Synthesis of One-Dimensional Binary Scope-2 Flexible Cellular Systems from Initial Final Configuration Pairs* PATRICK E. WHITE

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1 Reprinted from INFORMATION AND CONTROL Vol. 46, No.3, September 1980 All Rights Reserved by Academic Press, New York and London Printed in Belgium Synthesis of One-Dimensional Binary Scope-2 Flexible Cellular Systems from Initial Final Configuration Pairs* PATRICK E. WHITE Bell Telephone Laboratories, Naperville, Illinois AND JON T. BUTLER Department ofelectrical Engineering and Computer Science, Northwestern University, Evanston, Illinois This paper focuses on I-dimell,sional, binary, scope-2 cellular systems which are flexible; that is, at each point in time, the local transition function can be chosen from a set of functions. It is shown that Garden-of-Eden configurations can be uniquely characterized in such systems and that approximately 50 % of the set of large configurations are Garden-of-Eden. Further, a procedure is shown which produces the sequence of local transition functions which generates some given fmal configuration froni a given initial configuration, if indeed such a sequence exists. If the derivation is monotonic increasing (decreasing), that is, follow-on configurations are always larger (smaller) than or equal to the previous configuration, the procedure is all, algorithm. I. INTRODUCTION The inspiration for most of the work in cellular automata systems is the system developed by von Neumann (1966), which was capable of any computation and which could construct any automaton including itself, given a description of that automaton. Cellular systems have since been used for information storage and retrieval applications as in Lee and Paull (1963), the simultaneous execution of subprograms as in Holland (1970), biological modeling as in Stahl (1967), heart tissue modeling as in Moe et al. (1964), and pattern processing as in Unger (1959). Moore (1962) answered the question of whether there existed * This research has been sponsored in part by NSF Grant Mes 76oo326-AOl /80/ $02.00/0 Copyright <C 1980 by Academic Press, Inc. All rights of repro<iu.:tion in any form reserved.

2 242 WHITE AND BUTLER configurations, called Garden-of-Eden configurations, which could only occur as the first configuration. That is, such configurations are never generated from any other. Recent works related to this idea have dealt with the surjectivity and injectivity of global transition functions with respect to the set of all and the set of finite configurations (cf. Amoroso and Patt (1972), Hedlund (1969), Marouka and Kimura (1976), Yaku (1974), and Nasu (1978»). This paper focuses on the problem of designing a cellular automata system which transforms a given finite initial configuration into a given final configuration. In particular, the cellular system of interest is I-dimensional and has both a two-state cell, and a two-cell neighborhood. Although the local transition function is applied at the same time and is the same for all cells, it can be ditterent at ditterent points in time. In this paper, we show that Garden-of-Eden configurations are uniquely characterized in such a system. This allows a counting of the percent of configurations which are Garden-of-Eden. Further, a procedure is shown that produces a cellular system of the desired type, if indeed such a system exists. Related to this topic is an algorithm by Butler (1979) for determining a sequence of local transition functions on a two-cell neighborhood, which is equivalent to some given function on a larger neighborhood, if indeed such a sequence exists. Other related works include the "completeness problem" by Yamada and Amoroso (1970) which deals with deciding whether a given finite configuration can be generated from a specified primitive pattern. II. NOTATION The I-dimensional cellular spaces considered in this paper are based on a general model proposed by Yamada and Amoroso (1970). A flexible cellular system, A, is a 5-tuple: V is a finite nonempty set called the state alphabet of A. For this paper, V is {O, I} and represents the states assumed by each cell in the space. Z denotes the set of integers that identify the cells in the space. Specifically, cells are numbered in ascending order from left to right. X is a p-tuple of integers called the neighborhood index. Let X = (0,1,2,..., P 1) and define: Nx: Z -). ZP by NxCk) (k, k + 1, k + 2,..., k + p - 1). Each component of Nx(k) is called a neighbor of cell k. Note that since 0 E each cell is its own neighbor. Cellular systems with neighborhood indices defined in this manner are called scope-p systems.

