Burton Voorhees- A Note on Injectivity of Additive Cellular Automata

8/3/2019 Burton Voorhees- A Note on Injectivity of Additive Cellular Automata

1/9

Complex Systems 8 (1994) 151- 159

A Note on Inj ectivity ofAdditive Cellular Automata

Burton VoorheesFaculty of Science, Athabasca University,Box 10,000, Athabasca, Alberta TOG 2RO, Canada

Abstract. Additive cellular automata on finite sequences with periodic boundary conditions are tr eated in terms of complex polynomialswhose arguments are roots of unity. It is shown that the condition fora binary one-dimensional additive cellular automaton to be injectiveis that the associated complex polynomial have no zeros that are rootsof unity.

1. IntroductionCellular automata are discrete symbolic dynamical systems defined in termsof a latt ice of sites , L; an alphabet of symbo ls, K ; and an evolut ion rule,X , which maps configurations at any given time t to new configurat ions att ime t + 1. A configurat ion, or state, is an assignment of a symbo l from Kto every site of the lat t ice L. The set of all possible configurations is calledthe configuration space , denoted by E in the generic case.

Given a configuration p(t), t he evolution rule generates a new configuration p(t + 1) by assigning to every site in the lat ti ce a symbol chosen fromthe alphabet on the basis of the symbols in a neighborhood at that site.

In t his note t he lat tice is taken as a finite set of n sites located on thecircumference of a circle. This gives what has been called a cylindrical cellularautoma ton [1], because the evolution can be visualized as occurring on acylinder. In this case , the configuration space En consists of all periodicsequences of symbols with periods t ha t divide n . In addition, considerationis rest ricted to binary cellular automata, for which the alphabet is the set{a, I}.

The neighborhood of a site consists of a consecut ive block of k sites withinwhich the given site occupies a designated position. Here th is position isassumed to be located at the left-hand endpoint of the neighborhood; thatis, th e neighborhoods are left just ified.The evolution ru le is defined locally by a rule table specifying th e symbols

that are assigned to the designated site, for every neighborhood. Th is also


2/9

152 Burton Voorheesdefines a unique global operator X : En ----> En. The global operator is represented in terms of local neighborhood maps by defining its ith componentas

X i = X (io . . . ik - I ) (1.1)where io . .. ik - 1, t he ith neighborhood, is the binary expression for theindex i . The component form of X is writ ten as a "vector" with respect tothe "neighborhood basis,"

(1.2)The map X is surject ive if for every configuration (3 there is a configuration J-tsuch t hat X (J-t) = (3. If, in addition, this predecessor configuration is unique,t hen the map X is inject ive. For cellular automata, injectivity is equivalentto reversibility. Hence, if X is injective, there is another cellular automatarule X -I such that if X (J-t) = (3, t hen X -I ((3) = J-t .

I t is known th at t he question of whether or not a particular cellularautomaton is injective is decidable only in dimension one [2, 3]. Recenttheoret ical studies of reversible cellular aut omata have been carried out byHead [4], Toffoli and Margolus [5], McInt osh [6], and Hillman [7]. Fredkin [8]has suggested that reversible rules may provide a basis for modeling reversiblephysical processes.In this paper considerat ions are restricted to addit ive cellular automataru les, that is, those that sat isfy the condition(1.3)

where all sums are computed modulo 2.Restriction of the configuration space to En rather than a set of infinite orhalf-infinite binary sequen ces, is not a serious constraint as far as injectivityis concerned since it is known that a cellular automata rule is injective ont hese larger spaces if and only if it is inject ive on all periodic configuratio ns[9] .

The addit ivity condit ion (1.3) requires that Xo = O. In addition, equation(1.1) for addit ive rules takes the formk- I

X( io. . . i k - I ) = L asiss=o (1.4)It also possible to give an expression for an addit ive rule X in terms of theleft shift opera tor (J" , defined by [(J" (J-t )] i = J-ti+ J-ti+ 1:

k- IX = L as(J" ss=o (1.5)The coefficients in (1.5) are easily determined in te rms of th e components ofX by solving equation 1.4 with X( io . .. i k- I ) = X i .


