a survey of automata on three-dimensional input … survey of automata on three-dimensional input...

A Survey of Automata on Three-Dimensional Input Tapes

MAKOTO SAKAMOTO, NAOKO TOMOZOE,HIROSHI FURUTANI,and MICHIO KONO

University of MiyazakiFaculty of Engineering

1-1, Gakuen Kibanadai Nishi, Miyazaki CityJAPAN 889-2192

[email protected]

TAKAO ITO ∗ and YASUO UCHIDA

Ube National College of TechnologyDept. of Business Administration

2-14-1, Tokiwadai Ube CityJAPAN 755-8555

[email protected] and [email protected]

HIDENOBU OKABENittetsu Hitachi Systems Engineering, Inc.

Financial Solutions Division8-1, Akashi-Chou, Chuou-Ku, Tokyo

JAPAN [email protected]

Abstract: The question of whether processing three-dimensional digital patterns is much more difficult than two-dimensional ones is of great interest from the theoretical and practical standpoints. Recently, due to the advancesin many application areas such as computer vision, robotics, and so forth, it has become increasingly apparentthat the study of three-dimensional pattern processing has been of crucial importance. Thus, the research of three-dimensional automata as computational models of three-dimensional pattern processing has also been meaningful.The main purpose of this paper is to survey the definitions and properties of various three-dimensional automata.

Key–Words:Computation, Constructibility, Finite Automaton, Inkdot, Marker, Recognizability, Three-Dimension,Turing Machine

1 Introduction

Blum and Hewitt first proposed two-dimensional au-tomata as computational models of two-dimensionalpattern processing, and investigated their patternrecognition abilities [3]. Since then, many researchersin this field have been investigating a lot of propertiesabout automata on a two-dimensional tape. Recently,due to the advances in computer vision, robotics, andso on, the study of three-dimensional information pro-cessing has been of great importance. For instance,three-dimensional image is now needed in visual com-munication, such as virtual reality systems. Even inthe Internet environment, new protocols have beenproposed for virtual reality communication on theWWW [53]. In the medical field, we can easily getthe precise three-dimensional volumetric image of ahuman body by excellent equipments such as X-rayCT scanner and MRI scanner. Thus, the study ofthree-dimensional automata has been meaningful asthe computational model of three-dimensional infor-

∗TakaoIto is a visiting research professor of School of Man-agement, New Jersey Institute of Technology in USA from March29, 2008 to March 28, 2009.

mation processing. In this paper, we show a surveyof three-dimensional automata. Section 2 observes ahistorical overview, and provides a background and amotive for the study of three-dimensional automata.Section 3 concerns deterministic, nondeterministic,alternating with only universal states, and alternat-ing three-dimensional Turing machines (including fi-nite automata). In Section 4, we deal with three-dimensionally space constructibility and space hier-archy. In Section 5, we show some results aboutrecognizability of three-dimensional connected pic-tures. In Section 6, we introduce other topics of three-dimensional automata. Finally, in Section 7, we con-clude this paper by summarizing the results.

2 Historical BackgroundComputer science has two major components : first,the fundamental mathematics and theories underlyingcomputing, and second, engineering techniques forthe design of computer systems― hardware and soft-ware.

Theoretical computer science falls under the firstarea of the two major components. It had its begin-

WSEAS TRANSACTIONS on COMPUTERS

Makoto Sakamoto, Naoko Tomozoe, Hiroshi Furutani, Michio Kono, Takao Ito and Yasuo Uchida

ISSN: 1109-2750 1638 Issue 10, Volume 7, October 2008

ningsin various field: physics, mathematics, linguis-tics, electric and electronic engineering, physiology,and so on. Out of these studies came important ideasand models that are central to theoretical computerscience [1,14].

In theoretical computer science, Turing machinehas played a number of important roles in understand-ing and exploiting basic concepts and mechanisms incomputing and information processing. It is a sim-ple mathematical model of computers which was in-troduced by Turing in 1936 to answer fundamentalproblems of computer science― ‘What kind of log-ical work can we effectively perform ?’ [59]. If therestrictions in its structure and move are placed onTuring machine, the restricted Turing machine is lesspowerful than the original one. However, it has be-come increasingly apparent that the characterizationand classification of powers of the restricted Turingmachines should be of great importance. Such a studywas active in 1950’s and 1960’s. On the other hand,many researchers have been making their efforts to in-vestigate another fundamental problems of computerscience― ‘How complicated is it to perform a givenlogical work?’. The concept of computational com-plexity is a formalization of such difficulty of logicalworks.

