R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. no. 66, (1966).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....volume [BoR1] For non tight radii none of the optimal packings have been published. For optimal density problems of polyhedra there is again the phenomenon of aperiodicity, exemplified by the kite dart tilings in E . Aperiodicity was discovered in the Euclidean context by Berger in 1966 [Berg], and has been of growing interest, from many perspectives (though mainly still in Euclidean spaces) ever since [Radi] We emphasize how short the above list is, how few optimum density problems have been solved. As part of this small but distinguished class, the aperiodic ones must play a ....

R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66(1966).

....which one should try first in every attempt to establish the undecidability of a theory or the high complexity of a problem. 14) Theorem ( GaJoPa77] Square tiling and Unconstrained square tiling are NP complete (15) Theorem ( Wa61] Origin constrained tiling is undecidable. 16) Theorem ( Ber66] Unconstrained tiling is undecidable. The unconstrained cases can be reduced via a sophisticated construction to the corresponding origin constrained cases. These and the square tiling are proved to be undecidable or NPhard via a naturaJ coding of Turing machines. During his research on the ....

R. Berger, 1966, The undecidability of the domino problem, Mem. Amer. Math. 72 pp.

....problems considered in the literature can be divided into two classes, unbounded and bounded. The following are perhaps the cleanest versions in each class: U 1: Given T, can T tile G B l: Given T and n (in unary) can T tile an n x n subgrid of G Problem U I is undecidable (cf. Berger [5], Robinson [31] and Lewis [25] It is, in fact, co r.e. and hence, using the hierarchy notation of Rogers [32] it is II complete. Problem B 1 is NP complete (cf. Lewis and Papadimitriou [26] and van Emde Boas [35] This behavior generalizes as one runs through some of the other problems that ....

....on a blank tape, and then the positive quadrant G can be tiled iff the given TM does not halt. As it turns out, U 1 is considerably more subtle, since there is no apparent way to force the appearance of any fixed part of any configuration, not even the very existence 227 of a state. See [5] and [31] The bounded versions are proved complete for their complexity classes by similarly considering time or space bounded computations. So much for background on domino problems. 3. Two Levels of Undecidability Although there exist many different levels of undecidability, there seem to ....

BERGER, R. The undecidability of the domino problem. Mere. Arner. Math. Soc. 66 (1966).

....iff for each t 2 T there is a T tiling of Z Z that uses t a map f : Z Z T with Right( f (i; j) Left( f (i 1; j) Up( f (i; j) Down( f (i; j 1) for all i; j 2 Z, and (without loss of generality) f (0;0) t. It is well known that this tiling problem is undecidable see, e.g. [2]. To code this problem into BQCTL F , we go via a construction originating in algebraic logic, which has already been used to prove undecidability of 3 dimensional modal logics [13] It was shown in [12] that there is no algorithm to decide whether a finite Tarskian relation algebra is ....

R. Berger. The undecidability of the domino problem, volume 66 of Memoirs. Amer. Math. Soc., Providence, RI, 1966.

....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. Berger, The undecidability of the domino problem, Mere. Amer. Math. Soc. 66 (1966).

....a periodic tiling of N Theta N , if there are integers h; v 0 such that for all i; j 2 N, i; j) i h; j) and (i; j) i; j v) We note the following lemma. 41 Lemma 11.2 Whether a given domino system D admits a periodic tiling of N Theta N is undecidable. Note that Berger [4] proved that the unconstrained domino problem is undecidable. The lemma is an immediate consequence of the stronger result that the classes of domino systems which do not admit tilings of N Theta N and which admit periodic tilings of N Theta N are recursively inseparable (see [6, Theorem ....

R. Berger. The Undecidability of the Domino Problem. Mem. AMS, 66, 1966.

.... 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. Berger, \The undecidability of the domino problem." Memoirs Amer. Math. Soc. 66 (1966) 1-72.

.... of this 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 ....

