Some Properties of Three-Dimensional

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp665-670)

Some Properties of Three-Dimensional Synchronized Alternating Turing Machines Takao ITO, Makoto SAKAMOTO, Naoko TOMOZOE, Kouichi IIHOSHI, Hiroshi FURUTANI, Michio KONO Department of Computer Science and Systems Engineering,University of Miyazaki, 1-1 Gakuen Kibanadai-Nishi, Miyazaki 889-2192, JAPAN Tatsuhiro TAMAKI Department of Business Administration, Ube National College of Technology, 14-1 Tokiwadai 2-chome, Ube 755-8555, JAPAN and Katsushi INOUE Department of Computer Science and Systems Engineering, Yamaguchi University, 6-1 Tokiwadai 2-chome, Ube 755-8611, JAPAN Abstract: - The recent advances in motion image processing, robotics and so on prompted us to analyze computational complexity of three-dimensional pattern processing. Thus, the research of three-dimensional automata as a computational model of three-dimensional pattern processing has also been meaningful. From this viewpoint, we introduced a three-dimensional alternating Turing machine (3-AT M ) operating in parallel. In this paper, we continue the investigations about 3-AT M ’s, introduce a three-dimensional synchronized alternating Turing machine (3-SAT M ), and investigate some properties of 3-SAT M ’s whose input tapes are restricted to cubic ones. The main topics of this paper are: (1) hierarchies based on the number of processes of 3-SAT M ’s, and (2) recognizability of connected pictures by 3-SAT M ’s. Key-Words: - alternation, recognizability, synchronization, three-dimensional automaton, Turing machine

1

Introduction and preliminaries

Synchronized alternating Turing machines were introduced in [1] to study the effect of allowing processes of an alternating Turing machine to communicate via synchronization. Informally, a synchronized alternating machine is an alternating machine with a special subset of internal states called synchronizing states. Each of these synchronizing states is associated with a synchronizing symbol. If, during the course of computation, some process enters a synchronizing state, then it has to wait until all other processes enter either an accepting state or a synchronizing state with the same synchronizing symbol. When this happens, all processes are allows to continue their computation; otherwise, the machine is said to have a deadlock. A computation is successful if no deadlocks occur and all processes terminate in accepting states. It turns out that synchronization significantly increases the computaional power of alternating Turing machines. On the other hand, recently, due to the advances in many application areas such as computer vision, robotics, and so forth, it has become increasingly apparent that the study of three-dimensional pattern

processing has been of crucial importance. Thus, the research of three-dimensional automata as a computational model of three-dimensional pattern processing has also been meaningful. From this viewpoint, we introduced a three-dimensional alternating Turing machine (3-AT M )[13]. In this paper, we continue the investigations about 3-AT M ’s, introduce a three-dimensional synchronized alternating Turing machine (3-SAT M ), and investigate some properties of 3-SAT M ’s whose input tapes are restricted to cubic ones. In this section, we observe a historical overview, and provide a background and a motive for our study of three-dimensional automata. Moreover, this section summarizes the formal definitions and notations necessary for this paper. Section 2 investigates hierarchies based on the number of processes of three-dimensional synchronized alternating finite automata, and shows that for three-dimensional synchronized alternating finite automata, k + 1 processes are more powerful than k processes for any k ≥ 1. Section 3 investigates recognizability of connected pictures by five-way three-dimensional synchronized alternating Turing machines with only universal states, and shows that (1) the necessary and sufficient


