Fixed point theorem and aperiodic tilings

Fixed point theorem and aperiodic tilings ⋆ Bruno Durand 1 , Andrei Romashchenko 1 , 2 , Alexander She n 1 , 2 1 LIF , CNRS & Uni v . de Prov ence, Marseille 2 Institute for Information T ransmission Problems, Mosco w Abstract. W e propose a ne w simple construction of an aperiodic ti le set based on self-referential (fixed point) argume nt. People often say about some discov ery that it appeared “ahead of time”, meaning that it could be fully understood only in the context of i deas deve loped later . For the topic of this note, the construction of an aperiodic ti le set based on the fi xed-p oint (self-referential) approach, the situation is exactly the opposite. It should have b een found in 19 60s w hen t he question about aperiodic tile sets was first asked: all the tools were quite standard and widely used at that time. Howe ver , the history had chosen a different path and many nice geometric ad hoc constructions were de velop ed instead (by B erger , Robinson, Penrose, Ammann and many others, see [6]; a popular exp osition of Robinson-style construction is given in [3]). I n this note we try to correct this error and present a construction that should hav e been disco vered first bu t seemed to be unnoticed for more that forty years. 1 The statem ent: aperiodic tile sets A tile is a s quare with colo red sides. Giv en a set of tiles, we want to find a tiling, i.e., t o cover the plane by (translated copies of) th ese tiles in such a way that colors match (a common side of two neighbor tiles has the same color in both). 3 For example, if tile set consists of two tiles (one has black lower and left side and white right 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 Fig. 1. T ile set that has only periodic tilings and top sides, the other has the oppos ite colors), it is easy to see that only periodic (checkerboard) ⋆ Partially su pported by ANR (Sycomore and Nafit grants) and RFBR (05-01-0280 3, 06-01-00 122a). 3 T iles appe ared first in the context of domino pr oblem posed by Hao W ang. Here is the origin al formulation from [10]: “ Assume we are gi ven a finite set of square plates of the same si ze with edges colored, each in a different manner . Suppose further there are infinitely many copies of each plate (plate type). W e are not permitted to rotate or reflect a plate. T he question is to fi nd an effe ctiv e procedure by which we can decide, for each giv en fi nite set of plates, whether we can cover up the whole plane (or , equi v alently , an infinite quadrant t hereof) wit h copies of t he plates subject to the restricti on that adjoining edges must have the same color . ” This question (domino problem) i s closely related to the existence of aperiodic tile sets: (1) if they did not exist, domino problem would be decidable for some simple reasons (one may look in parallel for a periodic ti ling or a fi nite region that cannot be tiled) and (2) the aperiodic t ile sets are used in the proof of the undecidability of domino problem. Howe ve r , in this note we concen trate on aperiodic tile sets only . tiling is possible. Howe ver , if we add some other tiles the resul ting ti le set m ay admit also non- periodic tilings (e.g., if we add all 16 p ossible tiles, any combination of edge colors becomes possible). It turns out that there are other tile set that ha ve only a periodic tiling s. Formally: l et C be a finite set of colors and let τ ⊂ C 4 be a set of t iles ; the com ponents of t he quadruple are int erpreted as upper/right/lower/left col ors of a tile. Our e xample tile set wi th two tiles is represented then as {h white , white , black , black i , h black , black , white , white i} . A τ -tiling is a m apping Z 2 → τ that sati sfies matching condi tions. T iling U is called periodic if it has a period , i.e., if there exists a non-zero ve ctor T ∈ Z 2 such that U ( x + T ) = U ( x ) for all x . Now we can form ulate the result (first prov en by Be r ger [1]): Pr oposition . Ther e e xists a finite til e set τ such that τ -tilings exist but all of them ar e aperio dic . There is a useful reformulation of this result. Instead of til ings we can consider two-dimensional infinite words in s ome finite alphabet A (i.e., mapp ings of type Z 2 → A ) and put some local con- straints on them. This means that we choose some positive in teger N and lo ok at the w ord through a wi ndow of si ze N × N . L ocal constraint then says which patterns of size N × N are allowed to appear in a windo w . Now we can reformul ate our Proposition as foll ows: ther e exists a local constraint that is consistent ( some infinite wor ds sati sfy i t ) but i mplies aperiodicity ( all s atisfying wor ds are ap eriodic ). It is easy to see that these two formulations are equiv alent. Indeed, the color matching condition is 2 × 2 checkable. On the other hand, any local const raint can be expressed in terms of tiles and colors if we use N × N -patterns as tiles and ( N − 1 ) × N -patt erns as colo rs; e.g., the righ t color of ( N × N ) -tile is t he tile except for its left column; i f i t m atches th e left color of the right neighbor , these two tiles ov erlap corre ctly . 