Effective closed subshifts in 1D can be implemented in 2D

Effective closed subshifts in 1D can be implemen ted in 2D Bruno Durand 1 , Andrei Romashchenko 1 , 2 , Alexander She n 1 , 2 1 LIF , CNRS & Uni v . de Prov ence, Marseille 2 On leav e from the Institute for Information T ransmission Problems, Mosc o w Abstract. In this paper we use fixed point tilings t o answer a question posed by Michael Hochman and show that e very on e-dimensional ef fectiv ely closed subshift can be implemented by a local rule in two dimensions. The proof uses the fixed-point construc tion of an aperiodic tile set and its extensions. 1 Introd uction Let A be a fi nite set ( alphabet ); its elements are called letters . By A - configuration we mean a mapping C : Z 2 → A . In geom etric terms: a cell with coordinates ( i , j ) contains letter C ( i , j ) . A local rule is defined by a p ositive integer M and a l ist of prohibit ed ( M × M ) - patterns ( M × M squares filled by letters). A configuration C satisfies a local rule R if none of the patterns li sted in R appears in C . Let A and B be two alphabets and let π : A → B be any mappi ng. Then ev ery A -configuration can be transformed int o a B -configuration (its homo morphic im age) by applying π to each letter . Assume th at the local rule R for A -configurations and m apping π are cho sen in such a way t hat local rule R p rohibits patterns where l etters a and a ′ with π ( a ) 6 = π ( a ′ ) are vertical neighbors. This guarantees that ev ery A -configuration that satis fies R has an image where vertically aligned B -letters are the same. Then for each B -configuration i n t he image every vertical lin e carries one single letter o f B . So we can s ay that π maps a 2-dimension al A -configuration sati sfying t he local rule R to a 1-dimensional B -configuration. Thus, for ev ery A , B , local rule R and π (with describ ed p roperties) we g et a s ubset L ( A , B , R , π ) of B Z (i.e., L ( A , B , R , π ) is the set of π -images of all A -configurations that satisfy the local rule R ). The following result (theorem 1) characterizes the sub sets that can be obtained in this w ay . Consider a product topology in B Z . Base open sets of this topology are intervals . Each interv al is obtain ed by fixing letters in finitely m any places (oth er places may contai n arbitrary letters). Each interv al is therefore a finite object and we can define e ffectively open subsets of B Z as unions of (computabl y) enumerable families of i ntervals. An effectively closed set is a com plement of an ef fecti vely open one. A subshi ft is a subset of B Z that is closed and in variant under left and right shifts. W e are mostly interested in subshifts that are not only closed b ut ef fecti vely clos ed sets. Theor em 1. The subset L ( A , B , R , π ) is an effectively clos ed s ubshift. F or every effectively closed subshift S ⊂ B Z one can find A, R, and π such that S = L ( A , B , R , π ) . The first part of the statement i s easy . The s et L ( A , B , R , π ) is evidently shift inv ariant; it remains to show that it is effecti vely closed. The set of all A -configurations that satisfy R is a closed subset 1 of a compact sp ace and therefore is a compact space i tself. The mapping of A -configurations i nto B -configurations is continuo us. Therefore the set L ( A , B , R , π ) is comp act (as a continuous image of a compact set). This argument can be eff ectivized in a st andard w ay . A B -string is declared bad if it cannot appear in the π -image of any A -configuration that satisfies R . The set of all bad strin gs is enumerable and L ( A , B , R , π ) is the set of all bi-infinite sequences that ha ve no bad f actors. The re verse im plication i s m ore diffic ult and is the main subject of this paper . It cannot be proven easily since it impli es the classical result of Berger [2]: the existence of a local rule that makes all configurations aperiod ic. Indeed, it is easy to construct an effe ctiv ely clos ed subshift S that has no periodi c points; if i t is represented as L ( A , B , R , π ) , then local rule R h as no periodic con- figurations (configuratio ns that ha ve two in dependent period ve ctors); indeed, those configuratio ns hav e a horizontal period v ector . So i t is natu ral to expect a proof of theorem 1 to b e obt ained by m odifying one of the existing constructions of an aperiodi c local rule. It is indeed the case: we use the fixed-point cons truction described in [4]. W e d o not repeat this constructio n (assuming th at the reader is familiar with t hat paper or has it at hand) and explain only the modifications t hat are needed in our case. This is done in sections 2 – 6; in the rest of this section we survey som e other steps in the same direction. M. Hochm an [7] proved a si milar result for 3 D i mplementatio ns of 1D subshifts (and, i n gen- eral, ( k + 2 ) -dimensional implement ation