space for these machines to accept the complement of Tc (where Tc denotes the set of all the connected pictures) is m, and (2) five-way three-dimensional synchronized alternating finite automata can accept Tc . Finally, Section 4 concludes this paper by giving several open problems. Let Σ be a finite set of symbols. A threedimensional input tape over Σ is a three-dimensional rectangular array of elements of Σ. The set of all the three-dimensional input tapes over Σ is denoted by Σ(3) . Given an input tape x ∈ Σ(3) , for each j (1 ≤ j ≤ 3), we let lj (x) be the length of x along the jth axis. The set of all x ∈ Σ(3) with l1 (x) = m1 ,l2 (x) = m2 and l3 (x)=m3 is denoted by Σ(m1 ,m2 ,m3 ) . If 1 ≤ ij ≤ lj (x) for each j (1 ≤ j ≤ 3), let x(i1 , i2 , i3 ) denote the symbol in x with coordinates (i1 , i2 , i3 ). Furthermore, we define x[(i1 , i2 , i3 ),(i01 , i02 , i03 )], when 1 ≤ ij ≤ i0j ≤ lj (x) for each integer j (1 ≤ j ≤ 3), as the three-dimensional input tape y satisfying the following; (1) for each j (1 ≤ j ≤ 3), lj (y) = i0j − ij + 1; (2) for each r1 , r2 , r3 (1 ≤ r1 ≤ l1 (y), 1 ≤ r2 ≤ l2 (y), 1 ≤ r3 ≤ l3 (y)), y (r1 ,r2 ,r3 )=x (r1 +i1 1,r2 +i2 -1,r3 +i3 -1). (We call x[(i1 ,i2 ,i3 ),(i01 ,i02 ,i03 )] the [(i1 ,i2 ,i3 ),(i01 ,i02 ,i03 )] segment of x.) For each x ∈ Σ(m1 ,m2 ,m3 ) and for each 1 ≤ i1 ≤ m1 , 1 ≤ i2 ≤ m2 , 1 ≤ i3 ≤ m3 , x[(i1 ,1,1),(i1 ,m2 ,m3 )], x[(1,i2 ,1),(m1 ,i2 ,m3 )], x[(1,1,i3 ),(m1 ,m2 ,i3 )], x[(i1 ,1, i3 ),(i1 ,m2 ,i3 )] and x[(1,i2 ,i3 ),(m1 ,i2 ,i3 )] are called the i1 th (2-3) plane of x, the i2 th (1-3) plane of x, the i3 th (1-2) plane of x, the i1 th row on the i3 th (12) plane of x and the i2 th column on the i3 th (1-2) plane of x, respectively. We now introduce a three-dimensional synchronized alternating Turing machine. A three-dimensional synchronized alternating Turing machine (denoted by 3-SAT M ) is a 10-tuple M = (Q, q0 , U , E, S, F , Σ, Π, Γ, δ), where (1) Q = U ∪ E ∪ S is a finite set of states, (2) q0 ∈ Q is the initial state, (3) U is the set of universal states, (4) E is the set of existential states, (5) S ⊆ {(q, s) : q ∈ U ∪ E, s ∈ Π} is the set of synchronizing states (s-states), (6) F ⊆ Q is the set of accepting states, (7) Σ is a finite input alphabet (# ∈ / Σ is the boundary symbol ), (8) Π is a finite alphabet of synchronizing symbols, (9) Γ is a finite storage tape alphabet containing the special blank symbol B, (10) δ ⊆ (Q × (Σ ∪ {#}) × Γ) × (Q× (Γ − {B}) × {east,west,south,north,top,down,no move} × {left,right,no move}) is the next move relation. As shown in Fig.1, M has a read-only cubic input tape with boundary symbols #’s (# ∈ / Σ) and one semi-infinite storage tape, initially filled with the blank symbols. M begins in state q0 . A position is assigned to each cell of the input tape and the storage

