Recognizable languages defined by twodimensional

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1 University of South Florid Scholr Commons Grdute Theses nd Disserttions Grdute School 2006 Recognizle lnguges defined y twodimensionl shift spces Joni Burnette Pirnot University of South Florid Follow this nd dditionl works t: Prt of the Americn Studies Commons Scholr Commons Cittion Pirnot, Joni Burnette, "Recognizle lnguges defined y two-dimensionl shift spces" (2006). Grdute Theses nd Disserttions. This Disserttion is rought to you for free nd open ccess y the Grdute School t Scholr Commons. It hs een ccepted for inclusion in Grdute Theses nd Disserttions y n uthorized dministrtor of Scholr Commons. For more informtion, plese contct scholrcommons@usf.edu.

2 Recognizle Lnguges Defined y Two-dimensionl Shift Spces y Joni Burnette Pirnot A disserttion sumitted in prtil fulfillment of the requirements for the degree of Doctor of Philosophy Deprtment of Mthemtics College of Arts nd Sciences University of South Florid Mjor Professor: Ntš Jonosk, Ph.D. Committee Chir: Lili Woods, Ph.D. Gregory McColm, Ph.D. Stephen Suen, Ph.D. Mshico Sito, Ph.D. Xing-Dong Hou, Ph.D. Dte of pprovl: Octoer 24, 2006 Keywords: Recognizle picture lnguges, Dot systems, Two-dimensionl finite utomt, Trnsitivity, Douly periodic c Copyright 2006, Joni Burnette Pirnot

3 Dediction For Holly, who once told me to never wste tlent.

4 Tle of Contents List of Figures Astrct Prefce iii v vi Introduction. History nd Overview Nottion nd Terminology Dot Systems Finite Automt Recognizing Two-dimensionl Shift Spces 7 2. Recognition of Shifts of Finite Type y M F(X) Recognition of Sofic Shifts y M Φ F(X) Foridden nd Forced Structures in M F(X) nd M Φ F(X) Trnsitivity Fctor Lnguges of Dot Systems Defined over Z/2Z Grph Representtions Periodicity Douly Periodic Points in Two-dimensionl Shift Spces Exmples Monoids nd Follower Sets Monoids for Dot Systems Follower Sets nd Predecessor Sets i

5 Conclusion 06 References 08 Aout the Author End Pge ii

6 List of Figures. Enclosure of two locks Different types of trnsitivity Shpes within miniml rectngles Extending lock upwrd Extending n llowed lock Smple set of llowed locks for n REC lnguge Horizontl nd verticl k-conctentions Full shift s vertex shift () nd edge shift () Grph representing the Three-dot System Comprison of q r s nd q r s An exmple of lock pth where k = Grph M F(X) representing the Digonl-shift System Set of five llowed locks for sushift of Exmple Shift spce where ny is surrounded entirely y s In M h F(X), grph dimonds of length 2 re foridden Horizontl grph tringle of Corollry The Wllpper Pttern The inductive step The process of filling up the lock B Extension of lock into tringle Possile positions for intersection of right isosceles tringles Full-squre System One-column extension hving even sum One-column extension hving odd sum iii

7 3.9 A 4 3 lock of s meets 4 3 lock of 0 s Uniform trnsitivity in direction (u, v) Blocks nd lock pths over 2 2 sttes Point with lest doule period (6, 4) Vertex shifts in one-dimensionl cse Points of doule period (2, ) nd (, 2) Sugrphs representing point of doule period (2, 2) Stte mlgmtion Non-isomorphic grphs representing points of period (3, 3) Amlgmtion of two pirs of sttes Sugrph showing forced lels Amlgmtion of three pirs of sttes Amlgmtion of sttes, 5, nd 9 dicttes other lels Multiple mlgmtions suggested y /5/9 mlgmtion Represented points hve period (, n) for ll n Represented points hve period (n, n) for ll n Horizontl trnsltions of S Strictly sofic shift with follower-seprted grph Grph of one-dimensionl strictly sofic shift Strictly sofic shift; grph not follower seprted Grph of reduced size recognizing Y iv

8 Recognizle Lnguges Defined y Two-dimensionl Shift Spces Joni Burnette Pirnot ABSTRACT There re numerous connections etween the theory of forml lnguges nd tht of symolic dynmics. In ech, the trnsition from one dimension to two dimensions is ccompnied y much difficulty due in lrge prt to the emptiness prolem, which is relted to the presence (or lck thereof) of periodic points nd is known to e undecidle. Here, we focus on two-dimensionl lnguges tht hve the property tht ll locks llowed y the lnguge cn e extended to configurtion of the plne stisfying the structure of the lnguge; for such lnguges the emptiness prolem is not n issue. We first show tht dot systems my e ssocited with two-dimensionl lnguges hving this property, so tht we might employ these lnguges s vried exmples. We next define new type of finite utomton nd with it, tool for recognizing two-dimensionl strings of dt. It is then shown tht these utomt correctly represent the sofic shift spces tht result from the ppliction of lock mps to shifts of finite type. Therefter, these utomt re utilized to investigte properties of trnsitivity in the two-dimensionl lnguges tht they represent. More specificlly, new definitions for different types of two-dimensionl trnsitivity re dpted from topologicl dynmics nd then illustrted through the use of dot systems. The ppernce of periodic points in the lnguges represented y these utomt is lso explored, with min result eing tht the existence of periodic point is gurnteed under certin conditions. Finlly, issues of equivlence re introduced in the two-dimensionl setting with regrds to forml lnguges (syntctic monoids) nd symolic dynmics (the follower sets of grph representing sofic shift spce). v