In the study of computational complexity, thecomplexity measures are of great importance. In gen-eral, it is well known that the computational complex-ity has originated in a study of considering how thecomputational powers of various types of automataare characterized by the complexity measures suchas space complexity, time complexity, or some otherrelated measures. Especially, the concept of com-plexity is very useful to characterize various types ofautomata from a point of view of memory require-ments [34]. This study was motivated by Stearns,Hartmanis, and Lewis in 1965 [54]. They introducedanL(m) space-bounded one-dimensional Turing ma-chine to formalize the notion of space complexity, andinvestigated its computing ability. Some results wererefined by Hopcroft and Ullman [12-14]. Moreover,Chandra, Kozen, and Stockmeyer introduced an alter-nating Turing machine as a theoretical model of par-allel computation in 1981 [5]. An alternating Turingmachine, whose state set is partitioned into two dis-joint sets, the set of universal states and the set of exis-tential states, is a generalization of a nondeterministicTuring machine. A nondeterministic Turing machineis an alternating Turing machine which has only ex-istential states. In related paper [11,15,29-31,39,40,45,55], several investigations of these machines havebeen continued.

After that, the growth of the processing of pic-torial information by computer was rapid in those

days. Therefore, the problem of computational com-plexity was also arisen in the two-dimensional in-formation processing. Blum and Hewitt first pro-posed two-dimensional automata― two-dimensionalfinite automata and marker automata, and investi-gated their abilities of pattern recognition in 1967[3]. Since then, many researchers in this fieldhave been investigating a lot of properties about au-tomata on a two-dimensional tape. For example,Morita, Umeo, and Sugata proposed anL(m,n)space-bounded two-dimensional Turing machine andits variants to formalize memory limited computa-tions in the two-dimensional information process-ing [33]. Inoue, Takanami, and Taniguchi intro-duced two-dimensional alternating Turing machinesas a generalization of two-dimensional nondetermin-istic Turing machines and as a mechanism to modelparallel computation. Restricted version of two-dimensional alternating Turing machines were inves-tigated [24,25]. Special types of two-dimensional Tur-ing machines (two-dimensional pushdown automata,stack automata, multicounter automata, multiheadautomata, and marker automata) were investigated[16,21,51,52,56]. Moreover, cellular automata on atwo-dimensional tape were investigated not only inthe viewpoint of formal language theory, but alsoin the viewpoint of pattern recognition. Cellularautomata on a two-dimensional tape can be classi-fied into three types. The first type, called a two-dimensional cellular automata, is investigated [2,6,7].Especially, many properties of two-dimensional on-line tessellation acceptors, which are restricted typeof two-dimensional cellular automata, are investigated[17,18,20]. The second type of cellular automata ona two-dimensional tape is investigated [19,42,44,58].Two typical models of this type are parallel/ sequen-tial array automata and one-dimensional bounded cel-lular acceptors. The third type, called a pyramid cellu-lar acceptor, is investigated [22]. More detailed surveyof two-dimensional automata theory is done by Inoueand Takanami [23].

By the way, the question of whether process-ing three-dimensional digital patters is much difficultthan two-dimensional ones is of great interest fromthe theoretical and practical standpoints. In recentyears, due to the advances in many application ar-eas such as computer graphics, computer-aided de-sign / manufacturing, computer vision, image pro-cessing, robotics, and so on, the study of three-dimensional pattern processing has been of crucialimportance [9,10,41,43]. Thus, the reseach of three-dimensional automata as the computational model ofthree-dimensional pattern processing has been mean-ingful. However, it is conjectured that the three-dimensional pattern processing has its own difficul-




tiesnot arising in two-dimensional case. One of thesedifficulties occurs in recognizing topological proper-ties of three-dimensional patterns because the three-dimensional neighborhood is more complicated thantwo-dimensional case. Generally speaking, a propertyor relationship is topological only if it is preservedwhen an arbitrary ‘rubber-sheet’ distortion is appliedto the pictures. For example, adjacency and connect-edness are topological; area, elongatedness, convex-ity, straightness, etc. are not.