tape, as shown in Fig.1. A step of M consists of reading one symbol from each tape, writing a symbol on the storage tape, moving the input and storage tape heads in specified directions, and entering a new state, according to the next move relation δ. When a process P enters a synchronizing state, it stops and waits until all the parallel processes either enter the states with the same synchronizing element or stop in accepting states. An instantaneous description (ID) of a 3-SAT M M = (Q, q0 , U , E, S, F , Σ, Π, Γ, δ) is a pair of an element of Σ(3) and an element of CM = (N ∪ {0})3 × SM , SM = Q × (Γ − {B})∗ × N, where N denotes the set of all positive integers. The first component of an ID I = (x, ((i1 , i2 , i3 ), (q, α, k))) represents the input to M , and the first component (i1 , i2 , i3 ) of the second component of I represents the input head position (0 ≤ i1 ≤ l1 (x) + 1, 0 ≤ i2 ≤ l2 (x)+1, 0 ≤ i3 ≤ l3 (x)+1), and the second component (q, α, k) of the second component of I represents the state of the finite control, nonblank contents of the storage tape, and the storage head position (1 ≤ k ≤ |α| + 1). An element of CM is called a configuration of M , and an element of SM is called a storage state of M . An ID is universal (existential, synchronizing, accepting) depending on the type of the state of the finite control. The initial ID of M on input x is IM (x) = (x, ((1, 1, 1), (q0 , ², 1))), where ² is the null word. Suppose I1 and I2 are two ID’s of M and I2 follows from I1 in one step according to the next move relation δ. Then we write I1 `M I2 and say that I2 is a successor of I1 . The reflexive and transive closure of `M is denoted by `∗M . A sequence of ID’s of M , I0 , I1 , · · · , Im (m ≥ 0), is called a sequential computation of M if I0 `M I1 `M · · · `M IM . If I0 = IM (x) for some x, we call this sequence a computation path of M on x. The full computation tree of M on an input tape x is a (possibly infinite) labeled tree `M x such that (1) each node v is labeled by some ID Iv of M , (2) the root is labeled by IM (x), (3) v2 is a direct descendant of v1 iff Iv1 `M Iv2 . (Each branch of `M x is called a process.) The synchronizing sequence (s-sequence) of a node v in a full computation tree T with root v0 is the sequence of synchronizing symbols occuring in labels of the nodes on the path from v0 to v. Two s-sequences are compatible if one is a prefix of the other. If s1 and s2 are two compatible s-sequences, and s2 is longer than s1 , then we use s2 -s1 to denote their difference. A computation tree of M on an input x is a (possibly infinite) subtree T 0 of the full computation tree TxM satisfying the following conditions:


Fig. 1: Three-dimensional synchronized alternating Turing machine.

(1) if u is an internal (non-leaf) node of the tree T 0 , Iu is universal and {I | Iu `M I} = {I1 , · · · , Im }, then u has exactly m children v1 , · · · , vm , such that Ivi = Ii , 1 ≤ i ≤ m, (2) if u is an internal node of the tree and Iu is existential, then u has exactly one chiled v such that Iu ` Iv , (3) For arbitrary nodes u and v of T 0 , the ssequences of u and v are compatible. If M on input x has no computation trees, then any subtree of TxM that satisfies the first two conditions above must have two processes with incompatible ssequences. In this case, we say M deadlocks on x. The two processes with incompatible s-sequences are called deadlock processes and the nonmatching s-states causing the deadlock are called deadlock states. The longest synchronizing sequence of a node in the computation tree T is called the synchronizing sequence of the computation tree T . An accepting computation tree of M on an input x is a finite computation tree of M on x such that each leaf node is labeled by an accepting ID. We say that M accepts x if there is an accepting computation tree of M on x. Let T (M ) = {x ∈ Σ(3) | M accepts x}. We next introduce a five-way three-dimensional synchronized alternating Turing machine which can be considered as a synchronized version of five-way three-dimensional alternating Turing machine [13]. A five-way three-dimensional synchronized alternating Turing machine (denoted by F V 3-SAT M ) is a 3-SAT M M = (Q, q0 , U, E, S, F, Σ, Π, Γ, δ), such that δ ⊆ (Q × (Σ ∪ {#}) × Γ) × (Q × Γ − {B})× {east, west, south, north, down, no move} × {left,right, no move}). That is, an F V 3-SAT M is a 3-SAT M whose input head can move east, west, south, north, or down, but