of k -dim ensional subshifts) and asked whether a st ronger statement is true where 3D is replaced by 2D. As we ha ve menti oned, i t is indeed true and can be achiev ed by the technique of fixed point self-similar tilings. T he detailed exposition of this techni que and its appl ications, includ ing an answer to Hochman’ s qu estion, is give n in o ur paper [5]. Since this paper contain s many other results (most boring are related to er ror -prone ti le sets), we think t hat a sel f-contained (modulo [4]) exposition could be useful for readers t hat are prim arily int erested in thi s result, and p rovide such an exposition in the current paper . In fact, the fixed point construction of algorithm s and m achines is an old and well known tool (us ed, e.g., for Kleene’ s fixed po int theorem and von N eumann’ s self-reproducing auto mata) that goes back to s elf-referential paradoxes and G ¨ odel’ s incompleteness theorem. One may only wonder why it was not used in 1960s to construct an aperiodic ti le set. In a context of hierarchical constructions in the plane this technique was used by P . G ´ acs in a much mo re com plicated si tuation (see [6]); howe ver , G ´ acs did not bother to mention explicit ly that t his techni que can be applied t o construct aperiodic tile sets. Fixed point til ings are not t he only to ol that can be used to implement su bshifts. In [3] a m ore classical (Berger –Robins on style) cons truction o f an aperiodi c ti le set is m odified in several ways to implement o ne specific s hift: th e family of bi-infinite bit sequences ω such that all sufficiently long substring s x of ω hav e com plexity greater than α | x | or at l east ω can be cut into two pieces (left- a nd right-infinite) that ha ve this property . (Here α is som e c onstant less t han 1, and | x | stands for t he l ength of x .) In fact, the construction used there i s fairly general and can be applied t o any enumerable s et F of forbid den substring s: one m ay im plement a shift th at consists of bi-in finite sequences that hav e no subs trings in F or at l east can be cut into two p arts wit h t his property . Recently N. Aubrun and M . Sablik found a more ingeni ous const ruction t hat is f ree of this problem (splitting into two parts) and therefore provides a nother proof of theorem 1 (see [1]). 2 The authors thank their LIF col leagues, especially E. J eandel wh o point ed out th at their result answers a question posed in [7]. 2 The idea of the construction W e do not refer explicit ly to our paper [4] but use t he not ions and constructions from t hat paper freely . In that paper we used local rules of sp ecial type (each lett er was called a til e and has four colors at i ts sides; the l ocal rule says that colors of the neighbor t ile should match). In fact, any local rule can be reduced to thi s type by extending the alphabet; howe ver , we do not need to worry about this since we construct a local rule and may restrict ourselves to tilings. W e superim pose t wo layers in ou r tiling. One of the layers contain s B -letters; the local rule guarantees that each vertical line carries one B -lett er . (V ertical neighbors should be i dentical.) For simplicit y we ass ume that B = { 0 , 1 } , so B -letters are just bits , but t his is not really imp ortant for the argument. The second layer con tains an aperiodic ti le set constructed i n a way si milar to [4]. T hen s ome rules are used to organize the interaction between the layers; com putations in the second layer are fed wit h the data from the first layer and check th at t he first layer does not contain any forbidden string. Indeed, the m acro-tiles (at every level) in ou r construction contai n s ome comput ation used to guarantee their beha vior as b uilding blocks for the next l e vel. Could we run this computat ion in parallel with som e ot her one that enumerates b ad p atterns and termi nates the computati on (creating a violation of the rules) if a bad pattern appears? This idea immediately faces e vident problems: – Th e computation performed in macro-tiles (in [4]) was limited in time and s pace (and we need unlimited computations since we ha ve infini tely man y forbidden substrings and no li mit on the computational resources used to enumerate them). – Com putations o n hi gh levels do not ha ve access to bi t sequ ence they n eed to check: the b its that go through t hese m acro-tiles are “deep in th e su bconscious”, since macro-tiles operate on the lev el of their sons (cells of the computati on that are macro-tiles of t he previous l e vel), not individual bits. – Even if ev ery m acro-tile checks all the bi ts that go through i t (in so me mysterious way), a “degenerate case” could happen where an infinite v ertical lin e is not crossed by any macro-tile. Imagine a tile that is a left-most son of a father macro-tile who in its turn is the left-mos t son of its father and so on (see Fig. 2). They fill the right half-plane; the left half-plane is filled in a sym metric way , and the vertical dividing line between then is not crossed by any til e. Then, if each macro-tile takes care o f forbidden su bstrings inside its zone (bi ts that cross this macro-tile), some substrings (that cross the dividing lin e) remain unchecke d. These problems are discussed in the following section s o ne after anot her; w e apolo gize if th e description of them seemed to be quite informal and v ague and hope that the y would become more clear when their solution is discussed. 