A note on bottom-up pyramid acceptors

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  • Volume 8, number 1 INFORMATION PROCESSING LETTERS January 1979


    h&U&i INOUE and Itsuo TAKANAMI Department of Electronics, Faculty of Engineering, Yamaguchi University, Ube, 7.55 Japan

    Received 7 July 1978; revised version received 28 August 1978

    Celluiar automata, two-dimensional automata, pattern recognition

    1. lntroduct ion

    In [ 13, Dyer and Kosenfeld introduced a new type of acceptor on a two-dimensional partern (or tape), called the pyramid cellular acceptor (denoted by PA), and demonstrated :hat many useful recogni- tion tasks are executed by PAs in time proportional to the logarithm of the diameter of the input.

    They also introduced a bottom-up pyramid acceptor (denoted by BPA) which is a restricted version of the PA, and proposed some interesting open problems about BPAs. We are interested in the following problem (which is one of the open prob- lems): Does the class of sets accepted by determin- istic BPAs include the class of sets accepted by de- terministic two-dimensional finite automata (denoted by ~-DAS) [2,3,4]?

    In this note, we show that the class of sets accepted by ~-DAS is incomparable with the class of sets ac- cepted by deterministic BPAs which operate in time of order lower than the diameter of the input.

    2. Preliminaries

    We first give some definitions and notations con- cerning two-dimensional tapes.

    Definition I. Let Z be a finite set of symbols. A two-

    i)l when 1 < i G < II(x)

    : Q& &N is state transition fine- tion, mapping the current state of M and its four son cells in a 2-by-2 block in the level below into Ms next state. As shown in Fig. 1, let c be some cell in the (i + 1)st layer (i 2 l), and let cl, ~2, cg and c4 be four son cells (in the ith layer) of c. Then

    4c(t + 1) = V&(t), 4& 4& 4& 4&) 9

    where for example qc(t) means the state of c at time t. (Especially,

  • Volume 8, Nur, ber 1 INFORMATION PROCESSING LETTERS Jazmary 1979

    i /7 F C the (i+l)st layer L ,,$ the i-th lay.3

    Fig. 1.

    tom array depends only on the current state of c.) At time t = 6, the input tape x E 07) (I,(X) =

    Z*(X) = 2, n 2 11 is stored as the initial states of the bottom array (in such a way that x(., i) is stored at the cell of the ith row and the jth column), and the other cells are initialized to a quiescent state qS (E QN - QT - A). As usu4 we let S(q,, qs, qs, qs, qs) =qs. The input is accepted if and only if the root cell ever enters an accepting state. This BPA is called deterministic. A nondeterministic bottom-up pyramid acceptor (NBPA) is defined as a BPA using 6 : & + ZQN instead of the state transition function of the deterministic BPA. Below, we denote the deterministic BPA by DBPA, and the nondeterministic BPA by NBPA.

    A DBPA (or NBPA) operates in time T(x) if for every tape of size 2 by 2 (n > 1) it accepts, there is an accepting computation which uses no more than time T(n). Especially, we say that a DBPA (or NBPA) operates in real time if it operates in time T{n) = n. Let & [DBPA(T(n))] (J [NBPA(T(n))]) denote the class of sets by DBPAs (NBPAs) which operate in time T(n).

    We next recall a deterministic two-dimensional finite automaton (20DA) [2,3,4]. A 2-DA is a Gtuple M = (K, C, 6, qo, qa, qr), where K is a finite set of states, E is a finite set of input symbols, qo (E K) is the start state, qa (E K) is the accepting state, qr (E K) is the rejecting state, and S : (K - (qa, qy)) X (Z: U {#)) + K X A is the control function (where # is the boundary symbol not in Z and A = (L, R, U, D) cm be thought of as a set of directions (left, right, up, down)). An input tape to M is a two-dimensional tape over Z surrounded by the boundary symbol #.

    6 (q, a) = (pt d) means that if M reads the symbol a in state q, it enters state p and moves in direction d Given an input tape x E ~(~1, M starts ~1 state q. on the upper left-hand corner. If M falls off the tape x, M can make no more move. We say that M accepts the tape x if M eventually halts in state q?a ; M rejects the tape x if it halts in state qr. Let Z[2-DA] denote the class of sets accepted by 2-DAs.

    2. Results

    In this section, we show that the class of sets accepted by ~-DAS is incomparable with the class of sets accepted by DBPAs which operate in time of order lower than the diameter of the input. We assume, in this paper, that the sizes of inputs to automata are restricted to 2n by 2 (n > 1).

    Lemma 1. Let U- (x E (0, 1)(2) 1 3n(n > l)[Z&) = 12(x)=2]andx(2n-1,2n-1)= l).Then,

    (1) U $ .-C[2-DA], and (2) U E d: [DBP.A(n)] .

    Roof. The proof of (1) is similar to that of Theorem 1 in [2]. So the proof is omitted here. Below, we prove (2). The set U is accepted by the following DBPA &which operates in real time. Each cell of B is defined as the following finite state machine M = (QN, QT, 6, A):

    (i) QN = (qs, 0, 1) U (Y, N) X (0, 11, vhre qs is the quiescent state;

    (ii) QT= (0, 1); (iii)A = ([Y, 11, [Y, 01); (iv)(lj Foranya, b, cE (O,l),

    S(qs, 1, II, b, c) = [Y, b], and 6(qs,0,cc b, c> = [N b];

    (2) For any 17, q, r E (Y, N) , and for my a, b, cE W,lL

    S(qs, [Y, 11, b, al, [q. bl, [r, cl> = PW, S(qs, [N, 11, CpI al, [q, bl, I_c ~1) = P, bl, S(& [Y, 01, I& al, [4, b3r I?, cl> = WJ4 and 6(qa [N, 01, b, 4 [a bl, [p, cl> = [N1 bl .