not up. Let L(m) : N → N be a function with one variable m. With each 3-SAT M (or F V 3-SAT M ) M we assosiate a space complexity function SPACE which takes ID’s to natural numbers. That is, for each ID I = (x,((i1 ,i2 ,i3 ),(q,α,k))), let SPACE(I) be the length of α. We say that M is “L(m) space-bounded ” if for all m and for all x with l1 (x) = l2 (x) = l3 (x) = m, if x is accepted by M , then there is an accepting computation tree of M on input x such that for each node π of the tree, SPACE (I(π)) ≤ L(m). By “3SAT M (L(m))” (“F V 3-SAT M (L(m))”) we denote an L(m) space-bounded 3-SAT M (F V 3-SAT M ) whose input tapes are restricted to cubic ones. Three-dimensional alternating Turing machines (3-AT M ’s) and five-way three-dimensional alternating Turing machines (F V 3-AT M ’s) in [13] are 3-SAT M ’s and F V 3-SAT M ’s, respectively, which have no synchronizing states. We use 3-SU T M (F V 3-SU T M , 3-U T M , F V 3-U T M ) to denote a 3SAT M (F V 3-SAT M , 3-AT M , F V 3-AT M ) which has no existential states. By 3-AT M (L(m)) (F V 3AT M (L(m)), 3-SU T M (L(m)), F V 3-SU T M (L(m)), 3-U T M (L(m)), F V 3-U T M (L(m))), we denote an L(m) space-bounded 3-AT M (F V 3-AT M , 3-SU T M , F V 3-SU T M , 3-U T M , F V 3-U T M ). A three-dimensional deterministic Turing machine (3-DT M ) (five-way three-dimensional deterministic Turing machine (F V 3-DT M )) is a 3-AT M (F V 3AT M ) whose ID’s each have at most one successor, and a three-dimensional nondeterministic Turing machine (3-N T M ) (five-way three-dimensional nondeterministic Turing machine (F V 3-N T M )) is a 3-AT M which has no universal states. We denote an L(m) space-bounded 3-DT M (3-N T M , F V 3-DT M , F V 3N T M ) by 3-DT M (L(m)) (3-N T M (L(m)), F V 3DT M (L(m)), F V 3-N T M (L(m))). We use 3SAF A (F V 3-SAF A, 3-AF A, F V 3-AF A, 3-N F A, F V 3-N F A, 3-DF A,F V 3-DF A) to denote a threedimensional synchronized alternating finite automaton (five-way three-dimensional synchronized alternating finite automaton, three-dimensional alternating finite automaton, five-way three-dimensional alternaing finite automaton, three-dimensional nondeterministic finite automaton, five-way three-dimensional nondeterministic finite automaton, three-dimensional determinnistic finite automaton, five-way three-dimensional deterministic finite automaton). That is, a 3-SAF A (F V 3-SAF A, 3-AF A, F V 3-AF A, 3-N F A, F V 3N F A, 3-DF A, F V 3-DF A) is a 3-SAT M (F V 3SAT M , 3-AT M , F V 3-AT M , 3-N T M , F V 3-N T M , 3-DT M , F V 3-DT M ) which doesn’t have storage tape. Similarly, we use 3-SU F A (F V 3-SU F A, 3U F A, F V 3-U F A) to denote a 3-SU T M (F V 3SU T M , 3-U T M , F V 3-U T M ) which doesn’t have the storage tape. Furthermore, for any integer k ≥ 1, 3SAT M (L(m))[k] is used to denote a 3-SAT M (L(m))


such that any computation tree of M on any input x has at most k leaves. F V 3-SAT M (L(m))[k], 3SU T M (L(m))[k], · · · , 3-SAF A(L(m))[k], etc. have the similar meaning. For any integer k ≥ 1, 3N F A(k-heads) (3-DF A(k-heads)) is used to denote a 3-NFA (3-DF A) which has k input heads. For any machine class C, let

head-position as the current plane number, respectively. The currently scanned symbols are available as the components of the symbol from Σ read by the head. ¤ It is shown in [8] that the following lemma holds.

L [C]={T | T = T (M ) for some M in C }.

Lemma 2.2. For each k ≥ 1, L[1-N F A(k-heads)] ( L[1-N F A(k + 1-heads)].

Thus, for example, L[3-SAT M (L(m))] denotes the class of sets accepted by 3-SAT M (L(m))’s.

From Lemmas 2.1 and 2.2, we can get the following theorem.

2

Theorem 2.1. For any integer k ≥ 1, L[3-N F A(k-heads)] ( L[3-N F A(k+1-heads)].

