A study of Inverse Ultra-discretization of cellular automata
aa r X i v : . [ n li n . C G ] A p r A study of Inverse Ultra-discretization of cellular automata
Norihito ToyotaFaculty of Business Administration and Information Science,Hokkaido Information University, Ebetsu, Nisinopporo 59-2, [email protected]
Abstract
In this article, I propose a systematic method for the inverse ultra-discretization of cellautomata using a functionally complete operation. We derive difference equations for the 256kinds of elementary cellular automata(ECA) introduced Wolfram[9] by the proposed meansof the inverse ultra-discretization. We show that the behaviors of ECAs can be completelyreproduced by numerically solving the obtained difference equations. keyword; elementary cellular automaton, functionally complete operation,inverse ultra-discretization, difference equation
It is known that we can obtain difference equations by discretizing independenent variables,time and space, of differential equations. Moreover the method of the ultra-discretization hasbeen established [2][3][4] where the dependent variable is also discretized to get cellular au-tomata (CA)[1]. This method has mainly developed in a field of traffic flow. For an example,We can derive the descritized Burgers’ equation by descritizing the Burgers’ equation. More-over we obtain the ultra-discretized Burgers’ equation by descritizing the dependent valiableof it[5], which corresponds to rule 184 ECA[6]. Recently the method of the inverse ultra-discretization has developed[7]. Some CAs are lead to the corresponding difference equationsby an inverse process of the ultra-discretization.In this article, we propose a systematic method to inversely ultra-discretize CA by us-ing a functionally complete operation, for example, And-gate, OR-gate and NOT-gate. Toinversely ultra-discretize CA, the method using local linear equations and Max Plus algebraare developed in as ever[7]. In such a method, it is not clear how to derive the local linearequations. It is not seemed that there is any general algorithm in the sutudies. On the otherhand, Beniura and Nakano have proposed a method to inversely ultra-discretize ECAs sys-tematically by studying CAs connected in some transformations[8]. This method, however,is not clear to work in other CAs except for ECA. The method proposed in this article issystematic and solid because the method is based on functionally complete operations. Usingthis method, I inversely ultra-discretize 256 kinds of ECA introduced by Worfram[6]. Then Inumerically solve the difference equations given by the inverse ultra-discretization and showthat the behaviors of the solutions reproduce the properties of the original ECAs wholly .
In this article, t represents a discrete time, n is a position of a cell and u ∈ { , } is a statevariable. Thus u tn represents a sate in n -th cell at time t . A presented method of the inverseultra-discretization is described below.1. derive a logical expression consist of ANDi ∧ )COR( ∨ ) and NOT(¯ u ) which reproducesthe outputs of a CAFor example, focusing the following rule of a CA with 2 states and 3 neighborhoods u tn − u tn u tn +1 = 101 → u t +1 n = 1 , we obtain the following logical expression; u tn − ∧ u tn ∧ u tn +1 . hus we derive logical expressions for all outputs of the CA and connect their logical expres-sions by OR (principal disjunctive canonical form).2. rewrite the constructed logical expressions using AND and NOT as a sort of a minimalfunctionally complete operation.By using the de Morgan’s theorem, OR is rewritten by x ∨ y = ¯ x ∧ ¯ y. (1)The following formula is also useful for the exclusiveOR on occasion. x ⊕ y = x ∧ y ∧ ¯ x ∧ ¯ y. (2)3. transform the presented logical formula to Max operation.First the following transformation is made for the presented formula consist of AND and NOT; u tn −→ (1 − u tn ) ,u ∧ v −→ u × v, (3)where × represents a usual multiplication. Then we set up a formula max[ F ( u tn +1 , u tn , u tn − , F ( u tn +1 , u tn , u tn − ).4. construct a difference equation by using the following ultra-discretization formula forreal numberes A and B ;max [ A, B ] = lim ε → ε log (cid:20) exp (cid:18) Aε (cid:19) + exp (cid:18) Bε (cid:19)(cid:21) , (4)where the resltant formula is written as u t +1 n = E ε ( no.ofrule ) ( u tn − , u tn , u tn +1 ).Really we fix ε to a small positive finite value, because we cannot actually take ε → u tn , however, converges to 0 after many time steps and so the resultantdifference equation cannot support the properties of ECA. We need the following step 5 inorder to avoid this distress.5. set up simultaneous difference equations by using a kind of filter[7].First of all, replace the derived formula u t +1 n = E ε ( no.ofrule ) ( u tn − , u tn , u tn +1 ) for u t +1 n with v t definrd by v tn = F ε ( no.ofrule ) ( u tn ) = 11 + e − ( u tn − ∆) /ε , (5)where ∆ takes a very small positive value as ε . From (5), F ε ( no.ofrule ) ( u tn ) comes close to 0when u tn − ∆ < F ε ( no.ofrule ) ( v tn ) comes close to 1 when u tn − ∆ >
0. Thus ∆ paly a roleof a threshold.
We explain the presented method in the case of rule 161 among 256 ECAs as an example inthis section. The transition rule of the rule 161 ECA is given by Table 1.