    !t is straightforward to see that B accepts U, and so (2) of the lemma holds.

  • Volume 8, Number 1 INFORMATION PROCESSING LETTERS January 1979

    Lemma 2. Let V = (x E (0,l )(*)I 3n(n 2 1) [Z,(X) = By assumption, the root cell of I3 enters the same

    I*(x) = 2] and x[(l, l), (1,2)] =X [(2, I), (2? 2)] )- states until time T(n)g for the tapes z and z. Since Let -T(n j be a time function such that lim,+, [T(n)/ 2] = 0. Then,

    (1) VEL[Z-DA],and

    B operates in time T(n) and z is in V, it follows that z is also accepted by B. This contradicts the fact that z is not in V.]

    (2) V 4 d: PBPA(T(n))].

    proof. It is obvious that there is a 2-DA accepting K and SO (1) of the lemma holds. Below, we prove (2). Suppose that there is a DBPA B which accepts V and operates in time T(n), and that each cell of B has k states. For each n 2 2, let

    W(@ = {X E Zl(x) Z*(x) 2)) and

    W(n) Tx E {O, l)(*) I 21(x) = 12(X) = 2n-

    &x[(l, I), (1,291 E (0, l)(*)

    &x1(2, I), (2n-, 29] E (O)(2)}

    We consider the cases w1;en the tapes in W(n) are presented to B. Let c be the cell which is situated at the first row and the first column in the nth layer (Le., the layer just below the root cell). (Note that there are four cells in the nth layer.) For each x in W(n) such that x [( 1, l), (2-l, 2-)I E W(n), and for each r 2 1, let q,(x) be the state of c at time r when x is presented to B. Then, the following propo- sition must hold.

    Proposition 1. Let x, y be two different tapes in W(n) such that both x [(l , l), (2n- , 2n- )I and Y[(l, 0, (2V 2n-)] arein W(n)andx[(l, l), (2 -l, 2n-)] + y[(l, l), (2n-, 2n-1)]. Then, Mu), 42(x), l a9 47@)(x)) If (410% q2w9 -*=9

    4T(n) 0)).

    IFor suppose that (q&G, &), . . . . qT(n)(X)) =

    h I(.), hb), --. , qT(n)(Y))e We consider two tapes z, z in W(n) such that

    (i) z [( 1, I), (2~, 271

    =x[(l, I), (27 zn-l)]


    z[(l, l), (2~, P-l)]

    =.v[fl. l), (2n-1, 2n-1)] ,

    (ii) the part of z except for z [(l, 1); (2-l, 2-l)] is identical with the part of z except for z[(l, I), (27 2--)I, and

    (iii) z [(l, l), (1,2>] = 2 [(2n, l), (2, 2n)].


    Let t(n) be the number of different sequences of states which c enters until time T(n). Clearly, t(n) < kT@) On the other hand (for any set S, let lS1 denote the number of elements of S.), i W(n)] = 2*? Since lim, + 00 [T(n)/2n] = 0 (by assumption of the lemma), it follows that 1 FV(n)l~ > t(n) for large n. Therefore, it follows that for large n there must exist two different tapes x, y in W(n) such that

    (i) bothx[(l, l), (2-r, 2-)I and y[(l, l), (2-1, 2~91 are in W(n),

    (ii) x[(l, 1),(2+, 2"-')I #Y [(l, 0, (2-l, 2-91, and

    (iii) (41(x), 42(x), .a.~ 4T(n)(x)) = (410% 426% l **) 4T(n)W)* This contradicts the above Proposition 1, and thus the part (2) of the lemma holds.

    From lemmas 1 and 2, we can get the following theorem.

    Theorem 1. Let T(n) be a time function such that lim,+oo [T(n)/2n] = 0 and T(n) > n (n > 1). l%en d: [ 2=DA] is incomparable with 6: [ DBPA (T(n))].

    Corollary 1. d: [2-DA] is incomparable with X[DBPA(n)], which is the class o-f sets accepted by DBPAs operating in reaI time.

    For any NBPA M which operates in real time, we can construct a DBPA which accepts the same set as M and operates in real time, by using the well-known subset-construction method. Thus 63 [NBPA(n)] = E[DBPA(n)]. From this and Corollary 1, we get the following corollary.

    Corollary 2.6: [2-DA] is incomparable with 6: [NBPA(n)].

    It is still unknown whether the class of sets ac- cepted by DBPAs includes d: [2-DA].


    The authors would like to thank professors N. Honda, T. Fukumura of Nagoya University, Y. Inagaki of

  • Volume 8, Nurnfdr 1 INFORMATION PROCESSING LETTERS January 1979

    Mie University, A. Nakamura, N. Yoshida, T. Ae of Hiroshima Univzrcity for their hearty encouragement.


    References VI

    [ 1) C.R. Dyer and A. Rosenfeld, Cellular pyramids for image analysis, University of Maryland, Computer Science Center, TR-544, AFOSR-77-327 1.

    [2] M. Blum and C. Hewitt, Automata on a two-dimensional

    tape, IEEE Symp. on Switchiig and Automata Theory (1967) 155. K. Inoue and A. Nakamura, Some properties of two- dimensional on-line tessellation acceptors, Information Sci. 13 (1977) 95-121. K. Inoue, I. Takanami and A. Nakamura, A note on two-dimensional finite automata, Information Processing T,ett. 7 (1) (1978) 49. K. Inoue and I. Takanami, Remarks on two-dimensional finite automata with multiplication and bottom-up pyramid acceptors, Technical Report No. AL77-61, IECE of Japan (1978).