Hierarchy based on the number of processes

It is shown in [6] that for two-dimensional alternating finite automata, k + 1 processes are more powerful than k processes for any k ≥ 1. This section shows that a similar result holds also for three-dimensional synchronized alternating finite automata. We will need the following operation ρ mapping one-dimensional words over an alphabet Σ to cubic tapes over Σ × Σ × Σ. This operation was first introduced in [9]. Let w = a1 a2 · · · an be a word of length n. Then ρ (w) = x where x (i, j, k) = (ai , aj , ak ) for 1 ≤ i ≤ n, 1 ≤ j ≤ n and 1 ≤ k ≤ n. Thus a symbol of x in a certain row, column and plane has the corresponding symbol of w in the first, second and third component, respectively. A word w = a1 a2 · · · an is mapped to (a1 , a1 , a1 ) (a1 , a2 , a1 ) · · (a1 , an , a1 ) (a2 , a1 , a1 ) (a2 , a2 , a1 ) · · (a2 , an , a1 ) ··········· (an−1 , a1 , an ) (an−1 , a2 , an ) · · (an−1 , an , an ) (an , a1 , an ) (an , a2 , an ) · · (an , an , an ) This operation is extended in the usual way to languages. For each k > 1, let 1-N F A(k-heads) denote a onedimensional two-way nondeterministic k-heads finite automaton [8]. Lemma 2.1. For each k ≥ 1, a one-dimensional language L is accepted by a 1-N F A(3k-heads) if and only if ρ(L) ∈ L [3-N F A(k-heads)]. Proof: We only prove the lemma for the case of k = 1. The 1-N F A(3-heads) simulates the 3-N F A by storing the information of row, column and plane in its head positions. It assembles the trio from the symbols read by the heads. Conversely the 3-N F A verifies that the first (second, third) components of input symbols within every row (column, plane) agree. Then the 3-N F A starts a step by step simulation by storing the first headposition as the current row number, the second headposition as the current column number, and the third

Proof: Let us suppose that L∈L[1-N F A(3k+3-heads)]−L[1-N F A(3k-heads)] · · · (1). Then we have ρ(L) ∈ L[3-N F A(k + 1-heads)] from Lemma 2.1. Now, we assume that ρ(L) ∈ L[3-N F A(kheads)]. Then, we would have L ∈ L[1-N F A(3kheads)] from Lemma 2.1. This contradicts (1), and thus we have ρ(L) ∈ / L[3-N F A(k-heads)]. This completes the proof of the theorem. ¤ From Theorems 5.2 (1) of [7] and Theorem 2.1, we have Corollary 2.1. For any integer k ≥ 1, L[3-SAF A[k]] ( L[3-SAF A[k+1]].

3

Recognizability of connected pictures

There have been many interesting investigations on digital geometry [10-12]. These works form the theoretical foundation of digital image processing. Among them, the problem of recognizability of connectedness is one of the most interesting topics. This section investigates the recognizability of connected tapes by F V 3-SAF A’s and F V 3-SU T M ’s. (3)

Definitions 3.1. Let x be in {0, 1} . A maximal subset, P of N 3 satisfying the following conditions is called a 1-component of x. (i)For any (i1 ,i2 ,i3 ) ∈ P, we have 1 ≤ i1 ≤ l1 (x), 1 ≤ i2 ≤ l2 (x),1 ≤ i3 ≤ l3 (x), and x(i1 , i2 , i3 )=1. 0

0

0

(ii)For any (i1 ,i2 ,i3 ), (i1 ,i2 ,i3 ) ∈ P, there exists a sequence (i1,0 , i2,0 , i3,0 ), (i1,1 , i2,1 , i3,1 ),· · ·, (i1,n , i2,n , i3,n ) of elements in P such that (i1,0 , i2,0 , i3,0 ) = (i1 , 0 0 0 i2 , i3 ), (i1,n , i2,n , i3,n ) = (i1 ,i2 ,i3 ), and |i1,j − i1,j−1 | + |i2,j − i2,j−1 | + |i3,j − i3,j−1 | ≤ 1 (1 ≤ j ≤ n). A tape x ∈ {0, 1}3 is called connected if there exists exactly


the 2nd axis

one 1-compnent of x. We denote the set of all the cubic connected tapes by Tc . It is shown in [13] that a 3-AT M can accept Tc . From this fact and from the fact L[F V 3-SAF A] = L[3-SAF A] ⊇ L[3-AF A] by using a technique similar to that in Ref.[2], the following theorem holds.

the 1st axis the 3rd axis

‘0’ ‘1’

