R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177-209.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... be the maximum density of S satisfying x 2 S ) fx Gamma x : x 2 Sg B r 2 F : But this is probably too ambitious a generalization, because it should be possible to find Delta; F such that the question does D(F) Delta encodes the general tiling problem, and it is known (see e.g. [5]) that the solution of the general tiling problem is not computable. It should also be possible to find in the same way F such that D(F) is an arbitrary computable irrational, and or attained only by subsets of that are not even almost periodic. From these clouds of abstraction we return home to ....

Robinson, R.M.: Undecidability and nonperiodicity for tilings of the plane. Invent. Math. 12 (1971), 177--209. 28

....Unfortunately, full DLRUS , even restricted to atomic formulas, turns out to be undecidable. Theorem 1. The global satisfiability problem for DLRUS conceptual schemas containing only atomic formulas is undecidable. Proof. The proof is by reduction of the well known undecidable tiling problem [37]: given a finite set of square tiles of fixed orientation and with coloured edges, decide whether it can tile the grid Z N. Suppose T = fT 1 ; T k g is a set of tiles with colours left(T i ) right(T i ) up(T i ) and down(T i ) Consider the following schema , where D 1 ; D k ....

R. Robinson. Undecidability and non-periodicity for tilings of the plan. Invent. Math., 12:177--209, 1971.

....of the plane. Each tiled row of P corresponds to a legal configuration of the machine, and adjacent rows correspond to legal transitions of the machine. Exceptions are the unbounded constraint free versions of tiling problems (e.g. can T tile G ) which require a more complicated correspondence [1, 13]. Undecidability of the recurring versions is similarly proved using reductions from Turing machines. Here the machines are nondeterministic and the problem considered is the existence of an infinite computation that reenters a signaling situation infinitely often [7] Since tiling systems are ....

....of the general snake problem in the whole plane should also be contrasted with the undecidability of its tiling counterpart. It should be noted, though, that the general tiling problem in the whole plane is also unique among tiling problems, since its undecidability is much harder to prove [1, 13]. Infinite snake problems Recurring Miscellaneous Portions of the plane Problem 3.3 in PSPACE (infinite snake in a strip) hardness conjectured) Problem 4.1 open (unlimited infinite snake) Problem 3.6 co r.e. 3] strict, one way infinite snake) Problem 3.7 co r.e. strict, infinite ....

R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971) 177-209.

.... as many copies as we like of a nite set of shapes, can we ll a given region with them Tiling problems are potentially very hard: in fact, telling whether a nite set of tiles can tile the in nite plane is undecidable, since the non existence of a tiling is equivalent to the Halting Problem [1, 2]. For nite regions, this problem becomes NP complete [3] meaning that it is just as hard as combinatorial search problems like Traveling Salesman or Hamiltonian Path [4] In fact, tiling problems can be dicult even for small sets of very simple tiles, such as polyominoes [5] with as few as ....

R.M. Robinson, \Undecidability and nonperiodicity of tilings of the plane." Inventiones Math. 12 (1971) 177-209.

.... definition, and examples) The problem of determining for given data P and E whether # (P,E) is non empty is excluded by our definition; this question is known to be undecidable in general for d 1, because of the existence of Z subshifts of finite type without periodic points (see [B] [R]) The endomorphism semi group End(#) of the Z subshift of finite type # is defined to be the semi group of continuous surjective maps from # to # commuting with the action #. The automorphism group Aut(#) is the group of homeomorphisms of # commuting with #. For Z subshifts of finite ....

R. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177-209.

....floor floor FIGURE 1. The enumeration pair. 3 Linear orders with infinite ascending chains In this section we prove part (i) of Theorem 2. 2 by reducing the tiling problem for N N, which is known to be undecidable [22], to the satisfiability problem in C 1 C 2 . Let us recall first that tiles are 4 tuples of colours t = hleft(t) right(t) up(t) down(t)i : A finite set T of tiles is said to tile N N if there is a map : N N 7 T such that for each i; j 2 N, up( i; j) down( i; j 1) and ....

R. Robinson. Undecidability and nonperiodicity for tilings of the plane. Inventiones Math, 12:177--209, 1971.

....Unfortunately, full DLRUS , even restricted to atomic formulas, turns out to be undecidable. Theorem 1. The global satisfiability problem for DLRUS conceptual schemas containing only atomic formulas is undecidable. Proof The proof is by reduction of the well known undecidable tiling problem [31]: given a finite set of square tiles of fixed orientation and with coloured edges, decide whether it can tile the grid Z N. Suppose T = T 1 , T k is a set of tiles with colours left(T i ) right(T i ) up(T i ) and down(T i ) Consider the following schema #, where D 1 , ....