3 SYNTHESIS FROM INITIAL/FINAL PAIRS 243 Let(an assignment of states to cells be a configuration, c. Thus c: Z -li V. The image c(k) is referred to as the state of cell kin c. The state of any cell at time t + 1 can be uniquely determined from the states of neighboring cells at time t according to function (Jo'; VI' -l> V called the local transition function or local map. The local transition function is not assumed to be constant for all time. However, at any given instant in time, all cells have the same local transition function. I represents the set of all possible local transition functions which may be executed by A. A cellular system in which 2.< III is said to be flexible. Note that the restriction in which the neighborhood index is contiguous is not. a serious one. Any neighborhood index with noncontiguous cells can be represented by a contiguous one in which the local transition function is defined tobe independent of certain neighbors. However, it is assumed that a depends 0n the extreme elements in the neighborhood. That is, if X = (O,i"..., i p _ 1 ), there exists a set of assignments (1, a l,..., a p _ I ), and (0, a l ':, a p _ 1 ) such that a(l, a 1,..., a p _ l ) *' a(o, al"'" a p _ 1 )' Similarly, there exists a set of assignments, (bo, bl"'" b p _ 2 ' 0) and (bo' bl"'" b p _ 2 ' 1) to (0, i l,..., i p _ 1 ) such that a(b o ' b l,..., b p _ 2 ' 0) *' a(b o, b p..., b p _ 2, 1). qo E V is called the quiescent state of A and has the property that u(qo,"" qo} qo for all IT E 1. In this paper, qo is represented by O. Let c be a configuration at time t, and let c' be the new configuration at time t + 1 that is formed by applying u to the states of the neighborhood of cell k inc. The mapping.x.,,: C-l> c' is called a global transition function or parallel map. In other words,.x." is an operation on configurations which corresponds to the application of local map u on a neighborhood index X. c is said to be the predecessor of c'. The support of the function c: Z -l> A is the set S c Z, such that if k E S, then c(k)::/= O. The function c: Z -ta has finite support if its support is a finite set. The set of all configurations with finite support is represented by C p The size of a configuration c of finite support, denoted as #c is r -I+ 1, where r and I are site indices of the rightmost and leftmost 1, respectively. Thus, the size of c is the number of cells it "occupies." In some cases, it will be convenient to ignore the position of a finite configuration. Two configurations C 1 and Cz are shift equivalent, c 1 ~C2' if they differ by a shift of position. That is, c i ~C2 if Ct(i) = cz(i+s} for all i E Z and some fixed s E Z. For such cases, it is convenient to represent configurations as 01w10, where {j = is the infinite string of O's and w is a string oro's and 1 's of length m ~ O. For example, 011 "10 is a string of k + 21's, where 1k 11 1 (k times). The configuration consisting of a single 1 is (ho. Since a(o, 0,..., O) 0, the application of a parallel map on a configuration with finite support produces a configuration which also has finite support.

4 244 WHITE AND BUlLER This paper focuses on scope-2 systems. The local transition functions considered are 0'(1) a(l)a(2} am 0(4) 0'(5) 0(6) 0(1) I I "(0) is the trivial local transition function mapping all configurations to O. 0'(3) is the identity local transition function and 0"(5) shifts all configurations left one position. Note that the other eight functi!,ns on two variables are not included because of the restriction 0(0, 0) = O. Let co. cl>... c F bea sequence of finite configurations such that c1='rx,u,(co), Cz 'rx,ui(cl),,,,,cf='rx,uf(cf-l) and F~l for oiel, 1 ~ i ~F. Then it is said that A generates c F from Co. The sequence, Co, C p., CF is called a computation of A. The configuration Co is termed the initial configuration, and cf' the final corifiguration. Note that the application of a sequence of local maps, 0 1,0"2,',', OF on X is equivalent to the application of a local map 0 on some larger neighborhood index X'. For example, if X = (0, 1) and a l' Oz,, OF are all two variable AND functions, 0'(1), thenx t = (0, 1,...,F) and 0 is the F + 1 variable AND. See Fig. 1.. The symbol " is used to denote the composition of parallel maps. Thus if )(1 )(2 x3 x4 xf_l. xf 'FH "1 &,,(1) "2 *,,(1) - "3.,,(1)? <IF',,(1) FIG. 1. Sequence of configurations in a cellular system.