3 3 V ariable zoom factor In our previous cons truction the macro-tiles of all levels were of the same size: each of them contained N × N m acro-tiles of the previous lev el for so me constant zoo m factor N . Now it is not enough any more, since we need to hos t arbit rarily long computations i n hi gh-lev el macro-tiles. So we need an increasing sequence of zoom f acto rs N 0 , N 1 , N 2 , . . . ; macr o-tiles of the first lev el a re blocks of N 0 × N 0 tiles; macro-tiles of the second le vel are blocks of N 1 × N 1 macro-tiles of le vel 1 (and have size of N 0 N 1 × N 0 N 1 if measured in i ndividual tiles). In general, macro-tiles of le vel k are made of N k − 1 × N k − 1 macro-tiles of leve l k − 1 and have s ide N 0 N 1 . . . N k − 1 measured in i ndividual tiles. Howe ver , al l the macro-tiles (of different levels) carry t he s ame program in their comp utation zone. The differe nce between their behavior is caused by t he data: each m acro-tiles “knows” i ts lev el (consciously , as a sequence of bi ts on its t ape). Th en this level k may be used to comput e N k which is then used as a modulus for coordin ates in th e father macro-tile. (Such a coordi nate is a number between 0 and N k − 1, and the addition is performed modul o N k .) Of course, we need to ensure that thi s informat ion is correct. T wo properties are required: (1) all macro-tiles of the same level have the same idea about their leve l, and (2) these ideas are consistent between le vels (each f ather is one lev el hi gher t han its sons). T he fi rst is easy t o achieve : the l e vel s hould be a part of the side m acro-color and shou ld match in neighbo r tiles. (In fact an explicit check that brot hers have the same idea a bout th eir leve ls is not really needed, since the first property follows from the second one: sin ce all tiles on the le vel zero “kno w” their le vel correctly , by indu ction we conclude that macro-tiles of all levels have correct information about their lev els.) T o achieve t he s econd prop erty (consi stency between level informatio n conscious ly known to a father and it s sons) is also easy , thou gh we n eed some construction. It goes as follows: each macro-tile knows it s place in the father , so it kno ws whether the f ather should keep so me bits of his lev el information in that macro-tile. If yes, the m acro-tile checks that this information is correct. Each macr o-tile checks only one bit of the le vel information, but wi th brothers’ h elp the y check all the bits. 3 There is on e mo re thi ng we need to take care of: the lev el i nformation shoul d fit into the tiles (and the computation needed to comp ute N k knowing k shou ld also fit into le vel k tile). This means that lo g k , log N k and the tim e needed t o compu te N k from k s hould be m uch less t han N k − 1 (since the computation zone is some fraction of N k − 1 ). So N k should not grow too slow (s ay , N k = log k is too slow), should not gro w to o f ast (say , N k = 2 N k − 1 is too f ast) and should not be too difficult to compute. Howe ver , these restriction sti ll leav e a lot of room for us : e.g., N k can be prop ortional to √ k , to k , to 2 k , or 2 2 k , or 2 2 2 k (any fixed height is OK). Recall that com putation deals with binary encodings of k and N k and normally is polynomi al in thei r lengths. In th is way we are no w able to embed computations of i ncreasing sizes in to the m acro-tiles. Now w e hav e to explain which data these compu tations would get and how the communicatio n between lev els i s organized. 3 People sitting on the stadium during the f ootball match and holding color sheets to create a slogan for their team can check t he correctness of the slogan by looking at the scheme and kno wing their row and seat coordinates; each person checks one pixel, but in cooperation they check the entire slogan. 