R. Robinson. Undecidability and non-periodicity for tilings of the plan. Invent. Math., 12:177--209, 1971.

.... i. Unfortunately, full DLRUS , even restricted to atomic formulas, turns out to be undecidable. Theorem 1. The global satisfiability problem for DLRUS conceptual schemas containing only atomic formulas is undecidable. Proof The proof is by reduction of the well known undecidable tiling problem [31]: given a finite set of square tiles of fixed orientation and with coloured edges, decide whether it can tile the grid Z N. Suppose T = fT 1 ; T k g is a set of tiles with colours left(T i ) right(T i ) up(T i ) and down(T i ) Consider the following schema , where D 1 ; D k ....

R. Robinson. Undecidability and non-periodicity for tilings of the plan. Invent. Math., 12:177--209, 1971.

....spectrum) Note that as in probability theory we are using measure theoretic chiefly spectral properties to analyze the order of the dynamical system. Examples of a) are easy to obtain. Examples of b) were first obtained by R. Berger in 1966 [1] with nicer examples by R. Robinson [10] and others. Examples of c) are due to S. Mozes in 1989 [4,7] It is unknown if there are examples of d) and this is an important open problem. There are several reasons for our introduction of the class of zero temperature systems. They constitute a natural generalization of systems of finite ....

R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177--209.

....the original examples it was natural (for their algorithmic interpretation) to use only polygons which would appear in the tilings in a single orientation, never rotated. But driven by the criterion of reducing the number of different polygons allowed, researchers (for instance, Raphael Robinson [7]) found it advantageous to have polygons appear in several orientations in the tilings. These were followed by the kite and dart tilings (Fig. 1) of Roger Penrose [3] The Penrose tilings consist of copies of only two polygons (Fig. 5) which appear in ten different orientations. They have an ....

R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177--209.

....by certain questions in logic) initiated the study of sets of prototiles which can tile the plane but only nonperiodically. A tiling is periodic if it consists of a lattice of translates of one of its swatches. Two of the early notable examples are the sets of prototiles discovered by Robinson [14] and by Penrose [4] which can tile the plane but only nonperiodically. The work of Wang eventually led to a succession of e#orts to find sets of prototiles for which the tilings are of necessity (that is, for which all tilings are) more and more disordered a notion which is inherently vague, ....

R. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177--209.

....in the case where the alphabet A is finite. The most notorious of these di#culties is the following: there is no algorithm which determines, given a finite alphabet A, a nonempty finite set F # Z d and a nonempty set P # A F , whether the space X (F,P ) in (2. 2) is nonempty or not ( 4] [50], 67] This undecidability problem is closely related to the existence of nonempty shifts of finite type X (F,P ) without periodic points (cf. Definition 1.1 and Example 2.6 (2) Suppose for simplicity that d = 2, F = 0, 1 2 # Z 2 and P # A F . If every nonempty shift of finite ....

....3. Examples 2. 6 (Higher dimensional shifts of finite type) 1) Wang tilings) Let T be a finite nonempty set of distinct, closed 1 1 squares (tiles) with coloured edges such that no horizontal edge has the same colour as a vertical edge: such a set T is called a collection of Wang tiles (cf. [50], 67] For each # # T we denote by r(#) t(#) l(# ) b(#) the colours of the right, top, left and bottom edges of # , and we write C(T) r(#) t(#) l(# ) b(#) # # T for the set of colours occurring on the tiles in T. A Wang tiling w by T is a covering of R 2 by such that (i) ....

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R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177--209.

....allowed block is globally allowed. Consequently there exists no finite time algorithm for determining whether a given subshift of finite type is the empty set or not. Further discussion of these extension and emptiness problems can be found in Berger [1] Kitchens Schmidt [14] Robinson [22] and Wang [29] 5 Definition 4. Let A = f0; k Gamma 1g, and suppose MH ; M V are k Theta k zero one matrices. We define the matrix subshift X A Z 2 by X = n x 2 A Z 2 : MH (x (m;n) x (m 1;n) 1; M V (x (m;n) x (m;n 1) 1 8(m; n) 2 Z 2 o : Section 3. Semi safe ....

R. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177--209.

....on the mathematics which they have inspired, scattered in totally unexpected directions. The story began in the early 1960 s, with the philosopher Hao Wang modeling certain problems in logic [Wan] It slowly evolved into geometry, in large part from influential work of Raphael Robinson [Rob] and the kite dart tiling of Roger Penrose [Gar] We will pick up the story there, and follow it through the twists and turns it has undergone, to the new mathematics that is emerging. It has been a highly interdisciplinary journey, and though we will strongly emphasize the mathematics (chiefly ....

