Lindgren, K., Moore, C., and Nordahl, M. (1998). Complexity of two-dimensional patterns. Journal of Statistical Physics 91, 909-951.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....we have introduced, we are going to treat the k queens problem by showing that the set of pictures which represent chessboards (squares) of any size where there are as many queens as rows and such that no queen strike another queen is a recognizable picture language. This example is inspired by [17] where the 8 queens problem is treated. For instance, with the alphabet Sigma = f ; g, the two following pictures are in this language we denote L: For the moment, we forget the colors of the chessboard and we study only the positions of the queens. In the associated local language, denoted ....

K. Lindgren, C. Moore, and M. Nordahl. Complexity of two-dimensional patterns. Technical Report 97--03--023, Santa Fe Institute, Unit. States, 1997.

....its northern, southern, western, and eastern border, respectively. In the literature, the term picture is often used as a substitute for twodimensional or planar word, especially in the context of picture languages. Useful surveys on two dimensional languages may be found in [InT91] GiR97] or [LMN98]. Local Lattice Languages (LLL s, for short) also called tile systems, are speci ed using a nite set of nite (rectangular) words that are allowed as sub words of the recognized planar words. If one combines this with homomorphisms, than the power of the mechanism does not change by ....

K. Lindgren, C. Moore, and M. Nordahl. Complexity of two-dimensional patterns. Journal of Statistical Physics 91:909-951, 1998.

.... computational mechanics of nonlinear processes (Crutchfield, 1994; Crutchfield Hanson, 1993) Moore and his colleagues work on understanding what dynamical systems can compute and extending computation theory to continuousvalued computation and two dimensional languages (Moore, 1990, 1996; Lindgren, Moore, Nordahl, 1997); Fontana and Buss s work on self organization and the development of hierarchies in an algorithmic chemistry (Fontana Buss, 1996) and Crutchfield, Mitchell, Das, and others work on the evolution of emergent computation in cellular automata (Crutchfield Mitchell, 1995; Das, Mitchell, ....

Lindgren, K., Moore, C., & Nordahl, M. (1997). Complexity of two-dimensional patterns. (Working Paper 97-03-023.) Santa Fe, NM: Santa Fe Institute.

....the rst hardness results about tiling relied on simulating steps of a Turing machine from row to row, here we will instead simulate Boolean circuits, where wires with two possible tilings carry truth values, and junctions in these wires simulate logical gates. Similar approaches are taken in [6, 9, 11 13]. The question of whether a tiling exists then corresponds to the canonical NP complete problem Satis ability, which asks whether a set of truth values for the inputs exists that makes the output true [4] Our wire is shown in Figure 2. It moves in knight s moves in eight possible directions ....

K. Lindgren, C. Moore and M.G. Nordahl, \Complexity of two-dimensional patterns." Journal of Statistical Physics 91 (1998) 909-951.

....and in nite time. We hope that we have given the reader some conceptual tools for the classi cation of two or more dimensional patterns found in her research; or at least that she has found the examples and distinctions we have made enjoyable. A preliminary version of this work appeared in [36]. Acknowledgements. We are grateful to the Niels Bohr Institute where this work was begun, the Santa Fe Institute where it was continued, and the Bellairs Research Institute of McGill University where it was nally put to rest. We also thank Jean Camille Birget, Marek Chrobak, Tao Jiang, Oliver ....

K. Lindgren, C. Moore, and M.G. Nordahl, \Complexity of two-dimensional patterns." J. Unpub. Res. 1 (1990) 1-32.

....a language under an alphabetic homomorphism is tiling recognizable. Without loss of generality we may assume that all forbidden blocks have size 2 2, or are 1 2 and 2 1 dominoes [GR92] Tiling recognizable languages have also been called homomorphisms of local lattice languages or h(LLL)s [LMN98] or the languages recognizable by nondeterministic on line tessellation acceptors [IN77] We will follow [GR92] and denote this set of languages REC. While DFAs, NFAs, AFAs and REC are all equivalent to the regular languages in one dimension, in two or more dimensions they become distinct: DFA ....

....[IN77] We will follow [GR92] and denote this set of languages REC. While DFAs, NFAs, AFAs and REC are all equivalent to the regular languages in one dimension, in two or more dimensions they become distinct: DFA NFA AFA REC where all of these inclusions are strict. We recommend [LMN98,GR96,IT91,Ros79] for reviews of these classes. A bibliography of papers in the subject is maintained by Borchert at [BB] Note that we restrict our automata to move within the picture they are trying to recognize. For DFAs, it is known that allowing them to move outside the picture into a eld of blanks does ....

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K. Lindgren, C. Moore and M.G. Nordahl, \Complexity of two-dimensional patterns." Journal of Statistical Physics 91 (1998) 909-951.

.... AND MOORE deterministic, and DFAs and NFAs of this kind were introduced by Blum and Hewitt [1] Similarly, we can forbid a finite number of subblocks and then project onto a smaller alphabet, obtaining a class of picture languages which we call homomorphisms of local lattice languages, or h(LLL)s [8]. These are also called the recognizable languages [3] or the languages recognizable by non deterministic online tesselation acceptors [5] While DFAs, NFAs and h(LLL)s are equivalent in one dimension, in two or more they become distinct: DFA # NFA # h(LLL) where these inclusions are ....

....languages [3] or the languages recognizable by non deterministic online tesselation acceptors [5] While DFAs, NFAs and h(LLL)s are equivalent in one dimension, in two or more they become distinct: DFA # NFA # h(LLL) where these inclusions are strict. Reviews of these classes are given in [8, 4, 7, 11], and a bibliography of papers in the subject is maintained by Borchert at [2] A fair amount is known about the closure properties of these classes as well. The DFA, NFA, and h(LLL) languages are all closed under intersection and union using straightforward constructions. The situation for ....

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K. Lindgren, C. Moore and M.G. Nordahl (1998) Complexity of two-dimensional patterns. Journal of Statistical Physics 91 909--951. TWO-DIMENSIONAL AUTOMATA 13

....and infinite time. We hope that we have given the reader some conceptual tools for the classification of two or more dimensional patterns found in her research; or at least that she has found the examples and distinctions we have made enjoyable. A preliminary version of this work appeared in [31]. Acknowledgements. We are grateful to the Niels Bohr Institute where this work was begun, the Santa Fe Institute where it was continued, and the Bellairs Research Institute of McGill University where it was finally put to rest. We also thank Jean Camille Birget, Marek Chrobak, Tao Jiang, and ....

K. Lindgren, C. Moore, and M.G. Nordahl, "Complexity of two-dimensional patterns." J. Unpub. Res. 1 (1990) 1--32.

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Lindgren, K., Moore, C., and Nordahl, M. (1998). Complexity of two-dimensional patterns. Journal of Statistical Physics 91, 909-951.

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K. Lindgren, C. Moore, and M. Nordahl, "Complexity of two-dimensional patterns," Journal of Statistical Physics 91, pp. 909--951, 1998. URL http://arxiv.org/abs/cond-mat/9804071.

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K. Lindgren, C. Moore and M.G. Nordahl, "Complexity of two-dimensional patterns." Journal of Statistical Physics 91 (1998) 909-951.

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K. Lindgren, C. Moore and M.G. Nordahl, "Complexity of two-dimensional patterns." Journal of Statistical Physics 91 (1998) 909-951.

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