3/9

A Note on Injectivity of Additive Cellular A utom ata 153In section 2 a representa tion of addit ive cellular automata defined on En

is given in terms of complex polynomials. Sect ion 3 proves that an addit ivecellular automaton rule X : En --+ En is injective if and only if its associat edcomplex polynom ial has no zeros that are nt h roots of unity. Fin ally, in section 4, a restatement is given of a theorem of Martin , Odlyzko, and Wolfram[10] rela ting injectivity and reachability of configurations.2. R epresent a tions of addit ive rulesIn their classic st udy of addit ive cellular automa ta , Mart in, Odlyzko, andWolfram [10] made use of a dipolynomial representation, that is , states i.L EEn were represented as polynomials of the form

n'" t S - 1i.L --+ i.Ls .s=l (2.1)

The act ion of t he cellular automaton rule was represented as multiplicationby a dipolynomial of the formk-1r: L cst Ss=o (2.2)

with all indices and powers reduced modulo n . This corresponds to the shiftrepresent ati on (1.5) when r = a since left-just ified neighborhoods are beingused.Taking a different approach to additive rules, Guan and He [1] representedconfigurat ions as n-dimensional vectors and evolution rules as multiplicationof these vectors by certain circulant matrices, with all terms reduced modulo2. They also made use of left-justified neighborhoods, and t he circulantrepresentat ion of a rule given in the form of equation (1.5) is obtained bysubst itut ion of the circulant form for the left shift operator:

(

0 1 a aa ""1010 0) !!:l

100 a(2.3)

A connection between these different approaches can be made in termsof a complex polynomial p associat ed to each rule. It t urns out that whatis important are the values p(wn ) where W n = exp(27ri /n) is an nth root ofunity. In what follows t he subscript on W n will generally be supressed, withthe understanding that W is defined in terms of whatever value of n is underconsideration.Configurations are now represented as polynomials in the roots of unity:n

i.L =L i.LsW S- 1 .8= 1

(2.4)


4/9

154 Burton VoorheesA cellular automat on rul e X t akes t he form of multiplication by the complexconjugate of t he polynomial

n -lp(W) = 2::'>sws

s=o(2.5)

where the coefficients as are the entr ies in the circulant representation ofX : circ (aOal ' " an-d , and all sums are t aken modulo 2. Redu ction modulon, necessary in the dipolynomial approach, is au t omat ic since wn = 1.Much is known about circulants and their relation t o roots of unity, anda bri ef summary of resul t s that will be useful in this note concludes t hissection. These results are taken from the detailed st udy of circulant matricesby Davis [12].Lemma 2.1.

1. An n x n matrix A is circulant i f and only if it commutes with the shiftoperator.2. An n x n matrix A is circulant if and only if it has the form A = PA((J )where (J is the shift .

Definition 2.2. Th e Fourier matrix of order n is the matrix1 1 1 1 11 wn - 1 wn - 2 wn - 3 W

F = _I_ I wn - 2 wn - 4 wn - 6 w2 (2.6)yin1 w2 w4 w6 wn - 21 W w2 w3 n -lW .

Th e Hermitian conjugate of this matrix (i.e., the transpose of the complexconjugate) is denoted F *. This matrix is unitary, that is, F F* = F *F = I ,and its eigenvalues are 1 and i with multiplicity depending on the valueofn.Lemma 2.3. Let A = circ(aOal . . . an- I) have associated polynomial PA(W )and let A(A ) be the diagonal matrix

A(A) = diag(PA(I ),PA(w), .. . ,PA(Wn - 1)).Th en A = F *A(A )F .Corollary 2.4. The eigenvalues of A are A i = PA(W i ) .Remark. Since [(J(p.) ]i = P.i+l , the sh ift is equivalent to mul tiplication byw n - 1 , the complex conjugate of w . Hence the act ion of a rul e X, representedby circulant matrix A, on a state p.(w), is ob tained by multiplying p.(w) byPA(Wn - 1), t he nth eigenvalue of A. .Corollary 2.5. I f A is non-singular, then A -I = F *A- I(A) F .


5/9

A Note on Injectivity of Additive Cellular Automata 1553. Injectivi ty of additive rulesSince reversibility and inject ivity are equivalent , an addit ive cellular au tomaton rule X : En ---> En represented by a circulant matrix A will be injectiveif and only if A- I exists. From Corollary 2.5 we see that t his will be the caseif and only if none of the diagonal ent ries of A(A ) are zero . Recalling thatthese entries are reduced modulo 2, and not ing that PA(1) = I as> thisyields the conditions for injectivity of additive cellular automat a rules.Theorem 3.1. Let X : En ---> En be an additive cellular automaton represented by a circulant matrix A = circ(aO al .. .an-I). The rule X is injectiveif and only if no nth root of uni ty is a roo t of the complex polynomial PAmodulo 2.R emark: Since ui" = 1 is an nth root of unity, t his condit ion requires thatan odd number of the coefficients as be nonzero. We also not e that the rootsof complex polynomials come in complex conjugate pairs. Hence if wT is aroot , then so is ur" ,