9 Prefce It hs een sid tht the sum of the prts is greter thn the whole. Aleit difficult for mthemticin to condone such illogicl rhetoric, I must cknowledge tht in my cse I find truth in this sttement. The development of my cognitive skills, this disserttion, nd my outlook on mthemtics nd life in generl hve een motivted, encourged, nd sustined y mny people over the course of my life. I m grteful for the opportunity to thnk some of those people here, lthough few will never hve the opportunity to red here wht lsting impression they hve left on me. From n erly ge, my mother, fther, nd sisters were kind enough to tolerte my ookishness nd to overlook my lck of common sense. My mother plnted love of mthemtics in my young hert when she explined percents, decimls, nd frctions to me in wy tht illustrted the eutiful intercomplexity of mthemticl truths, nd my fther tught me tht quiet moments deep in thought could e some of the most vlule times in person s life. Dee nd Leh hve lwys hd words of encourgement for me, Holly strted me on n cdemic journey with words I will never forget, nd Amy ccompnied me on prt of tht journey s she studied our culture s most memorle words. When I ws young student, Mike Frme first tught me how to do serious mthemtics, nd when I thought I hd lost my wy, Soo Bong Che ecme my guide. It ws he who propelled me on to grdute school. Only one other person ws s dmnt out my ttending grdute school s Soo Bong, nd tht ws John. I look forwrd to shring the title of Dr. J with him. When my grdute-school life cme to hlt in order to mke wy for new life, my 03-yer-old friend Muggie told me tht Life is wht hppens while you re usy mking plns. Zen nd Mon were orn in 990 nd 992, respectively, nd they quickly ecme my rison d être. I thnk them oth for the mny importnt lessons they hve tught me. vi

10 When I finlly mde my wy ck to USF fter five-yer hitus, I met the womn who would chnge my life. Dr. Ntš Jonosk ssisted me cdemiclly nd emotionlly, she inspired me mthemticlly nd professionlly, nd she supported me personlly nd finncilly. (I m grteful for funding I received through the grnts CCF # nd EIA # ) I would never hve ttempted entering the Ph.D. progrm without her guidnce. There re not enough words to thnk her properly. I would lso like to thnk the other memers of my committee who devoted their time to the reding nd editing of this disserttion. In prticulr, Dr. Gregory McColm hs een n integrl prt of my eduction, nd I thnk him for his input nd insight. Additionlly, I thnk Dr. Jrkko Kri, who on severl occsions offered thoughtful dilogue with respect to symolic dynmics nd utomt theory. My work could not hve een completed without the support of Mntee Community College. Dr. John Rosen, Dr. Mike Mers, Dr. Dennis Runde, nd Alty Özgener hve een especilly supportive nd helpful these pst severl yers. Finlly, I wish to thnk my psychitrist, physicin, rtender, nker, chuffeur, msseuse, chef, nd est friend. The sum of these prts is one whole person - my husnd, Steve Pirnot - nd I thnk him for the mny, mny prts he hs plyed over the yers. vii

11 Introduction In this chpter, the required concepts from the fields of symolic dynmics nd forml lnguge theory re introduced. First we outline history of the reserch done in these fields in the two-dimensionl cse, nd we give n overview of the min ides nd results found within this pper. Next we provide nottion nd terminology required for discussion of the two-dimensionl cse; in prticulr, we define severl types of two-dimensionl trnsitivity y modifying similr notions found in the study of topologicl dynmicl systems. Finlly we explin dot systems s they exist in the literture, nd we then prove tht ny finite lock llowed y the structure of dot system lso ppers s sulock of some configurtion of the plne found within tht dot system.. History nd Overview One wy to study chnging (dynmicl) system is to mke the time discrete so s to study the itertes of single ction on the system. The field of symolic dynmics tkes this ide step further y lso mking the spce tht represents the system discrete. The generl ide is to use finite set of sttes to record the ction on the spce y first ssigning symol to ech stte nd then keeping trck of these sttes in discrete time steps y representtion vi n infinite sequence of symols [25]. So symolic dynmicl system studies spce of infinite sequences eing cted upon y trnsltion mp tht shifts oservtion from one prt of sequence to nother prt of the sequence. Such system is referred to s shift spce, nd the infinite sequences re referred to s points of the shift spce. One wy to keep trck of the ction on certin shift spces is to use finite utomton, which is directed grph with trnsitions etween the sttes tht represent the discrete time steps. This links the study of symolic dynmics with tht of forml lnguges, since recognizle lnguges re precisely those tht cn e represented y finite utomton.

12 While mny well-developed theories exist in one dimension for the study of symolic dynmics [25], utomt theory [2, 28], nd forml lnguges [0, 22, 29], mny of these results do not hold true in two dimensions. The trnsition from one to two dimensions is complicted y severl fctors, one of which is relted to the possile non-existence of periodic points in two-dimensionl shift spce. As n illustrtion, consider the following question posed y Ho Wng [3] in 96: Is it decidle whether finite set of equl-sized squre tiles with colors on ech edge cn tile the plne in such wy tht contiguous edges will lwys hve the sme color? Wng nswered his own question in the ffirmtive (see, for exmple, [32]) y constructing n lgorithm tht hinged on the ssumption tht ny set of tiles cple of tiling the plne would dmit periodic tiling. At the time, it seemed resonle ssumption: the set of ll possile tilings of the plne using given set of Wng tiles defines two-dimensionl shift of finite type, nd in the lnguge of one-dimensionl symolic dynmics, shift of finite type is nonempty if nd only if it contins periodic point [25]. However, in 966 puliction, Roert Berger shows tht Wng s solution is incorrect [2] y demonstrting the existence of set of Wng tiles tht cn only tile the plne periodiclly. Wng s question hs come to e known s the emptiness prolem. It is now known [7] tht the emptiness prolem is equivlent to the hlting prolem for Turing mchines nd is therefore undecidle. Efforts to investigte two-dimensionl recognizle lnguges do so in the context of finite rectngulr pictures (rrys of symols) nd non-rectngulr shpes []. In [7], Gimmrresi nd Restivo introduce the clss REC of recognizle picture lnguges s those tht cn e otined y projection of locl picture lnguge. (A locl picture lnguge over the lphet Σ is defined s one tht cn e completely descried y set of llowed 2 2 tiles over Σ {#}, with # eing non-lphet symol plced round the order of ech picture.) It is known tht the clss REC is not closed under complementtion [8], which motivtes the discussion of hierrchy within the fmily of twodimensionl lnguges. For exmple, Kri nd Moore [8] show tht lnguges recognized y 4-wy lternting finite utomt re incomprle to REC. It is demonstrted in [23] tht every recognizle picture lnguge cn e otined s the projection of n hvlocl picture lnguge. (In n hv-locl picture lnguge, the 2 2 tiles tht descrie the lnguge re replced y horizontl nd verticl dominoes - 2 tiles nd 2 tiles, respectively - so tht horizontl nd verticl reding of the pictures cn e ccomplished 2