During the past about thirty years, automataon a three-dimensional tape have been proposedand several properties of such automata have beenobtained. Inoue and Nakamura proposed an n-dimensional on-line tessellation acceptor which candetermine whether an n-dimensional tape is acceptedor not by the on-line and parallel processing [17].Blum and Sakoda investigated the capability of finiteautomata in two-dimensional and three-dimensionalspace [4]. Yamamoto, Morita, and Sugata intro-duced a three-dimensional k-marker automaton, anL(m) space-bounded three-dimensional Turing ma-chine and anL(m) space-bounded five-way three-dimensional Turing machine [60]. They studiedthe problem of recognizing connectedness of three-dimensional patterns by these machines. Taniguchi,Inoue, and Takanami investigated the relationship be-tween the accepting powers of three-dimensional fi-nite automata and five-way three-dimensional Turingmachines [57]. They also proposed a k-neighborhoodtemplateA-type two-dimensional bounded cellularacceptor which consists of a pair of a converter anda configuration-reader, as the computational modelof three-dimensional pattern process. The converterconverts the given three-dimensional tape to the two-dimensional configuration, and the configuration-reader determines the acceptance or nonacceptance ofgiven three-dimensional tape, depending on whetheror not the derived two-dimensional configuration isaccepted [58]. Nakamura and Aizawa proposed theinterlocking component which is a chainlike connec-tivity a new topological property of three-dimensionaldigital pictures, and investigated the recognizabilityof interlocking components [37]. Sakamoto proposedmulti-dimensional automata, and showed their severalproperties [46,47,49,50]. Moreover, Ito et al. investi-gated about synchronized alternation and parallelismfor three-dimensional automata [27, 28].

3 Three-Dimensional Turing Ma-chines

This section concerns alternating, nondeterminis-tic, and deterministic three-dimensional Turing ma-

chines, including three-dimensional finite automataand marker automata.

3.1 Preliminaries

Definition 3.1. Let Σ be a finite set of symbols.A three-dimensional input tapeover Σ is a three-dimensional rectangular array of elements ofΣ. Theset of all the three-dimensional input tapes overΣ isdenoted byΣ(3).

Given an input tapex ∈ Σ(3), for each inte-ger j(1 ≤ j ≤ 3), we let lj(x) be the length ofx along thejth axis. The set of allx ∈ Σ(3) withl1(x) = n1, l2(x) = n2 and l3(x) = n3 is de-noted byΣ(n1,n2,n3). When1 ≤ ij ≤ lj(x) for eachj(1 ≤ j ≤ 3), let x(i1, i2, i3) denote the symbol inx with coordinates (i1, i2, i3). Furthermore, we definex[(i1, i2, i3),(i′1, i

′2, i

′3)], when1 ≤ ij ≤ i′j ≤ lj(x)

for integerj(1 ≤ j ≤ 3), as the three-dimensionalinput tapey satisfying the following conditions:

(i) for eachj (1 ≤ j ≤ 3), lj(y) = i′j − ij + 1;(ii) for eachr1, 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).

Definition 3.2. A three-dimensional alternatingTuring machine(3-ATM ) is a seven-tupleM =(Q, q0, U, F,Σ, Γ, δ), where

(i) Q is a finite set ofstates;(ii) q0 ∈ Q is theinitial state;(iii) U ⊆ Q is the set ofuniversal states;(iv) F ⊆ Q is the set ofaccepting states;(v) Σ is a finiteinput alphabet(# /∈ Σ is thebound-ary symbol);(vi) Γ is a finitestorage tape alphabet(B ∈ Γ is theblank symbol; and(vii) δ ⊆ (Q × (Σ ∪ {#}) × Γ) × (Q × (Γ −{B}) × (east,west,south,north,up,down,no move)×(right,left,no move)is thenext move relation.

A stateq in Q − U is said to beexistential. Asshown in Fig.1, the machineM has a read-only rect-angular input tape with boundary symbols ‘#’ andone semi-infinite storage tape, initially blank. Ofcourse,M has a finite control, an input head, and astorage-tape head. A position is assigned to each cellof the storage tape, as shown in Fig.1.

A stepof M consists of reading one symbol fromeach tape, writing a symbol on the storage tape, mov-ing the input and storage heads in specified directions




Fig. 1: Three-Dimensional Alternating Turing Ma-chine.