....classify all the Morse tilings, and is a useful way 25 to understand the Morse tilings, or indeed any substitution tiling (at least when Lemma 1. 9 applies) We have reviewed the Morse tilings sufficiently, and now we return to our promise to exhibit a new set of tiles (due to Raphael Robinson ([Rob, GrS]) which can only tile the plane in the manner of the Morse tilings. That is, we will construct a finite alphabet A of square like tiles, and consider the set XA of all tilings that can be made by congruent copies of the letters in A, and show that in an appropriate sense these are the same as ....

R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971) 177-209.

.... related to tilings [13] 14] 18] and involve many additional subtleties; see, for example [16] 15] Of particular interest is The Nontriviality Problem: For a given set W, is the corresponding set hWi is even nonempty The Nontriviality Problem is known to be formally undecidable; see [19], 2] or [9] Theorem 1: Let U and W be as above. hWi is nontrivial if and only if there is some locally stationary probability measure U on A U , with supp [U ] ae hWi, such that U has a stationary extension. Proof: Suppose that such a U existed, and let be a stationary extension. Clearly, ....

....per EAS Theta A U is also a convex set. 8.3 Essentially Aperiodic measures Not every extendible measure has a periodic extension. This follows from the existence of essentially aperiodic tile systems that is, sets of tiles which can tile the plane, but only in an aperiodic fashion. In [19], Raphael Robinson exhibits a collection of six notched square tiles, which, along with their 4 rotations, will tile the plane, but only in 26 an aperiodic fashion. We can code these six tiles as six 3 Theta 3 matrices in the alphabet A : f0; a; A; b; B; c; Cg A C A B 0 d A B A a c a c 0 c a ....

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Raphael M. Robinson. Undecidability and nonperiodicity for tilings of the plane. Invent. Math, 12:177--209, 1971.

....a coloring map which defines it. In practice, finding such coloring map can be complicated. We leave the proof to the reader. 3. Complexity aspects It is well known that the tileability problem is NP complete when Gamma is finite [GJ] It is also undecidable when Gamma is the whole plane [Be,Ri]. We shall prove that the similar situation holds for GI Problem. But first we need to state it as a decision problem. GI rank Problem: Given T, r, decide whether rk G (T; B) r. Bounded GI rank Problem: Given T, r, N , decide whether rk G (T; BN ) r. Theorem 3.1 The GI rank Problem is ....

R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177--209.

....of the plane is valid if and only if all pairs of adjacent sides have the same color. Note that it is not allowed to turn tiles. Berger proved in 1966 that Given a tile set, it is undecidable whether this tile set can be used to tile the plane [3] a simplified proof was given in 1971 by Robinson [16]. We can also define finite tilings. We assume that the set of colors contains a special blank color and that the set of tiles contains a blank tile i.e. a tile whose all sides are blank. A finite tiling is an almost everywhere blank tiling of the plane. If there exist two integers i and j ....

....tiling is lower than i Theta j. Note that inside the i Theta j rectangle, there can be blank and non blank tiles. If there is at least one non blank tile, then the tiling is called non trivial. Another undecidability result can be proved simply by using a construction presented by Robinson in [16] which reduces the undecidability of the halting problem for Turing Machines into it: given a tile set with a blank tile, it is undecidable whether this tile set can be used to form a valid finite non trivial tiling of the plane. In the following, we are mainly interested in complexity results; we ....

R.M. Robinson. Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae, 12:177--209, 1971.

....is that the reverse CA of a reversible CA cannot be found by algorithm: its size can be greater than any computable function of the size of the reversible CA. The proof of this result consists in transforming the tiling problem of the plane which has been proved undecidable in 1966 by Berger [3, 11] into the reversibility problem on an adequate family of CA. The main goal of this paper is to prove that if we restrain the field of action of the two dimensional CA to finite configuration bounded in size, it is still difficult to prove that the CA is (or is not) reversible. As far as we ....

....is valid if and only if a given relation holds between each cell and its neighbors. Theorem 1 Given a tile set, it is undecidable to know whether this tile set can be used to tile the plane. This well known theorem is due to Berger [3] in 1966 and a simplified proof was given in 1971 by Robinson [11]. We can also define finite tilings. We assume that the set of colors contains a special blank color and that the set of tiles contains a blank tile i.e. a tile whose all sides are blank. A finite tiling is an almost everywhere blank tiling of the plane. If there exist two integers i and j ....