5 SYNTHESIS FROM INITIAL/FINAL PAIRS 245 'X'.u can be expressed as a composition of the parallel maps 'X,UF' 'X,UF_I"'" 'X,UI' then an expression for 'X'.", is given by where <x,u, is applied first and 'x,ur- is applied last. If X is an arbitrary neighborhood index such that X c XI, then X is said to be simpler than XI. The parallel map 'x,,,: Co ~ c F is said to be decomposable if there exists a simpler neighborhood index X such that Conversely, 'X',a is said to be indecomposable, if it cannot be represented by any composition of two or more parallel maps on simpler neighborhood indices. The synthesis problem may now be stated. Given two finite configurations Co and c[' such that c F = 'x,.,.{co), where (J is single valued, determine whether 'x,a is decomposable, and, if so, identify a sequence of parallel maps. x',o'" 'X"0"2''' '.X'.UF_. such that With respect to the local transition functions a decomposition corresponding to that in the above expression is given as III. GARDEN-OF-EDEN THEOREM Since Moore's (1962) surprising discovery that certain configurations in a cellular system could exist only initially, (i.e., no configurations could lead to them), much work has been done in this aspect of cellular automata. It has been shown, for example, that in I-dimensional systems certain fmite configurations have only infinite predecessor configurations (Amoroso and Cooper (1970»). Yaku (1973) has shown that for 2- and higher-dimensional systems, it is undecidable whether arbitrary configurations are Garden-of Eden. In a flexible system a finite configuration c is Garden-of-Eden if for all local transition functions in I there are no finite configurations which map to c; With respect to such configurations, we have THEOREM 1. A finite configuration c is Garden-oi-Eden in a binary,

6 246 WHITE AND BUTLER scope-2, flexible cellular system with 1= {at!), 0'(2), 0'(4), 0'(6), a(7)} if and only if c satisfies all of the following conditions. 1. c contains c contains c contains 1 L 4. c contains an odd number of 1'so Let A be the cellular system described in Theorem L Consider local transition function 0'0), the AND function and a configuration c which does not contain 101. Then a predecessor c' to c such that fx,<r'l): C' -+ C can be constructed as follows. For each group of three adjacent states in c, construct c' as shown below. substring of c' aoob aoo HOb substring of c a and b either 0 or 1. For example, has as its predecessor. Note that no conflict arises in concatenating substrings of c'. Furthermore, if c is finite, so also is c'. Thus, if a finite configuration c does not contain 101 it has a finite predecessor under local transition function 0'0). Conversely, consider a finite configuration c which has a finite predecessor c'. Since each 1 in c is the result of a pair of adjacent 1 'sin c'. a substring in c of the form la1 corresponds to a substring 1111 in c'. But this implies a = L Thus, 101 is not a substring of c. This proves, For any finite c in cellular system A there exists afinite PROPosmON 1. c', such that t x.":(1): c' -t c ifand only ifc does not contain 101. By a similar argument, it can be shown for the OR function that, PROPOSITION 2. For any finite c in cellular system A, there exists a finite c' such that t x. ul7): c' -+ C ifand only ifcdoes not contain 010. With respect to the local transition functions 0'(2) and 0'(4), substring 11 in c precludes any predecessor. Specifically, PROPOSITION 3. For any finite c in cellular system A there exists a finite c' such that t x,a(2): C' -+ cor t X u (4): c' -+ C ifand only ifc does not contain 1 L Since 0'(6), the exclusive OR local transition function, satisfies the condition of Amoroso and Patt (1972) for surjectivity, every configuration c has a predecessor c' under 0'(6). Furthermore, exactly two predecessors exist,