Table 1: Rule 161 u tn − , u tn , u tn +1 u t +1 n For simplicity, We use the logical formula (6) given by Wolfram[9] instead of the way givenin the example in Step 1 of the previous section(Step 1). We can show that the way given inthe example also gives essentially the same results as the method proposed here. u t +1 n = u tn − ⊕ u tn +1 ⊕ ( u tn − ∨ u tn +1 ∨ u tn ) . (6) sing (2), we rewrite this logical formula with one represented by AND and NOT(Step 2). u t +1 n = (cid:16) u tn − ∧ u tn +1 ∧ u tn − ∧ u tn +1 ∧ u tn (cid:17) ∧ (cid:18) u tn − ∧ u tn +1 ∧ (cid:16) u tn − ∧ u tn +1 ∧ u tn (cid:17)(cid:19) ∧ (cid:18) u tn − ∧ u tn +1 ∧ (cid:16) u tn − ∧ u tn +1 ∧ u tn (cid:17)(cid:19) ∧ (cid:16) u tn − ∧ u tn +1 ∧ u tn − ∧ u tn +1 ∧ u tn (cid:17) . (7)We apply the transformation (3) to (7) and set up a formula using max-operation(Step 3).So we obtain the following defference equation from (7). u t +1 n = max[ (cid:16) − (1 − u tn − ) × (1 − u tn +1 ) × (1 − u tn − ) × (cid:0) − u tn +1 (cid:1) × u tn (cid:17) × (cid:16) − (cid:0) − u tn − (cid:1) × u tn +1 × (cid:0) − (cid:0) − u tn − (cid:1) × (cid:0) − u tn +1 (cid:1) × u tn (cid:1)(cid:17) × (cid:16) − u tn − × (cid:0) − u tn − (cid:1) × (cid:0) − (cid:0) − u tn − (cid:1) × (cid:0) − u tn +1 (cid:1) × u tn +1 (cid:1)(cid:17) × (cid:16) − u tn − × u tn +1 × (cid:0) − u tn − (cid:1) × (cid:0) − u tn +1 (cid:1) × u tn (cid:17) , , (8)where we interpret u tn as a real number u tn ∈ [0 , u t +1 n = E ε (161) (cid:0) u tn − , u tn , u tn +1 (cid:1) , (9) E ε (161) ( x, y, z ) ≡ ε log (cid:20)
12 exp n(cid:16) − (1 − x ) × (1 − z )(1 − x )(1 − z ) y (cid:17) exp (cid:16) − (1 − x ) × z (1 − (1 − x )(1 − z ) y ) (cid:17) exp (cid:16) − x ) × (1 − z )((1 − (1 − x )(1 − z ) y ) (cid:17) exp (cid:16) − xz (1 − x )(1 − z ) y ) (cid:17) /ε o + 1 / , (10)where we use the following notations; x = u tn − , y = u tn , z = u tn +1 . u t +1 n comes close to 0 with time steps as mentiond above. This behavior is shown in theleft side of Fig.1.Thus we introduced the filter as explained in the step 5 and obtain (Step 5) F ε (161) ( u ) ≡
11 + e − ( u − ∆) /ε (11) (cid:26) u t +1 n = E ε (161) (cid:0) v tn − , v tn , v tn +1 (cid:1) v tn = F ε (161) (cid:0) u tn (cid:1) . (12) Figure 1: Behaviors of Rule 161 CA(right) and the solution of the corresponding difference equationwithout the filter(left). 3igure 2: Behaviors of Rule 161 CA(right) and the solution of the corresponding difference equationwithout the filter(left).
We numerically solve the difference equations obtained by the proposed procedure under aperiodic boundary condition. The behaviors of the numerical solutions are shown on a lattice,where the vertical direction represents time axis. The black cells show the state to be 1 andthe white ones show the state to be 0. The initial state is taken to be homogeneously random,the size of the cell number (horizontal size) is 1000 and it takes 1000 time steps.The right side of Fig.1 shows time developments for rule 161. The left side of Fig.1 showsthe numerical solution of (10) without filter where the behabvior is almost same as one of theright figure up to a few steps, but the states of all cells rapidly converse to 0-state after sometime steps.The left side of Fig.2 shows the numerical solution of (11) and (12) with the filter. Com-pared with the right of Fig.2 which shows the result of rule 161, we observe that both resultsagree perfectly.We carried out these procedures for all ECAs and confirmed that the behaviors of ECAsand those of the solutions of the corresponding difference equations are completely same forin all cases. These observations show that our method of the inverse ultra-discretization iscorrect.
I propose a systematic method for the inverse ultra-discretization via the logical formulas,especially a functionally complete operation, in this article. It is characteristic to adopt aminimal functionally complete operation, AND and NOT in the presented method. So itguarantees that this procedure stands good for all CAs with 2 states. We carried out ourprocedure for all ECAs and confirmed that the behaviors of ECAs and the solutions of dif-ference equations completely agree in all cases. This fact justifies our method of the inverseultra-discretization.For inversely ultra-discretizing CA, the method using local linear equations and Max Plusalgebra have been studied in a conventional manner[7]. In such method, it is not clear howto derive the local linear equations. Any general algorithm to get local linear equations seemsnot to be given in the sutudies. Beniura and Nakano have proposed a systematic methodto inversely ultra-discretize CA by studying CAs connected in some transformations[8]. Thismethod is not clear to work in other CAs except for ECA. Their method leads to defferenceequations that differ from the ones in this article, but the behaviors of numerical solutions ofthe both difference equations are the same for time developments. This fact shows that therecan be several corresponding difference equations for a CA. This may be effective to studythe equivalence class of difference equations. Furthermore to derive corresponding partialdifferential equations from the defference equationsis is worth studing. eferences