13 seprtely.) This suggests representtion of two-dimensionl lnguges through the use of two seprte grphs or mtrices [4, 26]. However, the min drwck to hving seprte grphs for horizontl nd verticl movement is tht when lock mp is pplied to the grph representing the locl lnguge, the newly-leled grph fils to correspond to the sofic lnguge intended [5]: the inherent properties of the symols tht result from interlcing horizontl nd verticl movement cnnot e descried. Other ttempts to represent picture lnguges focus on the use of prticulr kind of cellulr utomton [, 3]. A comprehensive survey of the reserch done in two-dimensionl finite utomt during the time period eginning with Blum nd Hewitt in 967 [4] nd up to 99 cn e found in [2], while n excellent survey of more recent results cn e found in [8]. The focus of this disserttion is on two-dimensionl recognizle lnguges hving the property tht ny finite picture llowed y the structure of the lnguge my e infinitely extended to some configurtion of the plne tht lso stisfies the structure of the lnguge; tht is, where the emptiness prolem is solvle since the lnguge is prolongle. Furthermore, we shll e interested in two-dimensionl lnguges tht re fctoril; tht is, lnguges hving the property tht for ny lock found in the lnguge, ll of its sulocks re lso found in the lnguge. In one dimension, there re welldeveloped theories concerning lnguges tht re fctoril, prolongle, nd recognizle (FPR-lnguges). Here we develop theory for two-dimensionl fctoril, prolongle, nd recognizle lnguges (2DFPR-lnguges) which re the fctor lnguges of certin two-dimensionl shift spces. Of further interest in one dimension is the set of fctoril, trnsitive, nd recognizle lnguges (FTR-lnguges) tht re suset of the FPR-lnguges. For one-dimensionl lnguges, there is only one notion of trnsitivity: given ny two locks found in the lnguge, there exists third lock, lso in the lnguge, which contins the given locks s sulocks. In Section.2 it is demonstrted tht severl different notions of trnsitivity exist for two-dimensionl lnguges nd tht ech defines n invrint property for two-dimensionl shift spces. Dot systems, which re initited in [9], re descried in Section.3 nd it is then shown tht these shift spces elong to the clss of 2DFPRlnguges. Dot systems cn therefore provide rich exmples for the theory found in susequent chpters. In Section 2., we define new type of utomton tht is cple of recognizing 3

14 2DFPR-lnguges. By dispensing with the oundry symol tht forms the order of the pictures found in the two-dimensionl lnguges elonging to the clss REC, this new type of construction llows the utomton to generte two-dimensionl shift spces (nd therefore, two-dimensionl fctor lnguges) in wy quite similr to the wy in which one-dimensionl shift spces re generted. It is explined tht the crucil component of the utomton s construction is definition of cceptnce tht gives the utomton specific instructions regrding the dimensions nd structure of locks tht re deemed to e recognizle. In Section 2.2, it is verified tht this new type of utomton correctly represents the imge of shift spce (of finite type) under lock code; no such grph representtion with this cpility exists in the literture. Throughout Chpter 2, the clss of dot systems is employed to generte some mngele exmples of two-dimensionl shift spces nd their grph representtions. Section 2.3 closes the chpter with severl Propositions nd Corollries tht will e of use in the discussion of periodicity tht is found in Chpter 4. Chpter 3 revisits the notion of different types of trnsitivity existing in two-dimensionl lnguges. Most of the results on trnsitivity found within this chpter hve lredy een pulished in [6]. In Section 3., s n illustrtion of the vrious types of two-dimensionl trnsitivity, dot systems re prtilly ctegorized sed on the shpes tht define them. In Section 3.2, it is shown how the type of grph representtion defined in Chpter 2 cn revel informtion regrding trnsitivity in the relted fctor lnguges. In prticulr, min result of the chpter nd of the disserttion is tht for 2DFPR-lnguge, there is n lgorithm tht decides whether the given lnguge exhiits prticulr type of trnsitivity. In Section 4., the min result from Chpter 3 is linked to the existence of periodic points under certin conditions. Section 4.2 then offers detiled exmples of periodic points found in two-dimensionl shift spces with respect to the ppernce of such points in corresponding grph representtions. Finlly, issues of equivlence for the locks of two-dimensionl lnguge re discussed in Chpter 5. In Section 5. monoid is defined sed on equivlence clsses for the locks of picture lnguge, nd then some prtil results re chieved with respect to dot systems. A different pproch is suggested in Section 5.2 y investigting the follower sets of locks tht ct s the sttes of grph representing two-dimensionl sofic shift 4

15 spce. These finl topics serve s n introduction to the mny open questions tht exist in the field of two-dimensionl forml lnguge theory with regrds to symolic dynmics nd finite utomt..2 Nottion nd Terminology For nottion, terminology, nd sic results of one-dimensionl symolic dynmicl systems, see [25]. For nottion, terminology, nd sic results of one-dimensionl forml lnguge theory, see [0]. Some dditionl nottion nd terminology will e required for the discussion of the two-dimensionl cse. Given finite lphet Σ, define the two-dimensionl full Σ-shift to e Σ Z2. A point x Σ Z2 is function x : Z 2 Σ, tht is, configurtion of the plne where the integer lttice Z 2 hs een populted with choices from the lphet Σ. For x Σ Z2 nd w Z 2, we will sometimes denote x(w) s x w nd my refer to the coordinte point w Z 2 s cell. Similrly, for x Σ Z2 nd R Z 2, let x R denote the restriction of x to R. We cll R region, nd we cll finite region S Z 2 shpe. In prticulr, [ j, j] 2 is the squre shpe of size 2j + centered t the origin. The set Σ Z2 is compct metric spce under the metric ρ(x, y) = 2 j, where for x, y Σ Z2, j is the lrgest integer such tht x [ j,j] 2 = y [ j,j] 2. (When x = y, define ρ(x, y) = 0.) Informlly, the closer two points re to ech other, the lrger the centered squre shpe on which they gree. For v Z 2, define the two-dimensionl trnsltion in direction v s σ v where σ v is defined y (σ v (x)) w = x w+v. A suset X Σ Z2 is sid to e trnsltion invrint if for ll v Z 2, σ v (X) X. If X Σ Z2 is trnsltion invrint nd closed with respect to the metric ρ, we sy tht X is two-dimensionl shift spce (or sushift of the full shift). We define design γ on shpe S to e function γ : S Σ, where the given shpe S hs een normlized so tht min{i : (i, j) S} = 0 nd min{j : (i, j) S} = 0. In other words, the shpe hs only non-negtive integer coordintes with oundries lying on the coordinte xes. The numer of occurrences of the symol Σ in design γ 5