2 Why theory of computation? At first glance this proposition has nothing to d o with theory of computation. Howe ver , the ques- tion appeared in the context of the und ecidability of some log ical d ecision prob lems, and, as we shall see, can be so lved using theory of computation s. (A ra re chance to con vince “normal” math- ematicians that theory of computation s is us eful!) The reason why theory of compu tation comes into play is that rules that determ ine the behavior of a computation device — say , a T uring machin e with one-dim ensional tape — can be transformed into local constraints for the space-time diagram that represents com putation process. So we can try to prove the proposit ion as follows: cons ider a T urin g machine with a very complicated (and therefore aperiodic) behavior and translate its rul es into local constraints; then an y tiling represents a time-space diagram of a computation and therefore is aperiodic. Howe ver , this na ¨ ıve approach does not work si nce l ocal constraints are s atisfied also at th e places where no com putation happens (in the regions that do no t contain the head o f a T uring machine) and therefore allow periodic confi gurations. So a more so phisticated approach is needed. 2 3 Self-similarity The main idea of this more s ophisticated approach is to construct a “self-sim ilar” set o f tiles. Informally speaking, this means that any tiling can be uniquely split by vertical and horizontal lines i nto M × M blocks that beha ve exactly like the individual tiles. Then, if we see a til ing and zoom out with scale 1 : M , we get a t iling with the same tile set. Let us give a form al definitio n. Assume th at a non-empt y set o f tiles τ and positive in teger M > 1 are fix ed. A macr o-t ile is a square of size M × M filled with matching tiles from τ . Let ρ be a non-empty set of macro-tiles. Definition . W e say th at τ implements ρ if any τ -tiling can be u niquely s plit by horizontal and vertical lines into macro-tiles fr om ρ . Now we give two examples that illustrate this definition: one nega tiv e and one posit iv e. Negative example : Consider a set τ that consists of one til e wi th all whit e si des. Then there is onl y one macro-tile (of give n size M × M ). Let ρ be a one-element set that con sists of this macro-tile. Any τ -tiling (i .e., t he o nly possible τ -tiling) can be sp lit i nto ρ -macro-tiles. Howe ver , the splitti ng lin es are not unique, so τ does not i mplements ρ . Positiv e example : Let τ is a set of M 2 tiles th at are i ndexed by pairs of integers modulo M : The colo rs are pairs of i ntegers modulo M arranged as shown (Fig. 2). Th en there exists only one ( i + 1 , j ) ( i , j ) ( i , j ) ( i , j + 1 ) Fig. 2. Elements of τ (here i , j are integers modulo M ) τ -tiling (up to translations), and this tiling can be uniq uely split into M × M squares whose borders hav e colors ( 0 , j ) and ( i , 0 ) . Therefore, τ implements a set ρ that consist s of one macro-tile (Fig. 3). Definition . A s et of til es τ is self-simi lar i f it implem ents some set of macro-tiles ρ that i s isomorphic to τ . This means that th ere exist a 1-1-correspondence between τ and ρ such that matchin g pairs of τ -tiles correspond exactly to matching pairs of ρ -macro-tiles. The following st atement follows directly from the definition : Pr oposition . A self-simi lar tile set τ has only aperiodic tilings . Pr oof . Let T be a period of s ome τ -tiling U . By definition U can be uniquely spl it into ρ - macro-tiles. Shift by T shou ld respect this splitting (ot herwise we get a dif ferent splitting), so T is a multiple of M . Zooming the til ing and replacing each ρ -macro-tile by a corresponding τ -tile, we 3 0 0 0 0 M Fig. 3. The only element of ρ : border colors are pairs that contain 0 get a T / M -shift of a τ -tiling. For the same reason T / M shoul d be a multi ple of M , then we zoom out again etc. W e conclude therefore that T i s a multiple of M k for any k , i.e., T is a zero vector .  