4 4 Conscious and subconscious bits The problem of communi cation between levels can be explained using t he following m etaphor . Imagine you are a m acro-tile; then you have some program, and p rocess the data according to the program; you “blow up” (i.e., your in terior cannot be correctly t iled) if some i nconsistency in th e data is found. Thi s program makes y ou perform as on e cell in t he next-level brain (in the compu- tation zone of t he father macro-tile), but you do not worry abo ut it: you just perform the program. At the same time e ach cell of yourself in fact is a son macro-tile, and elementary opera tions of thi s cell (the relation b etween signals on it s sides) are in fact performed by a lower -le vel c omputation. But this computation is your “sub-conscious”, you do not ha ve direct access to its data, though the correct functioning of the cells of your brain is guaranteed by the programs running in your sons. Please do not t ook this metaphor to o seriously and keep in m ind that the t ime axis of the computations i s just a vertical axis on the pl ane; con figurations are static and do not change with time. Howe ver , it could be useful while thinking about problems of inter-le vel communication. Let us decide th at for e ach macro-tile all th e bits (of the bit sequence that needs to be checke d) that cross this macro-tile form its r esponsibili ty z one . Moreover , one of th e bits o f this zone m ay be dele gated to the macro-tile, and in this case t he macro-tile cons ciously knows this bit (is r e- sponsible for t his bit). The choice of t his bit depends o n t he vertical positi on of t he macro-tile in its father . More technically , recall t hat a m acro-tile of level k is a square whose si de i s L k = N 0 · N 1 · . . . · N k − 1 , so there are L k bits of t he s equence that intersect this macro-tile. W e delegate each o f t hese bits to o ne of the macro-tiles it i ntersects. Note that ev ery macro-tile of the next level is made of N k × N k macro-tiles of level k . W e ass ume t hat N k is much big ger than L k (more abou t choice of N k later); this guarantees t hat there are enough m acro-tiles of level k (in th e next level macro-tile) to serv e all bits that int ersect th em. Let us decide that i th macro-tile of lev el k (from bottom to top) in a ( k + 1 ) -le vel macro-tile knows i th bit (from the left) in its zone. Since N k is greater than L k , we l ea ve some unused s pace in each macro-tile of level k + 1: many macro-tiles of l e vel k are not responsible for any bit, b ut thi s does not create any prob lems. · · · · · · · · · · · · N k tiles of size L k × L k Fig. 1. Bit delegation: bits assigned to vertical lines are distributed between k -leve l macro-tile (according to their positions in the father macro-tile of le vel k + 1). 5 This is our pl an; ho we ver , we need a mechanism that ensures th at the delegated bits are indeed represented correctly (are equal t o the corresponding bits “on the ground”, in the sequence th at forms the first level of our construct ion). This is done in the hi erarchical way: si nce ev ery bi t i s delegated t o macro-tiles of all levels, i t is enough to ensure that th e ideas about bit values are consistent between father and son. For this hierarchical check let us agree that ev ery macro-tile not only knows its own dele gated bit (or the fact that there is no delegated bit), b ut als o kn ows the bit delegated to its fa ther (if it exists) as well as father’ s coordinates (in th e grandfather macro-tile). This is sti ll an acceptable amount o f information (for keeping father’ s coordinates we need to ensure that log N k + 1 , t he s ize of father’ s coordin ate, is much smaller that N k − 1 ). T o make this information consis tent, we ensure that – t he data about the father’ s coordinates and bits are the same among brothers; – i f a macro-til e has t he same delegated bit as i ts father (this fact can be checked since a m acro- tile knows its coordinates in t he father and father’ s coordinates in th e grandfather), t hese two bits coincide; – i f a macro-tile is in the place w here its father keeps i ts delegated bit, the actual father’ s infor- mation is consistent with the information about what the father should ha ve. So the in formation t ransfer between levels is o r ganized as foll ows: the macro-tile that has the same dele gated bit as it s f ather , non-deterministically guesses this f act and distributes the informa- tion about father’ s coo rdinates and bi t among the brothers. Thos e of t he brot hers who are in the correct place, check that father indeed has correct information. On the l owest level w e hav e direct access to the bits o f the sequence, so the t ile that is above the correct b it can keep its value and transmit it together with (guessed) coordinates of its father macro-tile (in the grandfather’ s macro-tile) to all brothers, and some brothers a re in the right place and may check these va lues against the bits in the computation zone of the fa ther macro-tile. This const ruction makes all bits present at all le vels, but this is not enough for checking: we need to check not individual bit s, but bit g roups (against the lis t of forbidden subst rings). T o this end we need a special arrangement described in the next section. 