7 SYNTIIESIS FROM INITIAL/FINAL PAIRS 247 c' and E', a configuration obtained from c' by complementing all bits. If c' is finite, then E' is infinite. Thus, if c has a fmite predecessor it is unique. On the other hand, there exist configurations in which both predecessors are inlmite (e.g., the two predecessors of 010 are ot and TO). A predecessor c' of a finite configuration c= OaO can be constructed as follows. Choose each 0 on the left of c to be the result of the exclusive OR of 00 in c'. Moving left to right, let each 1 in c result in a change of state in c' and each 0 no change. By this method 010 and 01010, for example', produce as predecessor configurations ot and 0110, respectively. A predecessor configuration c' of c is finite only if the 0 on each side of c yield 0 on each side of c' and this in tum is true if and only if c has an even number of 1'so Thus, PROPosmON 4. For any finite c in cellular system A, there exists afinite c' such that!x.a(61: c' -> C if!c has an even number of 1's. It is interesting to note that Proposition 4 precludes the existence of symmetrical Garden-of-Eden configurations of even size. This is because in a symmetrical configuration of even size, there must be the same number of 1 's in each half, and thus an even number of 1 's in all. It follows from Propositions 1, 2, 3, and 4 that a configuration c in A containing 101, 010, 11, and an odd number of l's has n() predecessors. It also follows that the converse is true. This proves Theorem 1. The total number of configurations c(n) in the cellular system A qf size n is 2 u - 2 for n ~ 2, since a finite configuration is bounded at each end by a 1 and there are 2,,-2 ways to choose the intervening states. The fraction of these which are Garden-of-Eden can be calculated by counting the number of configurations which satisfy the conditions listed in Theorem 1. Let N123in) be the number of configurations of size n which satisfy conditions 1, 2, 3, and 4. A bar above a subscript will denote the property of not satisfying the indicated condition. For example, Ni(n) is the number of configurations not satisfying condition 1 (which mayor may not satisfy other conditions as well). Consider first the calculation of Nl2J(n). Note that N123(n) is strictly less than 2,,-2, the total number of configurations. Further, N123(n) is strictly larger than 2,,-2_ (N j (n)+n 1 (n) + Nl(n», since removal from the set of all configurations those which do not satisfy condition 1, those which do not satisfy condition 2, and those which do not satisfy condition 3 leaves the configurations which satisfy all conditions. N 12 in) is strictly larger than 2n-2_(Ni(n)+N~(n)+N'l(n» because included in tqe configurations counted in N 1 (n) are those which are also counted in N~(n), etc. Thus. N1(n) can be calculated as follows. A configuration which does not satisfy (1)

8 248 WHITE AND BUTLER condition 1 can be formed by placing a 1 to the left of a configuration of size n 1 which also does not satisfy condition l. Such a configuration has the form G1wO, where w is bounded on the right and left by 1 and is of size n - l. There are N1(n - 1) ways to choose w. There are no configurations of the form GlOwO, where w is now of size n 2, since the left 1 of w forms the subsequence 101 with the prefix 10. There are N 1 (n - 3) configurations counted in Nj(n) of the form 0100wO, N 1 (n - 4) of the form 01000w0, etc. Thus, /1-3 Nj(n) =N1(n - 1) + L NI(i), for n~4. (2) i=1 Substituting n - 1 for n in (2) and subtracting from (2) yields, for n ~ 4. (3) A closed~form solution for Ni(n) can be found by assuming that Ni(n) = yp" and substituting into (3). Doing this, and rearranging yields the charac~ teristic equation which has three solutions: Thus, PI 1.755, P2 = ( i), P3 ( i). where Y1> Y2' and "13 can be solved from the initial conditions Ni(l) = 3, N 1 (2) = 4, and N1(3) = 7. Thus, N1(n) = (0.234)(1.755t + ( ;)( it + ( i)( i)", for n): 4. (4) In a similar fashion, N'1.(n) and N3(n) are found to be and N'1.(n) = (0.134)(1.755)" + ( i)( it + ( i)( " for n):4 (5) N3(n) (0.171)(1.618)" (1.17)(-0.618)/1, for n~ 3. (6)

9 SYNTHESIS FROM INITIAL/FINAL PAIRS 249 TABLE I Number and Fraction of Configurations of Size n Which Are Garden-of-Eden in a Binary, Scope-2, ~Iexible Cellular System with 1= 111(1), 11(2), 11(4), 11(6), 11(7») Number of Fraction of n Garden-of-Eden configurations which configurations are Garden-of-Eden I to It is interesting to note that N,,(n) is F n - 3 the n - 3th Fibonacci number. For large n, NI(n), Nz(n), and Nl(n) are small compared to 2,,-2. Thus, from (1 ), where g(n) -- hen) means lim n... oo g{n)/h(n) = 1. However, since Nl234Cn) +Nm~(n) = N123Cn) '" 2,,-2 = N 4 (n) + NaCn) it follows that The number Nmin) of Garden-of-Eden configurations of LEMMA 1. size n in a binary, scope-2 flexible cellular automata system with 1= {aw, a(2,), a(4), a«;), a(1)} is N 1234 (n) '" Lemma 1 shows that for large n, the fraction of configurations of size n which are Garden-of-Eden is surprisingly high, about 50 %. Table I shows the number of Garden-of-Eden configurations for 1 ~ n ~ 8. For 11 = 1,2,3, and 5, there are no such configurations. For 11 4,6,7, and 8, the average fraction of configurations which are Garden-of-Eden is IV. SYNTHESIS PROCEDURE A motivation for the synthesis procedure is the problem in pattern recognition of transforming a given input pattern into a processed output