Theorem 3.1. Tc ∈ L[F V 3-SAF A]. It is shown in [13] that log m space is necessary and sufficient for F V 3-AT M ’s to accept Tc . We below show the necessary and sufficient space for F V 3SU T M ’s to accept T¯c (=the complement of Tc ). Theorem 3.2. m2 space is necessary and sufficient for F V 3-SU T M ’s to accept T¯c . Proof: (The proof of sufficiency) It is shown in [11] that Tc is accepted by a deterministic one-way parallel/sequential array acceptor (DOWPS), and it is shown in [5] that L[DOWPS] = L[T R2-DT M (m)] (T R2-DT M (m)) means m space-bounded three-way two-dimensional deterministic Turing machine). From these facts and the fact [3,4] that L[T R2-DT M (m)] is closed under complementation, it follows that T¯c is in L[T R2-DT M (m)], and thus in L[T R2-SU T M (m)]. By applying the same idea of such a two-dimensional case, we can easily get the fact that T¯c is in L[F V 3SU T M (m2 )]. (The proof of necessity) Suppose that there is an F V 3-SU T M (L(m)) M accepting T¯c , where L(m) = o(m2 ). We assume without loss of generality that M enters an accepting state only on the bottom boundary. Let Tc0 = { x ∈ {0, 1}(4m+1,4m+1,4m+1) | m ≥ 1 & ∀i1 (1 ≤ i1 ≤ m+1) ∀i2 (1 ≤ i2 ≤ 2m+1) [ x [(2i2 −1,1,2i1 −1),(2i2 −1,4m-2i1 +3,2i1 −1)], x[(2i2 −1,1,4m−2i1 +3), (2i2 −1,4m−2i1 +3,4m−2i1 +3)], x[(2i2 −1,4m−2i1 +3,2i1 −1), (2i2 −1,4m−2i1 +3,4m−2i1 +3)] ∈ {1}(3) ] & ∀i2 (1 ≤ i2 ≤ 2m) [ x [(2i2 ,1,2m+1),(2i2 ,2m+1,2m+1) ] ∈ {1}(3) ] & ∀i1 (1 ≤ i1 ≤ 2m) ∀i2 (1 ≤ i2 ≤ 2m+1) [ x(2i2 −1,1,2i1 ) = x(2i2 − 1, 1, 4m − 2i1 + 2) ] & (the other part of x consists of 0’s) }, where we define ¯0 = 1 and 1¯ = 0(See Fig. 2). Clearly Tc0 ⊆ Tc . Let s and t be the numbers of states (of the finite control) and storage tape symbols of M, respectively. For each m(m ≥ 1), let V (m) = { x ∈ Tc0 | l1 (x) = l2 (x) = l3 (x) = 4m+1 }.

Fig. 2: A tape in Tc0

For each x in V (m), let S(x) and C(x) be sets of configurations of M defined as follows: S(x) = {((i1 ,i2 ,2m+1),(q,α,k)) | there exists a computation path IM (x) `∗M (x,((i1 ,i2 ,2m),(q 0 ,α0 ,k 0 ))) `M (x,((i1 ,i2 ,2m+1),(q,α,k))) of M on x (that is, (x,((i1 ,i2 ,2m+1),(q,α,k))) is an ID of M just after the point where the input head left the (2m+1)th plane of x) }, C(x) = {{ρ1, ρ2} | ρ1 and ρ2 are configurations in S(x) such that (i)in case of ρ1 = ρ2, there exists a sequential computation of M which starts with ID(x,ρ1) and either terminates in a rejecting ID, or enters an infinite loop, and (ii)in case of ρ1 6= ρ2, there exist two sequential computations of M which start with ID’s(x,ρ1) and (x,ρ2), respectively, and terminate in sync ID’s with different sync elements}. (Note that, for each x in V (m), C(x) is not empty, since x is not in T¯c , and so not accepted by M .) Then the following proposition must hold. Proposition 3.1. For any two different tapes x, y ∈ V (m), C(x) ∩ C(y)= φ. [Proof: For otherwise, suppose that x 6= y(x,y ∈ V (m)), C(x) ∩ C(y) 6= φ, and {ρ1 ,ρ2 } ∈ C(x) ∩ C(y). Let z (with l1 (x)=l2 (x)=l3 (x)=4m+1) be the tape such that (i)z [(1,1,1), (4m+1,4m+1,2m+1)] = x[(1,1,1), (4m+1,4m+1,2m+1)], and