(east, west, south, north, up, down, or no move for in-put head, and left, right, or no move for storage head),and entering a new state, in accordance with the nextmove relationδ.

Definition 3.3. A configuration of a 3-ATMM =(Q,q0,U ,F ,Σ,Γ,δ) is a pair of an element ofΣ(3)

and an element ofCM = (N ∪ {0})3 × SM , whereSM = Q × (Γ − {B})∗× N , andN denotes the setof all positive integers. The first componentx of aconfigurationc = (x, ((i1, i2, i3),(q,α,j))) representsthe input toM . The second component(i1, i2, i3) ofc represents the input-head position. The third com-ponent(q, α, j) of c represents the state of the finitecontrol, nonblank contents of the storage tape, and thestorage-head position. An element ofCM is calleda semi-configurationof M and an element ofSM iscalled a storage state ofM . If q is the state associatedwith configurationc, thenc is said to be a universal(existential, accepting) configuration ifq is auniversal(existential, accepting)state. The initial configurationof M on inputx is IM (x) = (x, ((1, 1, 1), (q0, λ, 1))),whereλ is the null string.

Definition 3.4. GivenM = (Q, q0, U, F,Σ, Γ, δ), wewrite c `M c′ and c′ is a successorof c if config-uration c′ follows from configurationc in one stepof M , according to the transition rulesδ. `∗M de-notes the reflexive transitive closure of`M . The re-lation`M is not necessarily single-valued, becauseδis not. A computation pathof M on x is a sequencec0 `M c1 `M · · · `M cn(n ≥ 0), wherec0 = IM (x).

A computation tree ofM is a finite, nonemptylabeled tree with the following properties:

(i) each nodev of the tree is labeled with a configura-tion l(v);(ii) if v is an internal node (a nonleaf) of the tree,l(v) is universal andc|l(v) `M c =c1,…,ck, thenvhas exactlyk childrenv1, · · ·, vk such thatl(vi) = ci

(1 ≤ i ≤ k); and(iii) if v is an internal node of the tree andl(v) is ex-istential, thenv has exactly one childu such that

l(v) `M l(u).

A computational tree ofM on inputx is a com-putation tree ofM whose root is labeled withIM (x).An accepting computation treeof M on x is a com-putation tree ofM onx whose leaves are labeled withaccepting configurations. We say thatM acceptsx ifthere is an accepting computation tree ofM on inputx. Define

T (M) = {x ∈ Σ(3) |M accepts x}.We next define a five-way three-dimensional al-

ternating Turing machine, which can be considered asan alternating version of a five-way three-dimensionalTuring machine [49].

Definition 3.5. A five-way three-dimensional alter-nating Turing machine (FV 3-ATM ) is a 3-ATMM = (Q, q0, U, F,Σ, Γ, δ) such that

δ ⊆ (Q × (Σ ∪ {#}) × Γ) × (Q × (Γ − B)×{east,west,south, north, down, no move} × {right,left, no move}).

That is, anFV 3-ATM is a3-ATM whose inputhead can move east, west, south, north, or down, butnot up.

The set of states of alternating automaton is par-titioned into two nonintersecting sets, i.e., the set ofexistential states and the set of universal states. Athree-dimensional nondeterministic Turing machine(3-NTM ) (a five-way three-dimensional nondeter-ministic Turing machine(FV 3-NTM )) is a3-ATM(FV 3-ATM ) which has no universal state. Con-versely, a3-ATM (FV 3-ATM ) which has no exis-tential states is called athree-dimensional alternatingTuring machine with only universal states(3-UTM )(a five-way three-dimensional alternating Turing ma-chine with only universal states(FV 3-UTM )). Ofcourse, athree-dimensional deterministic Turing ma-chine(3-DTM ) (a five-way three-dimensional deter-ministic Turing machine(FV 3-DTM )) is a specialcase of3-ATM (FV 3-ATM ), i.e., each of whoseconfigurations has at most one successor.

