficientto prove Conjecture 5.
Lemma 3
Suppose that ˜ f is a second-order reversible cellular automaton and1. γ ( y, γ ( y, x )) = x γ ( x, γ ( y, z )) = γ ( y, γ ( x, z )) .Then, if ǫ is an additive conserved quantity density for ˜ f and ˜ g we have that ǫ [ˆ x i ( t ) , γ ( f (ˆ x i ( t )) , x i ( t − ǫ [ˆ x i ( t ) , γ ( g (ˆ x i ( t )) , x i ( t − for any i ∈ { , , . . . , N − } , t ≥ . Proof . Since the system is reversible it is enough to prove the claim for case t = 1.First notice that, if f (ˆ x i (1)) = g (ˆ x i (1)), the claim is trivial. Then, let x (0) = { x (0) , x (0) , . . . , x N − (0) } and x (1) = { x (1) , x (1) , . . . , x N − (1) } be the initial con-ditions.We set x f (2) := ˜ F ( x (1) , x (0)) and define the change of energy forward in time∆ N,f ( x (1) , x (0)) = E N ( x f (2) , x (1)) − E N ( x (1) , x (0)) , nd the change of energy backward in time∆ ′ N,f ( x (1) , x (0)) = E N ( x (0) , x (1)) − E N ( x (1) , x f (2)) . Let I = { i , i , . . . , i k } be the set of all i such that f (ˆ x i (1)) = g (ˆ x i (1)) and fix anindex j ∈ I . Since ǫ is a conserved quantity for both ˜ f and ˜ g , we have that∆ ′ N,f ( x (1) , x f (2)) − ∆ ′ N,g ( x (1) , x f (2)) = 0 , and hence we have " N − X i =0 ǫ (ˆ x i (0) , x i (1)) − N − X i =0 ǫ (ˆ x i (1) , γ ( f (ˆ x i (1)) , x i (0))) − " N − X i =0 ǫ (ˆ x i (0) , x i (1)) − N − X i =0 ǫ (ˆ x i (1) , γ ( g (ˆ x i (1)) , x i (0)) = X i ∈ I \ j ǫ (ˆ x i (1) , γ ( f (ˆ x i (1)) , x i (0)) − ǫ (ˆ x i (1) , γ ( g (ˆ x i (1)) , x i (0))) + ǫ (ˆ x j (1) , γ ( f (ˆ x j (1)) , x j (0))) − ǫ (ˆ x j (1) , γ ( g (ˆ x j (1)) , x j (0)))= 0 . (7)Consider now x (0) and replace x j (0) by γ ( f (ˆ x j (1)) , γ ( g (ˆ x j (1)) , x j (0))). We denotethis new state by x (0) and consider the evolution with initial conditions ( x (1) , x (0))under the dynamics induced by f . At time t +2, the value of the j th dependent variableis equal to γ ( f (ˆ x j (1)) , γ ( f (ˆ x j (1)) , γ ( g (ˆ x j (1)) , x j (0)))) = γ ( g (ˆ x j (1)) , x j (0)) . (8)Here we have used γ ( y, γ ( y, x )) = x , with y = f (ˆ x j (1)) and x = γ ( g (ˆ x j (1)) , x j (0)).By a similar argument, if we consider the evolution with initial conditions ( x (1) , x (0))under the dynamics induced by g , at time t + 2, the value of the j th dependent variableis equal to γ ( g (ˆ x j (1)) , γ ( f (ˆ x j (1)) , γ ( g (ˆ x j (1)) , x j (0))))= γ ( g (ˆ x j (1)) , γ ( g (ˆ x j (1)) , γ ( f (ˆ x j (1)) , x j (0))))= γ ( f (ˆ x j (1)) , x j (0)) . (9)Here we have used the fact that, by hypothesis, γ ( x, γ ( y, z )) = γ ( y, γ ( x, z )) and there-fore γ ( f (ˆ x j (1)) , γ ( g (ˆ x j (1)) , x j (0))) = γ ( g (ˆ x j (1)) , γ ( f (ˆ x j (1)) , x j (0))) . Because of this, for initial conditions ( x (1) , x (0)), conservation of energy gives rise tothe identity ∆ ′ N,f ( x (1) , x (0)) − ∆ ′ N,g ( x (1) , x (0)) = 0 , or X i ∈ I \ j ǫ (ˆ x i (1) , γ ( f (ˆ x i (1)) , x i (0))) − ǫ (ˆ x i (1) , γ ( g (ˆ x i (1)) , x i (0))) +( ǫ (ˆ x j (1) , γ ( g (ˆ x j (1)) , x j (0))) − ǫ (ˆ x j (1) , γ ( f (ˆ x j (1)) , x j (0)))= 0 . (10) y subtracting (7) from (10) we obtain ǫ (ˆ x j (1) , γ ( f (ˆ x j (1)) , x j (0))) = ǫ (ˆ x j (1) , γ ( g (ˆ x j (1)) , x j (0))) . Since we can repeat the argument for any j ∈ I , the lemma follows. (cid:3) Let us denote by f ∨ g the component-wise maximum of f and g . Then we have: Theorem 1
Let ˜ f and ˜ g be second-order reversible cellular automata of neighborhoodsize β and reversible combiner γ , and suppose ǫ = ǫ (ˆ x i ( t ) , x i ( t − is an additiveconserved density of size α for ˜ f and ˜ g . Then ǫ is an additive conserved density alsofor ˜ h , where h = f ∨ g .Proof. Suppose that initial conditions x (0) = { x (0) , x (0) , . . . , x N − (0) } and x (1) = { x (1) , x (1) , . . . , x N − (1) } are given. Using the notation introduced in the previous section, we need to show that∆ ′ N,f ( x (1) , x (0)) = ∆ ′ N,f ∨ g ( x (1) , x (0)) = 0 . This amounts to showing that N − X i =0 ǫ (ˆ x i (0) , x i (1)) = N − X i =0 ǫ (ˆ x i (1) , γ (( f ∨ g )(ˆ x i (1)) , x i (0))) . (11)First notice that we can write the right-hand side of Eq.(11) as X i ∈ Λ ǫ (ˆ x i (1) , γ ( f (ˆ x i (1)) , x i (0)))+ X i ∈ Λ ǫ (ˆ x i (1) , γ ( g (ˆ x i (1)) , x i (0)))+ X i ∈ Λ ǫ (ˆ x i (1) , γ ( f (ˆ x i (1)) , x i (0))) , (12)where Λ := { i | f (ˆ x i (1)) > g (ˆ x i (1)) } Λ := { i | f (ˆ x i (1)) < g (ˆ x i (1)) } Λ := { i | f (ˆ x i (1)) = g (ˆ x i (1)) } . By Lemma 3 we have that ǫ (ˆ b ( i, , γ ( g (ˆ b ( i, , b ( i, ǫ (ˆ b ( i, , γ ( f (ˆ b ( i, , b ( i, , (13) nd therefore the expression in (12) is equal to: X i ∈ Λ ǫ (ˆ b ( i, , γ ( f (ˆ b ( i, , b ( i, X i ∈ Λ ǫ (ˆ b ( i, , γ ( f (ˆ b ( i, , b ( i, X i ∈ Λ ǫ (ˆ b ( i, , γ ( f (ˆ b ( i, , b ( i, N − X i =0 ǫ (ˆ b ( i, , γ ( f (ˆ b ( i, , b ( i, . Since ǫ is a density conserved quantity for f the above is equal to N − X i =0 ǫ (ˆ b ( i, , b ( i, , and therefore we have that N − X i =0 ǫ (ˆ b ( i, , b ( i, N − X i =0 ǫ (ˆ b ( i, , γ (( f ∨ g )(ˆ b ( i, , b ( i, . (cid:3) Example 2