16 shll e denoted γ. If Γ is set of designs on fixed shpe S, then the set X := {x Σ Z2 : v Z 2, (σ v (x)) S Γ} (.2.) is two-dimensionl shift spce tht is clled two-dimensionl shift of finite type. For shifts of finite type defined through finite set of severl different shpes, there is no loss of generlity in ssuming tht X is defined through single rectngulr shpe hving size sufficient to contin ll other shpes. Given design γ on rectngulr shpe T Z 2 hving m rows nd n columns, we cll γ n m n lock nd denote such designs y B m,n. For ese of nottion, we my sometimes drop the suscripts when the numer of rows nd columns is irrelevnt, nd we my refer to locks s β i when the index of the lock does not refer to its dimension. We shll sy tht n m n lock hs height m, length n, nd thickness k = mx{m, n}. If m = 0 or n = 0, then B m,n is the empty lock nd is denoted y ε. For design B : T Σ, sulock B of B is the restriction of the design to rectngulr suset T T Z 2. In such cses, we sometimes sy tht B encloses B nd denote this y B B. For fixed r nd c, the set of ll r c sulocks of B is denoted s F r,c (B), nd the set of ll rectngulr sulocks of B is denoted with F(B). The set of ll locks of fixed size m n over Σ is denoted Σ m,n, nd the set of ll locks of ny size over Σ is denoted y Σ. A lnguge L is ny suset of free monoid. (A monoid is set with inry ssocitive opertion nd n identity.) A picture lnguge over the lphet Σ is defined to e suset of Σ. In prticulr, locl picture lnguge L is one where B L if nd only if F k,k (B) Q, where Q is finite set of llowed k k locks. Also of interest will e those lnguges tht re fctoril: lnguge L is sid to e fctoril iff L = F(L) := {F(B) B L}. All lnguges under discussion in this pper re recognizle lnguges: tht is, lnguges tht cn e represented y finite utomton. In prticulr, Gimmrresi nd Restivo [7] introduce the clss REC of recognizle picture lnguges s those tht cn e otined y the projection of locl picture lnguge, where the locl lnguge is tken over the lphet Σ nd the set Q of llowed 2 2 tiles is tken over Σ {#}, with # eing non-lphet symol plced round the order of ech rectngulr picture. 6

17 There is lnguge ssocited to ech shift spce. We sy lock B : T Σ occurs in X Σ Z2 if there exists n x X such tht x T = B. The fctor lnguge of shift spce X is F(X) := {F m,n (x) : m, n 0, x X}, (.2.2) i.e. the collection of ll sulocks tht occur in points of X. For shift spces, the fctor lnguge of the shift spce uniquely determines the shift spce; tht is, for two shift spces X nd Y, X = Y if nd only if F(X)=F(Y ) [25]. For this reson, when grph represents the fctor lnguge of shift spce, we shll more generlly refer to the grph s representtion of X. Let X e two-dimensionl shift of finite type defined y set of designs Γ on normlized shpe S. For the shpe S, define the numer of rows in S to e r = +mx{j : (i, j) S} nd define the numer of columns in S to e c = + mx{i : (i, j) S}. We shll refer to shpe S s hving dimension r c lthough S my e proper suset of the cells tht comprise the normlized r c rectngle T. Cells tht pper in T ut not in S will e of prticulr interest. Definition.2. Given n r c shpe S nd the r c rectngle T tht contins it, w T is clled free cell if w / S. For thickness k = mx{r, c}, set Q = F k,k (X) nd let ψ e normlized k k squre shpe. There is no loss of generlity in ssuming tht the shift of finite type X is defined y Q rther thn y Γ, tht is, X := {x Σ Z2 : v Z 2, σ v (x) ψ Q}. (.2.3) Note lso tht if X is two-dimensionl shift of finite type defined through set of locks F k,k (X), then for ll K k, X my lso e defined through F K,K (X). In some cses, however, it will e preferle to employ the set of locks F r,c (X) = Ψ whose dimensions minimlly contin the shpe S. In prticulr, the llowed locks of shift of finite type X is defined here s the locl picture lnguge A(X) of ll locks B Σ tht stisfy the condition F r,c (B) Ψ. (For B = B m,n with m < r or n < c, we sy B m,n is in A(X) if there exists B m,n with m r nd n c such tht B m,n is in A(X) nd B m,n B m,n.) More specificlly, when discussing the locks of lnguge defined y two-dimensionl 7