R.M. Robinson. Undecidability and nonperiodicity for tilings of the plane. Inventiones Mathematicae, 12:177--209, 1971.

....constraints seems to be much more difficult than the one dimensional case. This is, in part, due to the fact that in the general constrained setting, there are problems that are easy to solve in the one dimensional case, yet they become undecidable when we shift to two dimensions [3] [24]. One important example of a two dimensional constraint is the two dimensional extension of the (1; 1) RLL constraint. This constraint, which is also referred to as the hard square model, has been treated in quite a few papers in the past several years; see, for example, 6] 10] 11] 20] ....

R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Inventiones Math., 12 (1971), 177--209.

....rules for perfect quasicrystals. The Sierpinski carpet, for instance, is generated by a 3 3 expansion rule as shown in gure 3. To construct an h(LLL) for these, we need an underlying hierarchical structure which identi es blocks with their parent blocks on larger and larger scales. Robinson [50, 18] has constructed a set of tiles which does just that, by running H s along the boundaries between blocks and connecting them with the larger H s of their parent block, in such a way as to enforce a hierarchical tree. By decorating his tiling as shown in gure 4, we can enforce 2 2 expansion ....

....subset is also, since if the nite automaton accepting it has n states, extensibility of a nite word depends only on its rst and last n symbols. The question of whether an LLL in two or more dimensions has any in nite allowed con gurations at all is also undecidable; this is the Tiling Problem [5, 50], in which we try to cover the plane with a set of interlocking tiles. In one dimension, a nite state automaton accepts an in nite word if and only if there are loops in its transition graph, which is easily decidable. Of course, there are classes of LLL s where every block is extensible, such ....

R.M. Robinson, \Undecidability and nonperiodicity of tilings of the plane." Inventiones Math. 12 (1971) 177{.

....# # # # There exist complete tiling problems for many complexity classes. In the proof that follows we make use of a certain 0 1 complete tiling problem. Theorem 3.1 If jLj 2, and A is countably in nite then the satis ability problem for L KR2 is 0 1 hard. Proof: As shown in [3] [39], the following problem is 0 1 complete: N N tiling: Given a nite set T of tiles, can T tile N N That is, does there exist a function t from N N to T such that: right(t(n; m) left(t(n 1; m) and up(t(n; m) down(t(n; m 1) Let T = fT 1 ; T k g be a set of tiles. We ....

Robinson, R.: 1971, Undecidability and Nonperiodicity for Tilings of the Plane, Inventiones Math. 12, 177-209.

....A point x 2 X is periodic if its orbit under is nite. In contrast to the case where d = 1, a higher dimensional SFT X may not contain any periodic points (we give an example below) This potential absence of periodic points is associated with certain undecidability problems (cf. e.g. 1] 9] [19] and [32] Mathematics Institute, University of Vienna, and Erwin Schr odinger Institute for Mathematical Physics, Boltzmanngasse 9, A 1090 Vienna, Austria. Email: klaus.schmidt univie.ac.at 1 2 KLAUS SCHMIDT (1) It is algorithmically undecidable if X(F; P ) 6= for given (F; P ) 2) It ....

....in [7] h( WD ) 1 4 Z 1 0 Z 1 0 (4 2 cos 2 s 2 cos 2 t) ds dt: The domino tilings again have frozen con gurations which look like brick walls . Example 3 (A shift of nite type without periodic points) Consider the following set T 0 of six polygonal tiles, introduced by Robinson in [19], each of which which should be thought of as a 1 1 square with various bumps and dents. We denote by T the set of all tiles which are obtained by allowing horizontal and vertical re ections as well as rotations of elements in T 0 by multiples of 2 . Again we consider the set W T T Z ....

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R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177-209.

....of the plane are of interest both to statistical physicists and recreational mathematicians. While the number of ways to tile a lattice with dominoes can be calculated by expressing it as a determinant [1, 2] telling whether a nite collection of shapes can tile the plane at all is undecidable [3, 4]. In the 1994 edition of his wonderful book Polyominoes [5] Solomon W. Golomb states that the problem of determining how many ways a 4 n rectangle can be tiled by right trominoes appears to be a challenging problem with reasonable hope of an attainable solution. Here we solve this problem, and ....

R.M. Robinson, \Undecidability and nonperiodicity of tilings of the plane." Inventiones Math. 12 (1971) 177{.

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R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12 (1971), 177-209.

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Raphael M. Robinson. Undecidability and nonperiodicity of tilings of the plane. Inventiones Math., 12:177--909, 1971.

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