10 250. WHITE AND BUTLER pattern. For example, the input pattern may be the digitized image of an object and the output pattern, a set of features of that object. The question of interest is whether a given cellular system can produce some desired transformation. The synthesis procedure is based on an algorithm for determining whether a given local transition function is decomposable. In particular, Algorithm 1 Inputs: A local transition function (I on neighborhood X = (0, 1,..., j). Outputs: A sequence of local maps, a p a 2,, a j _ 1 such that the application of a p followed by a 2, etc., on the two-cell neighborhood X = (0, 1) is equivalent to the single application of a on X, if indeed such a sequence exists. Algorithm 1 is described in Butler (1979), and applies to both completely and incompletely specified local transition functions. The procedure executes as follows: Step 1. Step 2. Step 3. If ro < r F, stop; (I is not decomposable. j +-- L X+-- (0,..., j). Step 4. Compute (I for the chosen X. Step 5. If (I is not single valued, j +- j + 1. Go to Step 3. Step 6. Apply Algorithm 1 to decompose (I. If (I is decomposable, output it and its decomposition, and halt. Otherwise let I+-- j + 1 and go to Step 3, where r0 and rf are the locations of the rightmost one cell in the initial and fmal configurations, respectively. A proof is now given for the first step in the procedure. It is shown that for certain initial/final configuration pairs, it is immediately decidable whether a decomposable map exists for transforming the initial configuration into the fmal one. THEOREM 2. A necessary condition for the existence of a decomposable parallel map tx,o: co... C F for some Co and c F ' where X = (0, 1,..., s), s ~ 1, is r F ~ ro, where ro and r F are the positions of the rightmost cells of the initial and final coyifigurations, respectively. Proof. Let tx.<t be a parallel map on X = (0, 1,..., s), s ~ 1, which maps co... c F From the quiescent condition, a maps (0,0,...,0) to O. Hence, all cells in c F with coordinates larger than ro must be O. Thus r F ~ roo Q.E.D. The next two lemmas show that conditions in the initialjfinal

11 SYNTHESIS FROM INITIAL/FINAL PAIRS 251 configuration pair restrict the number of certain local transition functions in any computation for the pair. LEMMA 2. Ifr.~,u: Co""; cffor X = (0,1,..., s), s;;?; 1, is decomposable as a sequence S ofnontrivial local maps on X' (0, 1), then a(1) and a(4) occur in S at most ro - rp times, where ro and rp are the positions of the rightmost cells of the initial and final configurations, respectively. Proof Consider the rightmost nonquiescent cell in the configurations generated by the application of local maps in S. Whenever a(2), a(6) and a(7) are applied, the rightmost nonquiescent cell remains stationary. However, when "aw and 0'(4) are applied the rightmost cell moves left at least one. Thus ro r F is an upper bound on thenumber of applications of (1(1) and 0'(4). Q.E.D. Note that the rightmost nonquiescent cell may coalesce with the leftmost nonquiescent cell such as when all) is applied to , or may disappear such as when a(l) is applied to Note also that if a(s) were to be an allowable local map, then (ro rp) is the number of applications of (1(l), 0'(41, and a{s). Lower bounds can also be established for the number of occurrences of local maps. LEMMA 3. [frx,u: Co""; cpfor X (0, 1,...,s), s;;?; 1, is decomposable as a sequence S of nontrivial local maps on X' (0, 1), then " 1. There are at least 10 IF maps of the form 0'(4\ 0'(6), 0'(7) in S for (10 IF) ;;?; There is at least one map of the form 0'(1) and 0'2 in S for (IF -/o);;?; 1, where 10 and IF are the positions of the leftmost 1 cell of the initial andfinal configurations, respectively. Proof This lemma follows from the observation that the application of 0'(4), (1(6 l, and a(7) to a configuration moves the leftmost nonquiescent cell left once. On the other hand, the application of 0'(1) and a(2) either moves the leftmost nonquiescent cell one or more positions to the right, or leaves it unchanged, depending on the state of the cell to the immediate right of the leftmost nonquiescent cell. Thus if (10 -IF) is positive, no fewer than (lo -IF) applications of 0'(4),0'(6), and 0'(7) have occurred. Similarly, if (IF -/0) ;;?; 0, at least one application of 0'(1) and 0'(2) has occurred. Q.E.D. Step 3 in the procedure is based on an assumption that when IXI becomes "large enough," the local map is single valued. This assumption is now stated formally and proven.