R. Berger, The undecidability of the Domino problem, Mem. Amer. Math. Soc. 66 (1966).

....corresponding to purely absolutely continuous 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 ....

R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66 (1966).

....by mathematicians. The book, Quasicrystals and geometry, by Marjorie Senechal, has an even broader goal: to present certain developments in crystallography from the past decade. The developments were generated in the wake of two profound discoveries. The first was the mathematics discovery [1] in 1966 of aperiodic tilings, the origin of Penrose s 1977 examples. The more recent one was the physics discovery [8] in 1984 of quasicrystals. These two discoveries have led to a large volume of interdisciplinary research among the fields of crystallography, physics, Quasicrystals ....

....such that congruent copies of them can tile the plane but only nonperiodically (A tiling is periodic if it consists of repetitions of a unit cell, as in Fig. 4. We have generalized Wang s question slightly, using hindsight. The unexpected answer of yes by Wang s student Robert Berger [1] led, over the last thirty years, to the new subject of aperiodic tiling. Treating nonperiodicity as a form of unexpected complication , the subject evolved to analyze what other forms of complication could be forced by a finite set of polygons; that is, what other aspects, not true of periodic ....

R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc., no. 66 (1966).

....even 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 ....

.... Wang tiles for which the following completion (or extension) problem is undecidable: for a given subset E # Z 2 and a given allowed configuration y # T E , does there exist a point x # W T with #E (x) y An earlier construction of a tiling system without periodic points appeared in [4], but the number of tiles used there was considerably greater. For a detailed discussion of tilings and shifts of finite type without periodic points we refer to [47] and [49] 14 DOUGLAS LIND AND KLAUS SCHMIDT Part 3. One dimensional shifts of finite type The classical theory of shifts of ....

R. Berger, The undecidability of the Domino Problem, Mem. Amer. Math. Soc. 66 (1966).

....tiling is periodic if there exists a translation that leaves the tiling invariant and it is non periodic in other case. A set of prototiles such that always tiles non periodically is called aperiodic. It was conjecture by Wang that there is no such a set of prototiles [9] However, in 1966, Berger [1] encountered the first one with around 20,000 prototiles and since then, many other sets have been obtained. We will prove in this paper that from any aperiodic set of prototiles, it is possible to construct infinitely many of those sets using Voronoi diagrams. As it is well known, given a ....

R. Berger, The undecidability of the domino problem, Memoirs Am. Math. Soc. 66 (1966).

....various restrictions on the allowed adjacencies. If these restrictions are given by forbidding a finite number of finite size configurations, then the tiling is said to be of finite type. An early indication of the difficulties faced in two dimensions goes back to the work of Wang [29] and Berger [1], who considered a particular class of finite type tilings (now called Wang tiles) Wang conjectured that there existed a finite time algorithm which, for any given tile set, would determine whether it was possible to tile the whole plane. His proposed algorithm was based on a further conjecture, ....

....for determining whether a given locally 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; ....

R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. No. 66, (1966).

....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. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66 (1966).

....two dimensional 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] ....

R. Berger, The Undecidability of the Domino Problem, Mem. Amer. Math. Soc., Vol. 66, 1966.

....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. Berger, \The undecidability of the domino problem." Memoirs Amer. Math. Soc. 66 (1966) 1-72.

....be a SFT. 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 ....

R. Berger, The undecidability of the Domino Problem, Mem. Amer. Math. Soc. 66 (1966).

....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. Berger, \The undecidability of the domino problem." Memoirs Amer. Math. Soc. 66 (1966) 1-72.

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R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. no. 66, (1966).

No context found.

R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. no. 66, (1966).

No context found.

BERGER, R. 1966. The undecidability of the domino problem. Memoirs American Mathematical Society, 66, 1--72.

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R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66 (1966).

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R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66, 1966.

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R. Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. 66 (1966).

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Robert Berger. The undecidability of the domino problem. Memiors of the AMS, 66:1--72, 1966.

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