The condition in Theorem 3.1 requires that PA be irreducible with respectto the nth roots of unity. If we are only interest ed in whether or not PA hasroots tha t are nth roots of unity for some n , rat her than for specified valuesof n , this can be determined from the contour integral

No = lim _1_ 1 p ~ ( z dz (3.1) E represented bya circulant matrix A = circ(aOal . .. an-d will be injective if and only ifpA(z )is irreducible with respect to all roots of unity.

As an example, consider the well-known three-site rule 150 defined by[X( J-i )Ji = J-ii+ J-iH I + J-iH 2 The act ion of this rule on a configuration J-i(w) isobtained by multiplicat ion of J-i(w) by PA(Wn- l ) = 1+wn- l +wn- 2 For thisrule PA(Z) = 1+ Z+ Z2, which has roots given by Z = + i These arepowers of w = exp(21ri/3) . Hence rule 150 is not injective when 3 I n , and isinjective on all periodic sequences for which n I- 3m for any int eger m. Thenext theorem extends this well-known result [11,13].


6/9

156 Burton Voorheesy

He-----1-+--+--- -+- - - -+- +- -+ - - - --> X

Figure 1: Con tour for computat ion of No in equation (3.1). Integration is counterclockwise around t he circle ofr adius 1+E, and clockwisearound t he circle of radius 1 - E.

Theorem 3.3 . Let X : En -+ En be k-site additive cellular automaton forwhich every coefficient as in equation (1.5) is equal to 1. I f k is even, X isnever injective. I f k is odd, X is injective for all values of n which are notdivisible by k .

P roof. If all coefficients in equation (1.5 ) are unity, then PA(Z) = 1+Z+ Z2+...+ z k - l . I f k is even , then there are an even number of non zero coefficientsas, and PA(1) = 0 (mod 2) . Hence X cannot be inje ctive in this case .I f k is odd , PA (1) = 1 (mod 2), but w = exp(21rri/k) is a root for1 :::; r :::; k. Hence for n = mk , exp (21rmi /n ) will be a root . Further , PA hasdegree k - 1, and hence has only k - 1 roo ts, so no other values of n can

yield roots . Thus, so long as n #- mk , the rule is injective.

Table 1 list s t he additive rul es for up to five site neighborhoods, andindica tes conditions for their reversibility.

In those cases where a rul e is injective, its inverse can be computed. Theexample of rul e 150 acting on E4 and E5 indi cates, however, that t his inversemust generally be expected to dep end on the period n . For n = 4, the inverseof rul e 150 is computed to be 1+ (J2 + (J 3, while for n = 5 it is ( J ( I + (J + (J3 ).


7/9

A Note on Injectivity of Additive Cellular Automata

as Shift Form of Number of Injectivitycoefficients Rule Sites Conditions00000 0 1 never00001 0-4 5 always00010 0-3 4 always00011 0-3 + 0-4 5 never00100 0-2 3 always00101 0-2 + 0-4 5 never00111 0-2 + 0-3 + 0-4 5 n i ' 3m01000 0- 2 always01001 0- + 0-4 5 never01010 0- + 0-3 4 never01011 0- + 0-3 + 0-4 5 always01100 0- + 0-2 3 never01101 0- + 0-2 + 0-4 5 always01110 0- + 0-2 + 0-3 4 n i ' 3m01111 0- + 0-2 + 0-3 + 0-4 5 never10000 1 1 always10001 1 + 0-4 5 never10010 1 + 0-3 4 never10011 1 + 0-3 + 0-4 5 always10100 1 + 0-2 3 never10101 1 + 0-2 + 0-4 5 n i ' 3m10110 1 + 0-2 + 0-3 4 always10111 1 + 0-2 + 0-3 + 0-4 5 never11000 1 + 0 - 2 never11001 1 + 0- + 0-4 5 always11010 1 + 0- + 0-3 4 always11011 1 + 0- + 0-3 + 0-4 5 never11100 1 + 0- + 0-2 3 n i ' 3m11101 1 + 0- + 0-2 + 0-4 5 never11110 1 + 0- + 0-2 + 0-3 4 never11111 1 + 0- + 0-2 + 0-3 + 0-4 5 n i ' 5m

Table 1: Injectivity of additive rules for five sites or less.