12 252 WHITE AND BUTLER LEMMA 4. Let.x,u:CO--+CF and ro~rf' The local transition function 0 can be realized as a single valued local transition function 1. IXi ~ 2(ro -/0) ~ 2 for (3/0-2ro) < IF or 2. IXI ~ 10 IF + 1 for IF <(310-2ro)' Further, these bounds on IXI are firm in the sense that c{j/c F pairs exist such that a is not single valued for IXI less than the bounds. Proof. Let Co = DIal a 2 an_lid and consider the two cases separately. Case 1. (3/0-2ro) <IF" Figure 2 shows the form of a when IXI = 2(ro -/0) and IX! > 2; The binary sequences shown include all neighborhood patterns in X and represent the truth table of an incompletely specified function. For example, if IF = I{J then all patterns from to 01al an_3an_ inclusive map to 0 while lala2" a,,_ and perhaps others below it map to 1. For any choice of a 1 a2... an-2' all patterns shown are unique and so for any choice of c F ' a is single valued. Further, this is true for all larger IXI. Thus,,XI 2(r0-1 0 ) is a lower bound on IXI, such that a is single valued. To show that this is firm, consider the case where IXI = 2(ro 1 0 ) - 1. If o 0 0 o 0 0 o 0 0 o o o o o 1 "I o 0 0 o 1 an_4an_3an_2 a l o "1'2 "n_3'n_ a 1... an_3an_.! "I 32 "n_ "I 2 a (J 0 0 0, ~~ FIG. 2. Local transition function (J for Case 1.

13 SYNTHESIS FROM INITIAL/FINAL PAIR'S 253 at = a 2 =... = a n _ 2 0, patterns 000 0la 1 an_4an_jan_2 'and a l aza), an_zio '" 000 for such an IXI are identical. Thus, a c F exists for which 0 is not single valued. In effect, the neighborhood is not large enough to distinguish between these two patterns. The range of IF which can be obtained for IXI = 2(ro 1 0 ) can be seen as follows. The leftmost 1 of c F (at IF) corresponds to the highest pattern in Fig. 2 which maps to 1. Because 0(0, 0,..., 0) = 0 this 1 can occur no higher than pattern For this case, if = 10 [2(ro -10) ~ 1] = 3/0 2ro + L Thus, 310-2ro + 1 is a lower bound on IF for Case L The arguments above apply whenlxi > 2. IXI = 2 is a special case but similar arguments apply. Case 2. IF ~ 3/0 2ro' Figure 3 shows an initial and final configuration for this case. If IXI 2(ro ), the neighborhood pattern for the leftmost,i of c F (at IF) consists entirely of O's and this conflicts with the quiescent condition 0(0, 0,...,0) = O. However, if IXI = 10 - IF + 1 the pattern corresponding to this 1 is and no conflict arises. This proves the bound is finite. FurtlJ,er, since 10 IF > 2(r0 -/0) for IF ~ 3(10-2ro), an argument similar to that presented for Case 1 applies and all neighborhood patterns are unique. Q.E.D. As an example of the synthesis procedure, consider the initial/final configuration pair Co c F From Lemma 2, it follows that 0(1) and 0(4) occur at most r0 rf = 1 times and from Lemma 3, there are at least 10 IF = 3 maps of the form 0(4), 0(6), and om, if indeed a computation exists. Thus, at least three local transition functions are needed to produce C F from Co' This can also be seen by the fact that choosing X = (0, 1,2) leads' to a contradiction in the composite local transition function on X. That is, 000 in Co at indices 0, 1, 2, and 7, 8, 9 map to 1 and 0, respectively. t IF 1 "I "2.. 'n_21~ Co /' 10 "0 I b I b z bm-zi C F FIG. 3. Initial final configuration pair for Case 2. t r F