Definition 3.6. Let L(m) : N→ R be a functionwith one variablem, where N is the set of all




positive integers andR is the set of all nonnegativereal numbers. With a3-ATM M we associatea space complexity functionSPACE that takesconfigurations to natural numbers. That is, foreach configurationc = (x, ((i1, i2, i3), (q, α, j))),let SPACE(c) = |α|. We say thatM is L(m)space-boundedif for all m ≥ 1 and for eachxwith l1(x) = l2(x) = l3(x) = m, if x is acceptedby M , then there is an accepting computation treeof M on input x such that for each nodev of thetree, SPACE(l(v)) is smaller than or equal to thesmallest integer which is greater than or equal toL(m). By ‘3-ATM (L(m))’ (‘ FV 3-ATM(L(m))’,‘3-UTM(L(m))’, ‘ FV 3-UTM(L(m))’,‘3-NTM(L(m))’, ‘ FV 3-NTM(L(m))’,‘3-DTM(L(m))’, (‘ FV 3-DTM(L(m))’) we de-note anL(m) space-bounded3-ATM (FV 3-ATM ,3-UTM , FV 3-UTM , 3-NTM , FV 3-NTM ,3-DTM , FV 3-DTM ).

A 3-ATM (0) (FV 3-ATM (0)) is called athree-dimensional alternating finite automaton (five-waythree-dimensional alternating finite automaton), anddenoted by3-AFA (FV 3-AFA). Similarly, a3-UTM (0) (3-NTM (0), 3-DTM (0)) is called athree-dimensional alternating finite automaton withonly universal states (three-dimensional nondeter-ministic finite automaton, three-dimensional deter-ministic finite automaton), denoted by ‘3-UFA’(‘3-NFA ’, ‘ 3-DFA’). A five-way 3-UFA (3-NFA,3-DFA) is denoted by ‘FV 3-UFA’ (‘ FV 3-NFA’,‘FV 3-DFA’).

Moreover, we introduce two automata for the im-provement of picture recognizability of the finite au-tomaton. One is a multi-marker automaton. It is a fi-nite automaton which keeps marks as ‘pebbles’ in thefinite control, and cannot rewrite any input symbolsbut can make marks on its input with the restrictionthat only a bounded number of these marks can existat any given time [32, 49]. The other is a multi-inkdotautomaton. This automaton is a conventional finiteautomaton capable of dropping an inkdot on a giveninput tape for a landmark, but unable to further pickit up. For a three-dimensionalk-marker automaton(three-dimensionalk-inkdot automaton) is denoted by3-XMAk (3-XIAk) for eachX ∈ {D, N, U,A},whereD means determinism,N means nondetermin-ism, U means alternation with only universal states,andA means alternation.

For each X ∈ {D,N,U,A}, we denotea 3-XTM (FV 3-XTM , 3-XTM(L(m)),FV 3-XTM(L(m)), 3-XFA, FV 3-XFA) whoseinput tapes are restricted to cubic ones by3-XTM c.FV 3-XTM c, 3-XTM c(L(m)), etc. have the samemeaning.

For each X ∈ {D,N,U,A}, we denote

by L[3-XTM ] the class of sets of all three-dimensional tapes accepted by3-XTM ’s . That is,L[3-XTM ] =T |T = T (M) for some3-XTM M .L[FV 3-XTM ], L[3-XTM(L(m))], etc. are definedsimilarly.

3.2 Main Accepting PowersThis subsection states the accepting powers of three-dimensional Turing machines [49].

Theorem 3.1. If L(m) = o(logm), thenL[3-DTM c(L(m))] ( L[3-UTM c(L(m))] (L[3-ATM c(L(m))].

Corollary 3.1. L[3-DFAc] ( L[3-NFAc] (L[3-AFAc].

Theorem 3.2. If (i) L(m)=o(m2),or (ii) L(m)≥logm and L(m)=o(m3),then L[FV 3-DTM c(L(m))] (L[FV 3-UTM c(L(m))] ( L[FV 3-ATM c(L(m))],and L[FV 3-UTM c(L(m))] andL[FV 3-NTM c(L(m))] are incomparable.

Corollary 3.2. (i) L[FV 3-UFAc] (L[FV 3-AFAc]. (ii) L[FV 3-UFAc] is in-comparable with L[FV 3-NFAc]. (iii)L[FV 3-DFAc] ( L[FV 3-UFAc].

Theorem 3.3. If (i) L(m) = o(m2), or (ii)L(m) ≥ logm and L(m) = o(m3), thenL[FV 3-UTM c(L(m))] ( L[3-UTM c(L(m))].