Note also t hat any self-simil ar set τ has at least one t iling. Indeed, by definition we can til e a M × M square (since m acro-tiles exist). Replacing each τ -tile by a corresponding macro-tile, we get a τ -tiling of M 2 × M 2 square, etc. In thi s way we can t ile an arbi trarily large finite region, and then standard compactness ar gument (K ¨ onig’ s lemma) shows th at we can tile the entire plane. So i t remains t o construct a self-simil ar set of tiles (a set of tiles th at implements itself, up to an isomorphis m). 4 Fixed points and self-refer ential constructions The constructi on of a s elf-similar tile set is do ne in t wo steps. First (in Section 5) we explain how to cons truct (for a giv en tile set σ ) another tile set τ that impl ements σ (i.e., im plements a set of macro-tiles isomorphi c to σ ). In this const ruction the tile set σ is give n as a program p σ that checks whether four bit strings (representing four side colors) appear in one σ -tile. The tile set τ then guarantees that each macro-tile encodes a com putation where p σ is applied to these four strings (“macro-colors”) and accepts them. This gives us a m apping: for ev ery σ we ha ve τ = τ ( σ ) that im plements σ and depends on σ . Now we need a fixed point of t his m apping wh ere τ ( σ ) is isomorph ic to σ . It is done (Section 6) by a classical self-referential trick that app eared as liar’ s paradox, Cantor’ s diagonal argument, Russell’ s paradox, G ¨ odel’ s (first) incompleteness theorem, T arsky’ s theorem, undecidabil ity of the Halting problem, K leene’ s fixed point (recursion) theorem and von Neumann’ s const ruction of self-reproducing automata — in all these cases the core ar gument is essentially the same. The same trick is used also in a classical programming challenge: to write a program that prints i ts own text. Of course, for e very stri ng s i t is trivial to write a program t ( s ) that prints s , but how do we get t ( s ) = s ? It seems at first that t ( s ) should incorporate the s tring s itself plus some overhead, so how t ( s ) can be equal to s ? Howe ver , this first imp ression is false. Imagi ne that our com putational device is a univ ersal Turing m achine U where t he program is written in a special read-only layer of t he tape. (Th is means that th e tape alphabet is a Cartesian product of two components , and on e of the compon ents is used for the p rogram and is n e ver changed by U .) Then the program can get access to i ts own text at any moment, and, in particular , can copy it to 4 the output tape. 4 Now we explain in more details how to get a self-similar tile set according to this scheme. 5 Implementing a given tile set In this section we show how one can implement a given tile s et σ , or , better to say , how to construct a tile set τ that implements some set of macro-tiles that is isomorphi c to σ . There are easy ways to do this. Though we canno t let τ = σ (recall that zoom factor M should be g reater than 1), we can do ess entially the same for ev ery M > 1. Let us extend our “positive” example (with one macro-tile and M 2 tiles) by superimpos ing addit ional colors. Superimposing two sets of colors means the we consider the Car tesian product of color sets (so each edge carries a pair of colors). One set of colors remains the same ( M 2 colors for M 2 pairs of int egers modulo M ). Let us describe addi tional (superimposed) colors. Internal edges of each macro-tile should have the same colo r and t his color shou ld be different for all m acro-tiles, so we allocate # σ colors for that. This giv es # σ macro-tiles that can be put into 1-1-correspondence wi th σ -tiles. It remains to provide correct bo rder colors , and this is easy to d o since each tile “knows” which σ -tile it simulates (due to t he internal color). In th is way we get M 2 # σ tiles t hat im plement the til e set σ with zoom factor M . Howe ver , this (trivial) simulation is not really us eful. Recall that our goal is to get isomorphic σ and τ , and in this implementati on τ -tiles have m ore colors that σ -tiles (and we ha ve more tiles, too). So we need a more creative encodi ng of σ -colors that makes use of the space a vailable: a si de of a macro-tile h as a “macro-color” that is a sequence of M ti le colors, and we can hav e a l ot of macro-colors in this way . So let us assume that colors in σ are k -bit strin gs for some k . Then the ti le set is a subs et S ⊂ B k × B k × B k × B k , i .e., a 4-ary predi cate on the s et B k of k -bit stri ngs. Ass ume that S i s presented by a program that computes Boolean value S ( x , y , z , w ) given four k -bit strings x , y , z , w . Then we can construct a tile set τ as follows. W e st art again with a set of M 2 tiles from our example and superimpose addi tional colors but use t hem in a more econom ical way . Assum ing that k ≪ M , we allocate k places in