(ii)z [(1,1,2m+2), (4m+1,4m+1,4m+1)] = y[(1,1, 2m+2), (4m+1,4m+1,4m+1)]. Since {ρ1 ,ρ2 } ∈ C(x), there exist computation paths IM (z) `∗M (z, ρ1) and IM (z) `∗M (z, ρ2). Since {ρ1 ,ρ2 } ∈ C(y), in case of ρ1 =ρ2 , there exists a sequential computation of M which starts with the ID (z,ρ1 ) and either terminates in a rejecting ID, or enters an infinite loop, and in case of ρ1 6= ρ2 , there exist two sequential computations of M which start with ID’s(z, ρ1 ) and (z, ρ2 ), respectively, and terminate in sync ID’s with different sync elements. This means that z is not accepted by M . This contradicts the fact that z is in T¯c = T (M ). ¤] Proof of Theorem 3.2(continued ): Let p(m) denote the number of pairs of possible configurations of M just after the point where the input head left the (2m+1)th planes of tapes in V (m). Then ¡ ¢ p(m) = K 2 +K where K = s(4m + 3)2 L(4m+1)tL(4m+1) . On the other hand, |V (m)| = 2m(2m+1) . Since L(m) = o(m), we have |V (m)| ≥ p(m) for large m. Therefore, it follows that for large m there must be two different tapes x, y in V (m) such that C(x) ∩ C(y) 6= φ. This contradicts Proposition 3.1 and completes the proof of necessity. ¤

4

Conclusions

In this section, we conclude this paper by giving several open problems. (1) For any function L(m) ≥ log m, L[3ATM(L(m))] ( L[3-SAT M (L(m))]? (2)For any integer k ≥ 1, L[3-SU F A[k]] ( L[3SU F A[k+1]]? and L[F V 3-SU F A[k]] ( L[F V 3SU F A[k+1]]? (3)Tc ∈ L[3-SU F A]? and Tc ∈ L[F V 3-SU F A]?

References [1] J. Hromkoviˇ c, “How to organize the commmunication among parallel processes in alternating computations”, manuscript, 1986. [2] J. Hromkoviˇ c, K.Inoue, B. Rovan, A.Slobodov´ a, I.Takanami and K.W. Wagner, “On the power of one-way synchronized alternating machines with small space”, Int. J. of Foundations of Computer Science, Vol.3, No.1, pp.65-79,1992. [3] K. Inoue and I. Takanami, “Three-way tapebounded two-dimensional Turing machines”, Information Sciences 17, No.3, pp.195-220,1979.

[4] K. Inoue, I. Takanami and H. Taniguchi, “Threeway two-dimensional simple multihead finite automata -Closure properties-”, Trans. IECE ’79/4, Vol62-D, No.4, pp.273-280, 1979. [5] K. Inoue and I. Takanami, “A note on deterministic three-way tape-bounded two-dimensional Turing machines”, Information Sciences 20, pp.4155,1980. [6] A. Ito, K. Inoue, I. Takanami and Y. Inagaki, “Constant leaf-size hierarchy of two-dimensional alternating Turing machines”, Inter. J. Patt. Rec. Arti. Intell., Vol.8, No.2, pp.509-524,1994. [7] T. Ito, M. Sakamoto, M. Saito, K.Iihoshi, H. Furutani, M. Kono and K.Inoue, “Three-dimensional synchronized alternating Turing machines”, Proc. 11th International Symp.on AROB, 2006. [8] B. Monien, “Two-way multihead automata over a one-leter alphabet”, RAIRO Inform. Theory 14, pp.67-82, 1980. [9] H. Petersen, “Some results concerning twodimensional Turing machines and finite automata”, Lecture Notes on Computer Science 965, FCT ’95, pp.394-382,1995. [10] A. Rosenfeld and A.C.Kak, “Digital Picture Processing”, Academic Press, New York,1976. [11] A. Rosenfeld, “Picture Languages (Formal Models for Picture Recognition)”, Academic Press, New York,1979. [12] A. Rosenfeld, “Three-dimensional digital topology”, Inform. and Control 50, pp.119-127, 1981. [13] M. Sakamoto, K.Inoue and I.Takanami, “A note on three-dimensional alternating Turing machines with space smaller than log m”, Information Sciences 72, pp.225–249, 1993.