Corollary 3.3. L[FV 3-UFAc] ( L[3-UFAc].

Theorem 3.4. L[FV 3-UFAc] (L[FV 3-DTM c(m2)], and spacem2 is necessary andsufficient for FV 3-DTM c’s and FV 3-NTM c’s tosimulateFV 3-UFAc’s.

Theorem 3.5.L[3-UFAc] ( L[FV 3-DTM c(m3)],and spacem3 is necessary and sufficient forFV 3-DTM c’s to simulate3-UFAc’s .

Remark 3.1. We conjecture thatL[3-UFAc] (L[FV 3-NTM c(m2)], but we have not completed theproof of this conjecture yet.

Theorem 3.6. Spacem3 is necessary and sufficientfor FV 3-DTM c’s to simulateFV 3-AFAc’s and3-AFAc’s .

Open Problems 3.1. (i) Is L[3-NTM c(L(m))] in-comparable withL[3-UTM c(L(m))] for anyL such




that L(m) = o(logm)? (ii) L[3-DTM c(L(m))] (L[3-NTM c(L(m))] ( L[3-ATM c(L(m))] for anyL(m) ≥ logm?

4 Three-Dimensionally Space Con-structibility and Space Hierarchy

This section concerns three-dimensionally space con-structible functions and space complexity hierarchy ofthree-dimensional Turing machines whose input tapesare restricted to cubic ones.

4.1 Main Accepting Powers

Definition 4.1. A function L(m): N → R is calledthree-dimensionally space constructibleif there is a3-DTM(L(m))c M such that for eachm ≥ 1, thereexists some input tapex with l1(x) = l2(x) =l3(x) = m on whichM halts after its storage headhas marked off exactly the greatest integer cells whichis smaller than or equal toL(m). (In this case, we saythatM constructs the functionL in the storage tape.)

Definition 4.2. A function L(m): N → R is calledthree-dimensionally fully space constructibleif thereis a3-DTM(L(m))c M which, for eachm ≥ 1 andfor each input tapex with l1(x) = l2(x) = l3(x) =m, makes use of exactly the greatest integer cellswhich is smaller than or equal toL(m) and halts.

4.2 Three-Dimensionally Space ConstrutibleFunctions and Complexity Results

In this subsection, we show three-dimensionally fullyspace constructibility and space complexity hierar-chies of three-dimensional Turing machines whose in-put tapes are restricted to cubic ones [49].

Theorem 4.1. We consider the following three func-tions :

log(0)m = m,log(k)m = log(log(k−1)m), for k ≥ 1, andlog∗m = min{x|log(k)m ≤ 1}.

Then, the functionslog(k)m (k: any natural number)and log∗m are three-dimensionally fully space con-structible.

Theorem 4.2. For anyX ∈ {D,N,U,A}, for anyfunctionL(m): N →R, and for any constantd > 0,

L[3-XTM c(L(m))] = L[3-XTM c(L(m)+d)].

Theorem 4.3. For anyX ∈ {D, N,U,A}, for anyfunctionL(m): N →R, and for any constantd > 0,

L[3-XTM c(L(m))] = L[3-XTM c(dL(m))].

Theorem 4.4. Let L1(m) andL2(m) be any func-tions such that (i)L2(m) is three-dimensionallyspace constructible, (ii)limi→∞L1(mi)/L2(mi) =0, and (iii) L2(mi)/logmi > k (i=1,2,...) forsome increasing sequence of natural numbersmi

and for some constantk > 0. Then there ex-ists a setT in L[3-XTM c(L2(m))], but not inL[3-XTM c(L1(m))] for anyX ∈ {D, N}.

Theorem 4.5. For any functions L1(m) andL2(m) such that (i)L2(m) is three-dimensionallyspace constructible, (ii)L1(m) = o(L2(m)), thereexists a set inL[3-DTM c(L2(m))], but not inL[3-NTM c(L1(m))].

Open Problems 4.1. (i) Are the functionslog(k)m(k ≥ 3) andlog∗m fully space constructible by one-dimensional deterministic two-head Turing machinesor by two-dimensional deterministic Turing machineswith square inputs? (ii) Is there any other unboundedfunction below logm which is three-dimensionallyfully space constructible? (iii) Is there an infinitetight hierarchy for3-ATM c(L(m))’s with L(m) ≥logm? (iv) Is there an infinite space hierarchy for3-ATM c(L(m))′ with L(m) ≤ loglogm?