5 Checkin g bit g r oups Here the main idea is: each macro-tile checks some s ubstring (bit group) that is very small com- pared to the s ize of this macro-tile. Howe ver , since the size of the computati on zone grows infinitely as the lev el increases, this does not pre vent com plicated checks (that may in v olve a long su bstring that appears very late in the enumeration of the forbidden patterns) from happening. The check is performed as fol lows: we do som e number of steps in the enumeration of for- bidden p atterns, and then check whether one of these patterns appears in th e bit group un der con- sideration (assigned t o this macro-til e). The number of enumeratio n steps can be also rather s mall compared to the macro-tile size. W e reserve also some ti me and space to check that all the patterns appeared d uring the enu- meration are not substri ngs of the bit group under consi deration. This i s n ot a serious time/s pace 6 overhea d since subs tring search in the gi ven bit group can be performed rather fa st, and th e size of the bit group and the number of enumeration steps are chosen small enough (having in m ind th is overhea d). Then in the limit any viol ation inside some macro-tile will be discovered (and onl y degenera te case problem remains: substrings that are not covered ent irely by any tile). T he degenerate case problem is considered in t he next section; it thi s s ection it remains to explain how the groups of (neighbor) bits are made av ailabl e to the computation and how they are assigned to macro-tiles. Let u s cons ider an infinite vertical stri pe of macro-tiles of level k that share the s ame L k = N 0 · . . . · N k − 1 columns. T ogether , these macro-tiles keep in t heir memory all L k bits of their common zone of responsi bility . Each o f them perform a check for a small bit group (of length l k , which increases extremely slowly with k and i n particular is much less than N k − 1 ). W e need to di stribute somehow these groups among macro-tiles of this infinit e stripe. It can be d one in many differe nt ways. For example, l et u s agree that the starting point of the bit group checked by a macro-tile is the vertical coo rdinate of th is macro-tile in it s father (if i t is not too big ; recall that N k ≫ N 0 N 1 . . . N k − 1 ). It remains to explain how groups of (neighbor) bits are made a v ailable to the computational zones of the corresponding macro-tiles. W e do it in the same way as fo r delegated bits ; the di f ference (and simplification) i s that n ow we may use only two lev els of hierarchy since all the bits are a vailable in t he p re vious level (and not only in the “deep unconscious”, at the ground le vel). W e require that this group and the coordinate that determines its posi tion are again known t o all the sons of t he macro-til e where th e group is checked. Then the sons shoul d ensure that (1) this informati on is consistent between brothers; (2) it is consistent wi th dele gated bi ts where dele gated bit s are in the group, and (3) it is c onsistent with the information in th e macro-tile (father of these brothers) itself. Since l k is small , th is is a small am ount of information so there is no problem of its dis tribution between macro-tiles of the preceding lev el. If a forbidden pattern belongs t o a zone of responsibility of macro-til es o f arbitrarily high le vel, then this vi olation is be discovered in side a macro-tile of some level, so the til ing of the plain cannot not exist. Onl y the d egenerate case prob lem remains : so far we cannot catch forbidden sub strings that are not covered entirely by any macro-tile. W e deal with th e degenerate case probl em in the next section. 6 Dealing with the degenerate case The problem we need to deal wit h: it c an happen that o ne vertical line is not cross ed by an y macro- tile of a ny lev el (see Fig. 2). In this case some substrings are not covere d entirely by any macro-tile, and we do not check t hem. After the probl em is realized, th e solution is not difficult. W e let every macro-tile check bit groups in it s e xtended r esponsib ility zone that is three times wider and covers not only the macro-tile itself b ut also its left and right neighbors. Now a macro-tile of l e vel k is g iv en a small bit group which is a substri ng o f its extended responsibilit y zone (the width of th e extended respons ibility zone i s 3 L k ; it is com posed of the zones of responsibil ity of the macro-tile itself and two its neighbors). Respectiv ely , a macro-tile of l e vel ( k − 1 ) keeps the inform ation about three groups of bits instead of one: for its father , left 7 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Fig. 2. Degenerate case uncle, and right uncle. Thi s inform ation sh ould be consi stent b etween brothers (since they have the same f ather and uncles). M oreove r , it should be checked across the bo undary between macro-tiles: if tw o macro-tiles A and B are neighbors but hav e different fathers ( B ’ s f ather is A ’ s right uncle a nd A ’ s father is B ’ s left uncle), then t hey should compare the i nformation they ha ve (about bit groups checked by f athers of A and B ) and ensure it is consistent. F or t his we need to increase the amount of information kept in a macro-tile by a const ant factor (a macro-tile k eeps three bit groups instead of one, etc.), but this is still acc eptable. It is easy to see that now e ven in the degenerate case ever y substring is entirely in t he e xtended responsibilit y zone of arbitrary large tiles, so all the forbidden patterns are checked e verywhere. 