157


8/9

158 Burton Voorhees

4 . Injectivi ty and reachabilityA quest ion of major interest for studies of cellular automata is whether ornot a given configurat ion p,has a predecessor. Clearly, if a rule X : En Enis injective, then all configurations have predecessors. In general, however,t his is not t he case . In t heir classic analysis of addit ive cellular automata ,Mart in, Odlyzko, and Wolfram [10] prove a lemma specifying the conditionsunder which a configuration is reachable, that is, has a predecessor. Usingthe dipolynomial notation of equations (2.1) and (2.2) th eir result is given inthe next lemma:Lemma 4 .1 (10, Lemma 4.4) A configuration p,(t ) is reachable in the evo-lu tion of a size n addi tive cellular automaton over Zp, as described by T (t),if and only i f p,(t ) is divisible by the greatest com m on divisor A 1 (t ) =gcd(xn - 1,T (x)).

In terms of the nth roots of unity, th is can be restated in a form th atmakes the connect ion to injectivity explicit . For simplicity, the alphabet isrestricted to Z2.Lemma 4.2 . Let X : En En be an additive cellular automaton repre-sented by the polynomial PA. Furth er, let PA(W) be decomposed into irre-ducible factors

r SPA(W) = II 'iT;(w) II ll j (w ) (4.1);= 1 j= l

where each 'iT (w) represen ts an injective rule and the llj(w) represent non-inj ecti ve rules.A configuration p,(w) is reachable i f and only if

SII ll j (w) I p,(w ). (4.2)j = l

Proof: If p,(w ) is reachable, then there is a p, (w ) such that PA(W)P,'(w) =p,(w) and (4.2) is sat isfied as a consequence of equation (4.1).Conversely, suppose that equat ion (4.2) is sat isfied. Since each 'iT; repre-sents an injective rule, th ere exists an inverse 'iTi1 that is also a polynomialin w . Thuss rII ll j (w) = PA(W) I I 'iTi 1 (w ). (4.3)

j = l ;= 1But (4.2) implies that

Sp,(w ) = II ll j(W)p(w) for some p(w) .j= l

Hence, by (4.3),r

p,(w ) = PA(W) I I 'iTi 1(w )p(w ),;= 1

which provides a predecessor for p,(w ). (4.4)


9/9

A Note on Injectivity of A dditive Cellular A utomataAcknowledgmentsT his work was supported by NSERC Operating Gr ant OPG 0024817.R eferences

159

[1] E. J en, "Cylindrical Cellular Automata," Communications in MathematicalPhysics, 11 8 (1988) 569-590.

[2] S. Amoroso and Y. N. Pat t , "Decision Procedures for Surjectivity and Inject ivity of Parallel Maps for Tessellation Structures," Journ al of Computer andSystem Science, 6 (1972) 448-464.

[3] J. Kari, "Reversibility and Surjectivity Problems of Cellular Automat a,"Journal of Computer and System Science (to app ear).

[4] T . Head, "One-Dimensional Cellular Automata: Injectivity from Unambiguity," Complex Syst ems, 3 (1989) 343- 348.

[5] T. Toffoli and N. Margolus, "Invert ible Cellular Automata: A Review," Phys-ica D, 45 (1990) 229-253.

[6] H. V. McIntosh , "Reversible Cellular Automata," Physica D (to appear).[7] D. Hillman , "The Structure of Reversible One-Dimensional Cellular Au

tomat a," Physi ca D, 52 (1991) 277-292.[8] E. Fredkin , "Digit al Mechanics: An Informational P rocess Based on Re

versible Universal Cellular Automata," Physica D, 45 (1990) 254-270.[9] K. Culik II , L. P. Hurd , and S. Yu, "Computat ion Theoretic Aspects of Cel

lular Automata," Physica D, 45 (1990) 357-378.[10] O. Mart in , A. M. Odlyzko, and S. Wolfram, "Algebra ic P ropert ies of Cellular

Automata," Communications in Math em atical Physics, 93 (1984) 219- 258.[11] P. Guan and Y. He, "Exact Results for Deterministic Cellular Automata,"

Journal of Statistical Physics, 43 (1986) 463-478.[12] P. J . Davis, Circulant Mat rices (New York: John Wiley, 1979).[13] B. Voorhees, "P redecessors of Cellular Automata Stat es: 1. Additive Au

tomata," Physica D, 68 (1993) 283-292.

Documents

Burton Voorhees- A Note on Injectivity of Additive Cellular Automata