t he m iddle of each side of a m acro-tile and allow each of them to carry an additional color bit ; then a macro- color represents a k -bit string. Then we need to arrange the internal colo rs in such a way that macro-colors ( k -bi t strings) x , y , z and w can appear on the four sides of a m acro-tile if and only if S ( x , y , z , w ) is true. T o achieve this go al, let us agree that the middle p art (of s ize, say , M / 2 × M / 2) i n every M × M - macro-tile is a “computation zone”. T iling rules (for superim posed colors) in this zone gu arantee that it represents a time-space d iagram of a comput ation o f so me (fixed) univer sal T uring machine. (W e assume that time goes up in a vertical direction and t he t ape is horizontal.) It is con venient 4 Of course, this looks like cheating: we use some very sp ecial univ ersal machine as an interpreter of our programs, and this makes our task easy . T eachers of programming that are seasoned enough may recall the B ASI C program 10 LIST that indeed prints its o wn te xt. Howe v er , this trick can be generalized enough to sho w that a self-printing program ex ists in ev ery language. 5 to assum e that prog ram of this machine is writt en on a special read-only layer o f the tape (see the discussion in Section 4). Outside the computation zone the tiling rules guar antee that bits are transmitted from the sides to the initial configuration of a computation . Uni ve rsal T uring machine program Fig. 4. k -macro-colors are transmitted to the computation zone where they are c hecked W e als o requi re that this machine shoul d accept its inpu t before running out of time (i.e., less than in M / 2 steps), o therwise the tiling is impossible. Note that in this descrip tion different parts of a macro-tile behave dif ferently; this is OK since we st art from our e xample where each tile “kno ws” its position in a macro-tile (keeps two inte- gers modulo M ). So the tiles in th e “wire” zone know that they shoul d transm it a bit, the tiles inside the computatio n zone know t hey shoul d obey the local rules for time-space diagram of the computation, etc. This con struction us es only bounded number of addition al colors since we have fix ed t he u ni- versal T uring machin e (including its alphabet and number of s tates); we do not need to increase the number of c olors when we increase M and k (though k should be small compared to M t o leave enough sp ace for the wires; we do not gi ve an exact positi on of the wires but i t is easy to see that if k / M is small e nough, there is enough space for them). So the const ruction us es O ( M 2 ) colors (and tiles). 6 A tile set that implements itself Now we come to the crucial point i n our argument: can we arrange things in such a way that the predicate S (i.e., the tile set it generates) is isom orphic to the set of tiles τ used to implement it? 6 Assume that k = 2 log M + O ( 1 ) ; then macro-colors hav e enough space to encode t he coordi- nates modulo M plus superim posed colors (which require O ( 1 ) bits for encoding). Note that many of t he rules that define τ do n ot depend on σ (i.e., on the predicate S ). So the program for the univ ersal T uring machine may start by checking these rules. It should check that – bits that represent coordinates (integers modulo M ) on the four sides of a macro-tile are related in the proper way (left and lower sides hav e identical coordinates, on the right/upp er si de on e of the coordinates increases modulo M ); – if the macro-tile is outside computation zone and the wires, it does not carry additional colors; – if t he macro-til e is a part of a wire, then i t transmits a bit in a required direction (of course, for th is we should fix t he pos ition of the wires by some formul as that are then checked by a program); – if the macro-tile is a part of the computation zone, it shou ld obey the local rules for the compu- tation zone (bits of the read-only layer should propagate vertically , bits that encode the content of the tape and the h ead of our u niv ersal T uri ng machine should change as tim e increases ac- cording to the beha vior of this machine, etc.) This guarantees that on the next l ayer macro-tiles are grouped into m acro-macro-tiles w here bits are transm itted correctly t o the compu tation zone of a m acro-macro-tile and some comput ation of the universal T uring machine is performed in this zone. But we n eed more: this computatio n should be the same computation that is performed on the macro-tile level (fixed poi nt!). This is also easy to achie ve s ince in our m odel the t ext of a running program is av ailable to it (recall the we assume that the program is wri tten in a read-only l ayer): the program should check also t hat if a macr o-ti le is in the comput ation zone, then the pro gram bit i t carries is cor r ect (program knows the x -coordinate o f a macro-tile, s o it can go at the correspondi ng place of i ts own tape to find o ut which program bit resides in this place). This sound like so me m agic, but we h ope t hat o ur pre vious example (a program for the UTM that prints its own text) makes this trick less magical (indeed, reliable and reusable m agic is called technology). 