7 Final adjustments W e finished our argument, but we was qui te vague about th e exact values of parameters saying only that som e quantities should be much less than ot hers. Now we need to check again the entire construction and see that t he relations b etween p arameters th at were needed at differe nt steps could be fulfilled together . Let us remi nd the parameters u sed at several steps o f the constructi on: macro-tiles of level k + 1 con sist of N k × N k macro-tile of le vel k ; thus, a k -level macro-tile consists of L k × L k tiles (of lev el 0), wh ere L k = N 0 · . . . · N k − 1 . M acro-tiles o f level k are responsible for checking bit blocks of length l k from their e xtended responsibility zone ( of width 3 L k ). W e ha ve sev eral const raints on the values of t hese parameters: 8 – l og N k + 1 ≪ N k and ev en log N k + 2 ≪ N k since ev ery macro-tile m ust be able to do simple arith- metic manipulations with its own coordinates in th e father and wit h coordinates of th e father in the grandfather; – N k ≫ L k since we need enough sons of a macro-tile of le vel k + 1 to keep all bits from its zone of responsibili ty (we u se one macro-tile of le vel k for each bit); – l k and e ven l k + 1 should be much less than N k − 1 since a macro-tile of le vel k must contain in its computational zone the b it block of length l k assigned to itself and th ree b it blocks of length l k + 1 assigned to its father and tw o uncles (the left and right neighbors of the father); – a k -level macro-til e should enumerate in its computati onal zone se veral forbidden patterns and check whether any of them is a subs tring o f the given (assigned to this macro-tile) l k -bits block; the numb er of steps i n this enu meration mu st be small compared to the size of the macro-tile; for example, let us agree that a macro-tile of le vel k runs this enumeration for exactly l k steps; – t he v alues N k and l k should be simple functions o f k : we want to c ompute l k in time polynom ial in k , and compute N k in ti me poly nomial in log N k (note that ty pically N k is m uch greater than k , so we cannot compute or e ven write do wn its binary representation in time pol ynomial in k ). W ith all these const raints we are stil l quite free in the choice of parameters. For example, we may let N k = 2 C 2 k (for some large enough constant C ) and l k = k . 8 Final remarks One may also use essentially t he same construction to implement k -dimensional ef fectively clo sed subshifts using ( k + 1 ) -dim ensional subshifts of finit e type. How far c an we go further? Can we implement e vert k -dim ensional effecti vely closed subshifts by a tili ng of the sam e d imension k ? Another question (posed in [7]): let us replace a finite alphabet by a Cantor space (with the stand ard to pology); can we represent eve ry k -dim ensional effecti vely closed subshifts over a Cantor s pace as a continuous image of the set of tili ngs of dimension k + 1 (for some finite til e s et)? E. Jeandel noticed that the answers to the both questio ns are negative (this fact is also a corollary of results from [3] and [8]). Refer ences 1. N. Aubrun, M. Sablik, personal communication (submitted for publication as of February 2010). 2. R. Berger , The Undecidability of the Domino Problem. Mem. Amer . Math. Soc. , 66 , 1966. 3. B. Durand, L. Le vin, A. Shen, Comple x Tilings. J. Symbo lic Logic , 73 (2), 593–61 3, 2008. 4. B. D urand, A. Romashchenko, A. Shen, F ixed point theorem and aperiodic tilings, Bulleti n of the EA TCS , no. 97 (2009), pp. 126–136 (T he Logic in Computer S cience Column by Y uri Gure vich). Electronic version: arXiv:1003.28 01 [cs.LO] , http://arx iv.org/ab s/1003.28 01 5. B. Durand, A. Romashchen ko, A. Shen, F ixed-po int tile sets and their applications. arXiv:0910.2415 [cs.CC] , 2009. http://arx iv.org/ab s/0910.24 15 6. P . G ´ acs, Reliable Computation with Cellular Automata. J. Comput. Syst. Sci. 32 (1), 15–78, 1986. 7. M. Hochm an, On the dynamic and recursiv e properties of multidimensional symbolic systems. In ventiones mathematicae , 176 , 131–16 7 (2009). 8. A. R umyantse v , M. Ushakov , Forbidde n Substrings, K olmogoro v Complexity and Almost Periodic Sequences. ST ACS 2006: 396-407. 9