14 254 WHITE AND BUTLER. However, for X = (0, 1,2,3) a is single valued and Step 6 of the procedure yields and If the synthesis problem is rephrased to that of determining a sequence of local maps apaz,..,a F, such that ix.,,:co-+c F for a=a F o af_1 0 a p with the restriction that the sequence of configurations is strictly monotonically increasing or decreasing in size, then the procedure becomes an algorithm. That is, if the computation from Co to consists of configurations cn,cp""c F, where either #Ci~#Ci+l or #C1~#Ci+l for o..;; i..;; F - 1, then the synthesis problem is decidable. This is because the restriction of a monotonic increase or decrease in size implies there can be only finitely many configurations between any given initial fmal configuration pair; Thus, all possible sequences of configurations can be enumerated and tested to see if they can be generated by a sequence of local transition functions from I. The set of monotonic derivations is a proper subset of the set of unrestricted derivations. For example, the initialjfmal configuration pair i Co c F requires at least two applications of a nontrivial local transition function, since the pairs 01 at indices 0, 1 and 3, 4 map to 1 and 0, respectively. If the first local map is a(6) or a(1), the size of the intermediate configuration is larger than that of Co and C F Thus, no monotonic derivation could have a(6) or a(7) as the first local transition function. However, if a(1), am, or a(4) is the frrst local transition function the intermediate configuration is of size less than Co and c/" Thus, no monotomic derivation exists. However, a derivation does exist, namely, a a(7) 0 a(1). The sequence of configurations is shown below. i Co c 1 a(l) a(7) c/,=c c F

15 SYNTHESIS FROM INITIAL/FINAL PAIRS 255 V. CONCLUDING REMARKS This paper has dealt with the I-dimensional binary scope-2 flexible cellular system. In particular, a synthesis procedure has ~been shown for a system which realizes a given initial/flllal configuration pair. If monotonic generation only is allowed, the procedure is an algorithm. It is an. open question whether the general synthesis procedure is also an algorithm. As it stands, there is no criteria to halt the procedure when rf~ ro or when the final configuration is not Garden-of-Eden. Because of the result by Yaku (1972) that it is undecidable whether a configuration in an arbitrary 2 dimensional cellular system is Garden-of-Eden or not, there is no synthesis algorithm for an arbitrary 2-dimensional cellular system. Necessary and sufficient conditions are shown for a configuration to be Garden~of-Eden. This allows an enumeration of such configurations, and it was shown that when size is sufficiently large almost half of the configurations are Garden-of-Eden. It is an open question how many of the remaining configurations are decomposable. RECEIVED: January 18, 1980; REVISED: October 22, 1980 REFERENCES AMOROSO, S. AND COOPER, G. (1970), The Garden-of-Eden theorem for finite configurations, Proc. Amer. Math. Soc. 26, AMOROSO, S. AND PATI, Y. N. (1972), Decision procedures for surjectivity and injectivity of parallel maps for tessellation structure, J. Comput. System Sci. 6, BUTLER, J. T. (1979), Synthesis of one-dimensional binary cellular automata systems from composite local maps, Inform. COnlr. 43, HEDLUND, G. A. (1969), Endomorpbisms and automorphisms of the shift dynamical systems, Math. Systems Theory 3, HOLLAND, J. H. (1970), A universal computer capable of executing an arbitrary number of subprograms simultaneously, in "Essays of Cellular Automata" (A. W. Burks, Ed.), Univ. of Illinois Press, Urbana. LEE, C. Y. AND PAULL, M. C. (1963), A content addressable distributed logic memory with applications to information retrieval, Proc. IRE 51, MAROUKA, A. AND KIMURA, M. (1974), Strong connectiveness of one-dimensional scope-3 tessellation automata, J. Comput. System Sci MOE, G. K., RHEINBOLT, W., AND ABILDSKOV, J. A. (1964), A computer model of atrial fibrillation, Amer. Heart J. 67-2, MOORE, E.F. (1962), Machine models of self-reproduction, Prec. Symp. Appl. Math. 14, NASU, M. (1978), Local maps inducing surjective global maps of one-dimensional tessellation automata, Math. Systems Theory 11, VON NEUMANN, J. (1966), "Theory of Self-Reproducfng Automata" (A. W. Burks, Ed.), Univ. of Illinois Press, Urbana.

16 256 WHITE AND BUTLER. STAHL, W. K. (1967), A computer model of cellular reproduction, J. Theoret. Bioi. 14, UNGER, S. H. (1959), Pattern detection and recognition, Froc. IRE 46, YAKU, T. (1973), The constructability of a configuration in a cellular automaton, J. Comput. System Sci. 7, YAMADA, H. AND AMOROSO, S. (1970), A completeness problem for pattern generation in tessellation automata. J. Comput. System Sci. 4, YAMADA, H. AND AMOROSO, S. (1971), Structural and behavioral equivalences of tessellation automata, Inform. Contr. 18, Printed by the St. Catherine Press Ltd., Tempelhof 41, Bruges, Belgium