As the size of the alphabet increases, the number of possible reversiblecombiners undergoes a combinatorial explosion, so computing the equivalence set F I becomes computationally impossible. What we can do, however, is to fix a reversiblecombiner and hunt for pairs ( f, g ) of CA with the same conserved quantities and checkthe validity of the theorem by observing that ψ ( f ) ⊆ ψ ( f ∨ g ) .For instance, for | S | = 3 and the reversible combiner γ such that γ (0, 0) = 0; γ (1, 0)= 1; γ (2, 0) = 2; γ (0, 1) = 1; γ (1, 1) = 0; γ (2, 1) = 2; γ (0, 2) = 2; γ (1, 2) = 1; γ (2, 2)= 0, we found that the CA (expressed in Wolfram’s notation in base 3) and belong to the same class. Then we computed ∨ and verified that ψ (1311051521973) ⊆ ψ (6589964923167) . Remark 3
The substance of the theorem is that of proving the existence of an orderrelation on CA which is preserved by the conserved quantities. Consider the set F I ofsecond-order reversible CA with a fixed window I with the partial order relation inducedby ∨ : f < g ⇐⇒ f ∨ g = g. Then, if we consider the set h C i = { f ∨ g } f,g ∈ C generated by any equivalence class C ∈ F I / ∼ , the theorem says that ψ is an order-preserving map. Indeed, since there isa natural partial order ⊆ between subspaces of R n , we have that f < g ⇒ ψ ( f ) ⊆ ψ ( g ) . In this article we have studied a certain class of second-order, reversible CA that werefirst considered by Takesue [1] for their ability to exhibit thermodynamic behavior.The inspiration for this work has, in part, come from Noether’s theorem, whichis one of the most celebrated results both in mathematics and physics. It asserts hat conserved quantities are in correspondence with symmetries of the laws of nature.Time-translation symmetry is associated with conservation of energy, space-translationsymmetry with conservation of momentum, and rotation-symmetry is associated withconservation of angular momentum.These results hold under the assumption that space and time are both continuous:to what extent do they still hold for discrete systems such as CA? The answer to thisquestion is still an open problem, and a fully satisfactory discrete version of Noether’stheorem has not yet been formulated. In this work, however, we show that there existsa logical organization of additive conserved quantities which reflects some symmetriesof the system, at least for a large class of second-order, reversible CA. In particular,we have shown that there is a partial order relation defined on CA which is preservedby the set of their respective conserved quantities.There are still many open problems in the theory of reversible CA which we hopeto approach using our result. In particular, there is an interesting conjecture claimedin [2] which states that “antisymmetric conserved quantities”, that is those such that E ( x ( t ) , x ( t + 1)) = − E ( x ( t + 1) , x ( t )), can only be associated to CA endowed withspecial symmetries (Rule 90, for instance, is one of them). More generally, it appearsthat, even in the discrete case, it is possible to classify conserved quantities accordingto symmetry groups.As in a previous paper of ours [8], we have herein suggested an approach to thisproblem which only uses combinatorial techniques. We hope that this work will con-tribute towards the formulation of a discrete version of Noether’s theorem, which soneatly classifies the conserved quantities of continuous systems according to the char-acteristics of their dynamics. Acknowledgments
This work was partially funded by ARO award number W911NF-04-1-0334, AFOSRaward number FA9550410176, and facilitated by scientific visualization equipmentfunded by NSF award number 0619447. The authors are grateful to Peter Love andZbigniew Nitecki for helpful conversations.
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