18 shift spce, we shll often use the set F r,c (X) = Ψ in order to simplify the proofs, ut when discussing the symolic dynmics of two-dimensionl shift spce, we shll find it preferle to employ the set F k,k (X) = Q. We point out tht for the set of locks F(X), F r,c (B) Ψ is necessry for B F(X) ut is not sufficient: B must lso occur in some point of X. On the other hnd, for the set of locks A(X), F r,c (B) Ψ is oth necessry nd sufficient for B A(X). For one-dimensionl shift of finite type, the fctor lnguge of the shift spce is lwys locl lnguge; tht is, A(X) = F(X) for ll one-dimensionl shifts of finite type [25]. In two-dimensionl shift of finite type, however, lock in A(X) need not pper s lock in F(X) since the emptiness prolem rises the question of whether the locl picture lnguge A(X) is prolongle. For exmple, Kri [7] provides smll periodic set of Wng tiles descriing shift of finite type X hving the property tht F(X) A(X): using the given set of Wng tiles, one cn construct locks tht conform to the structure of the lnguge yet cn not e extended ny frther in certin directions nd therefore cn not pper in the set F(X). Just s REC refers to the clss of lnguges tht cn e otined through the projection of locl picture lnguges, two-dimensionl shifts of finite type my e projected to form nother clss of shift spces. More specificlly, for given two-dimensionl shift of finite type X, we cn trnsform x X into new point y Y where Y Z2 employs some new lphet. For the k k squre shpe T, function Φ : F k,k (X) tht mps k k locks in X to symols in y Φ(x T+w ) = y w is clled k k-lock mp. The mp φ : X Z2 defined y y = φ(x) with y w induced y Φ is clled k k-lock code, nd its imge Y = φ(x) is clled two-dimensionl sofic shift. When the lock code φ is invertile, we refer to φ s conjugcy nd sy tht the spces X nd Y re conjugte. A key feture of lock codes is tht for lock code φ : X Y nd point x X, computing φ t the shifted point σ (i,j) (x) gives the sme result s shifting the imge φ(x) using σ (i,j) in the spce Y. Tht is, the digrm in (.2.4) commutes ([25].) In symolic dynmics, we re often interested in properties tht re invrint; tht is, properties tht hold true for ll shifts tht re conjugte to given shift. For twodimensionl lnguges, we will e interested in whether given pir of locks might coexist within single point of the shift spce. In one dimension, such question is one of trnsitivity. Unlike the one-dimensionl cse, however, there re severl types of 8

19 trnsitivity tht pper in two-dimensionl lnguges, ech of which defines n invrint property for conjugte shift spces (see Proposition.2.4 for proof). σ (i,j) X X φ φ Y Y σ (i,j) (.2.4) To discuss trnsitivity in the two-dimensionl cse, we first need to define distnce nd direction etween pir of locks in two-dimensionl spce. Definition.2.2 A lock B encloses the pir of locks B nd B if B, B F(B). Furthermore, lock B minimlly encloses B nd B if B encloses B nd B in such wy tht oth the ottom nd top rows of B s well s the left nd right columns of B ll intersect t lest one of the locks B, B. If lock B m,n minimlly encloses the pir of locks B p,q, B s,t we sy tht d(b, B ) = mx{0, m p s, n q t} is the distnce t which B meets B. B t (u,v ) B s d B p (u,v ) q Figure.: Enclosure of two locks Let L e two-dimensionl lnguge contining the locks B nd B. If there is lock B in L tht minimlly encloses B nd B without llowing them to overlp, where the ottom-left corners of B nd B pper t vertices (u, v ) nd (u, v ) respectively, we sy tht B meets B within L in direction (u, v) for ny (u, v) hving integer coordintes tht is non-zero multiple of (i.e., prllel to) the vector (u u, v v ). 9

20 Informlly, we sy tht B encloses B nd B if oth B nd B pper s sulocks of B. This is depicted in Figure., where the lock tht minimlly encloses B nd B is indicted with dotted lines. The direction (u, v) in which B nd B meet is determined y the ottom-left corners of B nd B, nd therefore u, v Z. Note tht it might e the cse tht the two locks touch, in which cse the distnce t which they meet would e 0. Definition.2.3 We sy tht two-dimensionl lnguge L is trnsitive in direction (u, v) if for every pir of locks B, B L the lock B meets B in direction (u, v) within L. trnsitive if for every pir of locks B, B L there is lock B L tht encloses B nd B. uniformly trnsitive if there is positive integer K such tht for every pir of locks B, B L there is lock B L tht minimlly encloses B nd B in wy tht d(b, B ) < K. mixing if for every pir of locks B, B L, there is positive integer K such tht the lock B meets B in every direction within L provided d(b, B ) > K. uniformly mixing if there is positive integer K such tht for every pir of locks B, B L, the lock B meets B in every direction within L provided d(b, B ) > K. Different types of trnsitivity re presented in Figure.2. Directionl trnsitivity is worth nming in two prticulr cses: If L is trnsitive in direction (, 0), we shll sy tht L is horizontlly trnsitive (see digrm () in Figure.2), nd similrly we sy tht L is verticlly trnsitive if L is trnsitive in direction (0, ). Trnsitivity is more generl thn uniform trnsitivity s there is no ound on how fr prt the locks B nd B might e: tht is, uniform trnsitivity ensures tht locks need not extend too fr in order to meet ech other (see digrm () in Figure.2). A notion of uniform directionl trnsitivity could e defined in the sme mnner. Mixing llows for two locks to meet everywhere outside of certin neighorhood, wheres uniform mixing gurntees tht there is ound on the rdius of this neighorhood regrdless of the size of the locks. For 0

21 exmple in digrm (c) of Figure.2, the shded region in (c) indictes the neighorhood defined y the constnt K; here, the lock B cn pper in ny direction from B provided it is outside this neighorhood. From the definitions proposed here, one cn see tht we hve the following implictions: mixing (u, v), trnsitive in direction(u, v) trnsitive (.2.5) Furthermore, the implictions in (.2.5) still hold true when we insert the word uniform in front of ech property. B B B () B B B () B < K B K (c) B Figure.2: Different types of trnsitivity In the cse of one-dimensionl recognizle lnguges, ll notions of (uniform) trnsitivity nd mixing coincide. Directionl trnsitivity most closely resemles the notion of one-dimensionl trnsitivity, s one cn exmine i-infinite sequence of locks in the specified direction. The definitions presented here for trnsitivity nd mixing in twodimensionl lnguges re similr to those defined for topologicl trnsformtion groups [27, 9] in the study of topologicl dynmics. Proposition.2.4 descries trnsitivity in the two-dimensionl cse s n invrint property for conjugte shift spces. Proofs for the invrince of mixing nd/or uniformity properties cn e ccomplished in similr fshion. Proposition.2.4 Let X nd Y e two-dimensionl shift spces, nd let φ : X Y e conjugcy from X to Y. If L = F(X) is trnsitive, then L = F(Y ) is lso trnsitive. Proof. Let B : T Σ nd B : T Σ e locks in the lnguge F(Y ) over the lphet Σ. We seek lock B F(Y ) tht encloses B nd B. Since φ is invertile, B nd B hve unique preimges, sy φ (B ) = β nd φ (B ) = β. Furthermore,