5 Recognizability of Connected Pic-tures

The recognition of the connectedness of digital pic-tures is one of the most fundamental problems in pic-ture processing. There have been various results re-lated to this problem. Especially, to recognize three-dimensional connectedness seems to be much moredifficult than the two-dimensional case, because ofintrinsic characteristics of three-dimensional pictures.This section mainly show the recognizability of three-dimensional connected tapes by three-dimensional au-tomata. LetTC be the set of all three-dimensionalconnected pictures. It is interesting to investigate howmuch space is required for three-dimensional Turingmachines to acceptTC . For this problem, we have

Theorem 5.1. (i) TC ∈ L[3-AFAc]. (ii) logm spaceis necessary and sufficient forFV 3-ATM ’s to recog-nizeTC .




Theorem 5.2.TC ∈ L[3-NMA1] [60].

Theorem 5.3. (i) the necessary and sufficientspace for FV 3-DTM ’s to simulate 3-DTM1’s(3-NTM1’s) is 2lmloglm (2k, wherek = l2m2). (ii)the necessary and sufficient space forFV 3-NTM ’sto simulate 3-DTM1’s (3-NTM1’s) is lmloglm(l2m2), wherel(m) is the number of rows (columns)on each plane of three-dimensional rectangular inputtapes.

Theorem 5.4.TC /∈ L[3-NIAk] [32].

Remark 5.1. [3-NIAk] ( L[3-AIAk] for any integerk.

Open Problems 5.1. (i) TC /∈ L[3-DTM(L(m))]or TC /∈ L[3-NTM(L(m))] for L(m) = o(logm)?(ii) TC ∈ L[3-UIA1]? (iii) Is TC accepted by a3-DTM1?

By the way,Digital geometryhas played an im-portant role in computer image analysis and recog-nition [43]. In particular, there is a well-developedtheory of topological properties such asconnected-nessandholesfor two-dimensional arrays [44]. Onthe other hand, three-dimensional information pro-cessing has also become of increasing interest with therapid growth of computed tomography, robotics, andso on. Thus it has become desirable to study the ge-ometrical properties such asinterlocking componentsandcavitiesfor three-dimensional arrays [37]. Inter-locking components was proposed as a new topolog-ical property of three-dimensional digital pictures in[37]. Let S1 andS2 be two subsets of the same three-dimensional digital picture.S1 and S2 are said tobe interlocked when they satisfy the following con-ditions:

(1) S1 andS2 aretori,(2) S1 goes through aholeof S2,(3) S2 goes through aholeof S1.

The interlocking ofS1 and S2 is illustrated inFig.2. This relation may be considered as a chainlikeconnectivity. Generally speaking, a property or rela-tionship istopologicalif it is preserved when an arbi-trary ‘rubber-sheet’distortion is applied to the pic-tures. For example, adjacency and connectedness aretopological; area, elongatedness, convexity, straight-ness, etc. are not.

It is proved that no one-marker automatoncan recognize interlocking components in a three-

Fig. 2: Interlocking Components.

dimensional input tape [37]. Moreover, in [35], three-dimensional one-marker automata are investigated interms of the space complexities that five-way three-dimensional Turing machines require and suffice torecognize interlocking components.

6 Other Topics

In this section, we list up other topics and related ref-erences about three-dimensional automata.

(i) Properties of special types of three-dimensionalTuring machines (leaf-size bounded automata, paral-lel automata, multi-counter automata, etc. on three-dimensional tapes) [28,48,49].(ii) Cellular types of three-dimensional automata[8,17,22,58].(iii) Closure properties [14,23,49].(iv) Recognizability of topological properties [36-38].(v) NP-complete problems [14,26,49].

7 Conclusions

In this paper, we surveyed several aspects of three-dimensional automata. Especially, we dealt withthree-dimensional Turing machines, including finiteautomata, three-dimensionally space constructability,recognizability of three-dimensional connected pic-tures, and so on. We believe that there are many prob-lems about three-dimensional automata to solve in thefuture. We hope that this survey will activate the in-vestigation of three-dimensional automata theory.




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