7 So what? W e believe that ou r proo f i s rather natural. If von Neumann live d few years more and were asked about aperiodic tile sets, he would probably immediately give t his ar gum ent as a solution. (He wa s especially well prepared to i t since he used very si milar self-referential tricks to construct a self- reproducing autom ata, see [9].) In fact this proof somehow appeared, though not v ery explicitly , in P . G ´ acs’ papers on ce llular automata [5]; the a ttempts to understand these papers were our startin g points. This proof is rath er flexible and can be adapted to get many results usually associated with aperiodic tiling s: undecidabil ity of domino problem (Ber ger [1]), recursive inseparabilit y of p e- riodic tile sets and inconsistent tile sets (Gurevich – K oryakov [7]), enforcing subst itution rules (Mozes [8]) and others (see [2,4]). But does it giv e som ething ne w? W e beli e ve t hat indeed there are some appl ications that hardly could b e achiev ed by previous ar guments. Let us conclude b y ment ioning two of them . First is th e constructi on of rob us t aperiodi c 7 tile sets. W e can consider tili ngs with holes (where no tiles are pl aced and therefore no m atching rules are checked). A robust aperiodic tile set should have the foll owing property: if the set of holes is “sparse enough”, then ti ling still should be far from any per iodic pattern (say , in the sense of Besicovitch dis tance, i.e., t he limsup of the fraction of mismatched posit ions in a centered s quare as the si ze of the square goes t o infinit y). The notion of “sparsity” s hould not be too restrictive here; we guarantee, for e xample, that a Bernoulli random set wi th small enough probability p (each cell belongs to a hole independently with probability p ) i s sparse. While the first example (robust aperiodic tile sets) i s rather technical (see [4] for details), the second is more basic. Let us split all tiles i n some ti le set into two classes , say , A- and B-tiles. Then we consider a fraction of A-tiles in a tili ng. If a tile set is not restrictive (allo ws many tilings), this fraction could vary from one tiling to anoth er . For classical aperiodic tilings this f raction is usually fixed: in a big tiled r egion the fraction of A-tiles is close to s ome limit value, usually an eigen value of a n inte ger matrix (and therefore an algebraic number). The fixed-point construction a llows us to get any computable number . Here is the formal statem ent: for any computable r eal α ∈ [ 0 , 1 ] ther e e xists a tile set τ divided into A- and B-tiles suc h that for any ε > 0 ther e e xists N su ch that for al l n > N the fraction of A-t iles in any τ -tiling of n × n-squ ar e is between α − ε and α + ε . Refer ences 1. R. Berger , The Undecidability of the Domino Problem. Mem. Amer . Math. Soc. , 66 , 1966. 2. Egon B ¨ orger , Erich Gr ¨ adel, Y uri Gu re vich, The Classical Decision Pro blem , Springer , 1987. IS BN 3-54 0-57073 -X 3. Bruno Durand, Leonid Le vin, Alexander Shen, Loc al Rules an d Gl obal Order, or Aperiodic T ilings, Mathematical Intellig encer , 27 (1), 64–68 (2005). 4. Bruno Durand, Andrei Romashchenko , Alexander Shen, Fixed P oint and Aperiodic T ilings, De velopments in Langua ge Theory , 12th International Confer ence , DLT 2008, K yoto, J apan, September 16–19, 2008. Pr oceedings , Springer , Lecture Notes in Computer Science, volu me 5257, 2008. ISBN: 978-3-540-85 779-2. 5. Peter G ´ acs, Reliable Cellular Automata with Self-Organization, J . Stat. Phys. , 103 (1/2), 45–267, 2001. 6. Branko Grunbau m, Geoffre y C. Shephard, T il ings an d P atterns , W . H.Freeman & Co, 1986. 7. Y uri Gurevich , Igor K oryak ov , A remark ob Berger’ s paper on the domino problem, Siberian Mathematical Jou rnal , 13 , 319– 321, 1972. 8. Shahar Mozes, Tilings, Substitution Systems and Dyn amical Systems Gen erated by Them, J . A nalyse Math. , 53 , 13 9–186, 1989. 9. John von Neu mann, Theory of Self-r epr oducing Automata , edited by A. Burks, Uni versity of Illinois Press, 196 6. 10. Hao W ang , Proving theorems by pattern recognition II, Bell System T echnical Jo urnal , 40 , 1–42 (1961). 8