22 since F(X) is trnsitive, there exists lock β F(X) tht encloses β nd β. Tht is, there must exist w, w Z 2 such tht β(t + w ) = β nd β(t + w ) = β. Therefore, φ(β) = B F(Y ) is the desired lock since B φ(β(t +w )) = φ(β ) = φ(φ (B )) = B nd B φ(β(t + w )) = φ(β ) = φ(φ (B )) = B..3 Dot Systems When G is finite group, G Z2 is group vi coordinte-wise product. Let X e two-dimensionl shift spce which is lso sugroup of G Z2 : Such sushift is clled two-dimensionl group shift. It hs een shown [9] tht ll two-dimensionl group shifts re shifts of finite type. In puliction dted 992, Kitchens nd Schmidt define type of group shift tht they cll dot system nd then show tht every two-dimensionl group shift is finite intersection of these dot systems [20, 30]. For the work herein, we shll employ dot systems tht re defined vi some shpe S over the group G = Z/2Z. Tht is, { } X := x {0, } Z2 : v Z 2, (x w ) = 0. (.3.6) w S+v Informlly, point x elongs to this type of dot system X if nd only if ll trnsltes of the shpe S in x contin n even numer of s. Dot systems my lso e defined over finite elin groups other thn Z/2Z in the ovious wy: tht is, the product of the symols tht pper within trnsltes of S should equte to the identity element for tht group. (See Lemm.3. nd Proposition.3.4 to follow.) It is useful to think of S s shpe of dots within defining rectngle T s illustrted in Figure.3. In prticulr, the sushift defined through shpe () is known in the literture s the Three-dot System. For convenience, we shll lso occsionlly refer to the elements of S simply s dots. Unless stted otherwise, we disregrd the singleton cse when S = (0, 0), s the shift spce resulting from this shpe is the trivil one comprised of the single point of ll zeros. Dot systems shre prticulrly nice property tht llows us to employ the set of llowed locks when investigting properties of the fctor lnguge of the shift spce: tht is, ny lock llowed y the structure of the dot system lso ppers s sulock of some 2

23 () () (c) (d) (e) Figure.3: Shpes within miniml rectngles point in the shift spce. The proof of such is fcilitted y the oservtion tht in dot systems, ny lock tht is too nrrow to contin the shpe S my e extended into lrger (llowed) lock through the ssignment of dditionl rows nd/or columns. Lemm.3. (Block Extension) For dot system X defined y the finite elin group G over n r c shpe S, {G m,n : m < r or n < c} A(X). Proof. Denote the identity element of G s e, use to denote repeted use of the group opertion, nd denote the inverse of g G s g. Suppose B m,n : T G is such tht m < r nd n < c. Fix dot from S with coordintes (i, j ) such tht i n or j m. We my then define B G r,c A(X) in the following wy: B(i, j) = B m,n(i, j) for 0 i n, 0 j m e for i n or j m, (i, j) (i, j ) ( (s,t) S T B m,n (s, t)) for (i, j) = (i, j ) (.3.7) (The ssignment of the identity element in (.3.7) is chosen only to ese nottion: one should note tht with the exception of the fixed dot, ll ssignments re ritrry.) The cse when only one of m or n is smller thn r or c, respectively, is treted similrly; however, cre should e tken in the choice of which dot from S to fix initilly. Without loss of generlity, suppose tht B m,n : T G is such tht m < r nd n c: refer to Figure.4 s needed, where the shded re represents S, the shpe T is comprised of the cells hving solid order, nd the dotted lines show the cells tht we intend to define s we extend lock B m,n upwrd. First, let i = mx{i : (i, r ) S} nd fix the dot (i, r ) S (indicted y drkly-shded dot in Figure.4). Next, egin to define B G r,n A(X) s detiled in (.3.8). (Here gin, the ssignment of the identity 3

24 element is chosen merely to ese nottion.) B(i, j) = B m,n (i, j) for 0 i n, 0 j m e for 0 i n, m j < r e for 0 i < i, j = r ( (s,t) S T B m,n (s, t)) for (i, j) = (i, r ) (.3.8) Note tht if i < c, tht is, if there exist free cells in the top right corner of the r c lock T contining S, then there will e V = (c ) i cells not yet defined in B T ; in ddition, there exists n undefined cell in B for ech of U = n c trnsltes of S. Therefore, in the definition of B, one should sequentilly ssign the cells on the top row of B such tht for 0 u U, B(i + u, r ) = ( (i,j) {S+(u,0)}\(i +u,r ) B(i, j)). Following U trnsltes, ny undefined cells of B tht remin in the top row my then e populted with the identity element, tht is: for v V, let B(i + U + v, r ) = e. e e 0 e e e e e e e e e e e e e Figure.4: Extending lock upwrd When B m,n : T G is such tht m r nd n < c, one cn nlogously extend B m,n to the right y fixing the point (c, j ) S for j = mx{j : (c, j) S}. Alterntively, we could extend locks to the left (nd/or downwrd) y vrying the choice of which dot to fix initilly efore pplying symmetric rgument. We refer to the type of construction found in the proof of Lemm.3. s lock extension. Lter, when we employ dot systems for the discussion of vrious types of twodimensionl trnsitivity, we shll elorte on the importnce of fixing n initil dot efore defining dditionl cells in prticulr direction. We shll lso mke use of the following results regrding crdinlity when the group G = Z/2Z. 4

25 Corollry.3.2 For dot system X defined y the group G = Z/2Z over some r c shpe S, F m,n (X) = 2 mn whenever m < r or n < c. Corollry.3.3 If X is dot system defined through some r c shpe S, nd k = mx{r, c}, then F k,k (X) = 2 k2 r c. Proof. If r = c = k, then the shpe S is minimlly enclosed y the k k squre shpe T. Therefore, within ny lock B F k,k (X) fix one dot of the shpe S, rndomly populte ll other cells of B, nd then define the fixed dot s needed for the pproprite sum. So when r = c, F k,k (X) = F r,c (X) = 2 rc = 2 k2 r c. Without loss of generlity then, ssume tht r > c. Similr to the proof of Lemm.3., one cn egin to define lock B F k,k (X) y fixing dot in the top row of the normlized shpe S efore ritrrily populting ll other cells of the normlized r c rectngulr shpe T. Then for ech of r c horizontl trnsltes of S, the shpe S remins enclosed y the k k squre shpe T, so tht ech cell in the empty column intersecting the newly-trnslted shpe S my e ritrrily populted - with the exception of the one cell contining the trnslte of the fixed dot whose vlue is dictted y the required sum. After r c horizontl trnsltes of the shpe S, the shpe is no longer enclosed y T, so tht ny remining cells of T my e ritrrily populted. Therefore, for the k 2 cells locted in shpe T, ll ut + r c my e ritrrily populted with either 0 or, nd the result follows. When c > r, the proof is nlogous. Proposition.3.4 is true for dot systems defined over ny finite elin group G. Proposition.3.4 For the dot system X defined over the finite elin group G, A(X) = F(X). Proof. Since ll dot systems re shifts of finite type, X dot system X is shift of finite type F(X) A(X). For the reverse inclusion, suppose B m,n : T G is such tht B m,n A(X). We will demonstrte tht there exists lock B m+2,n+2 A(X) such tht B m+2,n+2 (T + (, )) = B m,n (see Figure.5). Then y König s Lemm, B m,n must occur in point of the shift spce X, since there exists n infinite process y which one cn extend the lock vi 5

26 selection from finite set of cceptle cell ssignments. Without loss of generlity, we my ssume tht B m,n is such tht m r nd n c. (Otherwise, use the pproprite lock extension from Lemm.3. to construct some lrger lock in A(X) tht encloses B m,n s sulock.) Consider the sulock β = {B m,n (i, j) : 0 i n, m (r ) j m }. Then β hs dimension (r ) n, nd y Lemm.3., there exists n upwrd extension of β. Note tht trnsltes of S tht lie entirely in B m,n re not ffected y the one-row extension of β : therefore, this extension of β lso serves s n extension of B m,n. Denote the extended lock s B m+,n A(X), nd next consider the sulock β 2 = {B m+,n(i, j) : n (c ) i n, 0 j m} hving dimension (c ) (m+). By Lemm.3., there exists n extension to the right of β 2 resulting in the construction Figure.5: Extending n llowed lock of new lock B (3) m+,n+ A(X). However, to mintin construction in clockwise direction, we use j = min{j : (c, j) S} to fix the dot (c, (m + ) r + j ) for the trnslte S + (0, (m + ) r). We then ssign cell vlues for β 2, eginning t the top of the new column nd working our wy downwrd. To continue the process, consider the sulock β 3 = {B (3) m+,n+ (i, j) : 0 i n, 0 j r 2} hving dimension (r ) (n + ). We my extend β 3 elow y first tking i = min{i : (i, 0) S} nd fixing the dot ((n + ) c + i, 0) in the trnslte S + ((n + ) c, 0). We then egin to ssign lrger lock B (4) in such wy tht B (4) m+2,n+ (i, j) = B(3) m+,n+ (i, j ) for 0 i n, j m +, nd complete the ssignment y extending β 3 (nd hence B (4) m+2,n+) y sequentilly inspecting trnsltes of S to the left of the initil fixed dot. Finlly, using the sulock β 4 = {B (4) m+2,n+ (i, j) : 0 i c 2, 0 j m + }, B(4) m+2,n+ my e extended to the left y fixing dot (0, j ) S for j = mx{j : (0, j) S}, inspecting trnsltes of S upwrd from this dot, nd so on, to form B m+2,n+2. 6

27 2 Finite Automt Recognizing Two-dimensionl Shift Spces The grph construction defined in this chpter is sed on interlcing horizontl nd verticl movement in the grphs trnsitions. In this wy, the grph representtion tkes full dvntge of the rich complexity of those lnguges tht re the fctor lnguges of two-dimensionl shifts of finite type hving the property tht A(X) = F(X). The min component of the construction is tool for recognizing rectngulr locks tht conform to the structure of the lnguge. In Section 2.2 it is verified tht for two-dimensionl shift of finite type hving property A(X) = F(X), the constructed grphs ccurtely represent the shift spces tht result from pplying lock code to the shift spce X. In the proof of Proposition 2.3.3, n explicit exmple demonstrtes how to pply higher lock code to the sttes F k,k (X) = Q, which chnges the lphet ut not the dynmics, so tht the resulting shift spce is conjugte to the originl ut hs sttes of size 2 2. This in turn llows us to mke severl oservtions out certin structures within the grph nd out the loclized recognizle pictures tht they force. When Gimmrresi nd Restivo define the clss REC of recognizle picture lnguges, they do so in the context of finite rectngulr pictures tht cn e surrounded y nonlphet order symol. By going to higher lock code s needed, one my ssume tht ny locl lnguge in REC cn e defined through finite set of llowed 2 2 locks tht hve een populted y symols from the (new) lphet Σ nd the non-lphet symol #. For two-dimensionl shift spces, there is similr notion of the lnguge of shift of finite type eing contined within locl lnguge defined y finite set of 2 2 locks. Tht is, for ny shift spce X, there is locl lnguge A(X) such tht F(X) A(X), with the fctor lnguges of the shift spce eing locl in the cse when A(X) = F(X). If lnguge L is in REC ut is not locl, then L must e the projection of some locl lnguge tht is in REC. In the sme mnner, if sofic shift spce Y is the imge under lock code of shift of finite type X hving the property tht A(X) = F(X), then F(Y ) 7

28 is in REC even if Y is not shift of finite type. (When Y is sofic shift spce tht is not shift of finite type, F(Y ) is not locl lnguge). The following illustrtes sutle distinctions tht must e mde for two-dimensionl picture lnguge where oundry symol is inherent in the definition of the lnguge. Exmple Define L {, } to e two-dimensionl picture lnguge with L REC such tht for every lock B L, ny ppernce of is completely surrounded y s. The lnguge L cn e defined y the set of llowed locks depicted in Figure 2.. For exmple, the lock # # # # # # # B = # # (2.0.) # # # # # # # would not e in the lnguge, lthough B = # # # # # # # # # # # # # # # # # # # # # # # # (2.0.2) would e in the lnguge. # # # # # # # # # # # # # # # # # # # # Figure 2.: Smple set of llowed locks for n REC lnguge Blocks B nd B in (2.0.) nd (2.0.2), respectively, indicte tht F(L) L. Therefore, the lnguge L of Exmple is not fctoril lnguge. As we consider the fctor 8

29 lnguges of shifts spces, ll lnguges in the sequel re fctoril. In Section 2.2, we shll revisit Exmple in the context of two-dimensionl shift spces. 2. Recognition of Shifts of Finite Type y M F(X) Our gol is to construct finite utomton tht will llow the input dt to consist of m n locks tht cn e scnned loclly (nd intermittently) y oth horizontl nd verticl trnsitions. To do so would require distinct sets of edges for horizontl nd verticl trnsitions, sy E h nd E v, respectively, nd would require tht we define wht we men y cceptnce. The generl ide is tht given n input lock, we will consider sequences of symols tht pper in window of fixed size s we scn the input lock from the lower-left corner to the upper-right corner y trveling in two directions - up nd/or to the right - within the constrints of the lock s dimensions. If the utomton ccepts ll such sequences of symols ppering s result of such moves nd it is determined tht these sequences overlp progressively in some sense (to e mde cler lter), then we shll sy tht the lock itself is ccepted y the utomton. To ese the forml discussion of constructing finite utomt cple of recognizing two-dimensionl sushifts, we shll refer to the extension of design (not necessrily lock) y one row (column) of length (height) k s k-conctention. Informlly, eginning with k k lock, we llow sequence of conctentions consisting of k locks conctented horizontlly to the right of the upper-most symols in the existing design nd/or k locks conctented verticlly ove the right-most symols in the existing design. More formlly, suppose we egin with k k lock nd proceed to extend this lock y the verticl k-conctention of m -mny k locks. Let m = k + m nd let B m,k e the result of these conctentions. We next llow k lock B k, to e horizontlly k-conctented to the existing lock. The inry opertion is denoted y, nd the resulting design B m,k B k, (see the first digrm in Figure 2.2) is defined y B m,k (i, j) for 0 i k, 0 j m γ(i, j) = B k, (0, j (m k)) for m k j m. (2..3) In the sme wy, k n lock B k,n with n k, my e extended y verticl k- conctention with k lock B,k. The inry opertion is denoted y, nd the 9

30 resulting design B k,n B,k is defined y B k,n (i, j) for 0 i n, 0 j k γ(i, j) = B,k (i (n k), 0) for n k i n. (2..4) Finlly, k-conctention onto shpe other thn rectngle is llowed provided tht the shpe is geometriclly congruent to shpe tht resulted from finite sequence of k- conctentions. We cll such sequence of llowed k-conctentions k-phrse, resulting in set of progressively overlpping locks of size k k. We shll denote k k lock B tht occurs in the k-phrse Py B k,k P. With ech k-phrse P, we my lso ssocite two functions s nd t: ech k-phrse strts with s(p) = β α, where β α is k k lock; nd ech k-phrse termintes in t(p) = β ω, where β ω is lso k k lock. An exmple of n underlying shpe for k-phrse is depicted in Figure 2.2 to the right. k k β ω k β α k Figure 2.2: Horizontl nd verticl k-conctentions Now suppose two-dimensionl shift of finite type X is defined y Q = F k,k (X) for some k 2. If F(X) = A(X) so tht the fctor lnguge of the shift spce is locl, then the finite utomton M F(X) = (Q, E, s, t, λ) defined through Q is finite directed grph otined s follows. (Note tht if shift of finite type is completely descried y set of locks tht define the locl lnguge A(X) = F(X), then the shift spce must e the full shift, since this would imply tht ny two lphet symols my e plced next to ech other. In such cses, the full shift X over the lphet Σ my lso e defined through the set F 2,2 (X) = Q hving Q = Σ 4.) First, define the vertex set of M F(X) to e Q. For exmple, sy q = q (0,k )... q (k,k )..... nd r = q (0,0)... q (k,0) r (0,k ).... r (k,k ).... r (0,0)... r (k,0) 20

31 re two vertices in M F(X). Next, the edge set representing the trnsitions etween the sttes of M F(X) is defined to consist of horizontl nd verticl trnsitions, E h nd E v respectively, such tht E = E h E v nd E h E v =. For horizontl trnsitions, n edge from stte q to stte r is defined s e h E h if nd only if q (,k )... q (k,k )..... = q (,0)... q (k,0) r (0,k )... r (k 2,k )..... nd r (0,0)... r (k 2,0) q (0,k )... q (k,k ) r (k,k ) = q (0,0)... q (k,0) r (k,0) q (0,k ) r (0,k )... r (k,k ) F(X). q (0,0) r (0,0)... r (k,0) In this cse, the horizontl edge is denoted e h = q r nd is given the lel of the k (k + ) lock tht is the result of the horizontl k-conctention of q with r := {r(i, j) : i = k, 0 j k }. The k (k + ) lock q r shll e denoted λ(e h ). Similrly for verticl trnsitions, n edge from stte q to stte r is defined s e v E v if nd only if q (0,k )... q (k,k )..... = q (0,)... q (k,) r (0,k 2)... r (k,k 2)..... nd r (0,0)... r (k,0) r (0,k )... r (k,k ) r (0,k )... r (k,k ) q (0,k )... q (k,k )..... =..... r (0,0)... r (k,0) F(X). q (0,0)... q (k,0) q (0,0)... q (k,0) Here the verticl edge is denoted e v = q r nd is given the lel of the (k+) k lock tht is the result of the verticl k-conctention of q with r := {r(i, j) : 0 i k, j = k }, while the (k + ) k lock q r is denoted λ(e v ). In ddition to the leling function λ lredy descried, to ech grph we my ssocite two other functions s nd t: Ech edge e E hs the source t vertex denoted y s(e) Q, nd ech edge hs trget t vertex denoted t(e) Q. (The cse where t(e) = s(e) is permissile.) To egin the discussion of the lnguge recognized y M F(X), we must mke more precise the mening of pth nd its lel. Define pth in M F(X) to e sequence 2