In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several de nitions of "regular language" or "local rule" that are equivalent in d = 1 lead to distinct classes in d 2. We explore the closure properties and computational complexity of these classes, including undecidability and L, NL and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d 2 has a periodic point of a given peri...

We show that 2-tag systems efficiently simulate Turing machines. As a corollary we find that the small universal Turing machines of Rogozhin, Minsky and others simulate Turing machines in polynomial time. This is an exponential improvement on the previously known simulation time overhead and improves a forty year old result in the area of small universal Turing machines. 1

It is known that the tiling technique can be used to give simple proofs of undecidability of various two-dimensional modal and temporal logics. However, up until now, the simplest two-dimensional temporal logic, the compass logic of Venema, has eluded such treatment. We present a new coding of an enumeration of the tiling plane which enables us to show that the compass logic is undecidable.

Abstract. We show that the values of entropies of multidimensional shifts of finite type (SFTs) are characterized by a certain computation-theoretic property: a real number h≥0is the entropy of such an SFT if and only if it is right recursively enumerable, i.e. there is a computable sequence of rational numbers converging to h from above. The same characterization holds for the entropies of sofic shifts. On the other hand, the entropy of an irreducible SFT is computable. 1.

We develop a class of examples in the form of tiling dynamical systems for use as toy models in statistical mechanics, to analyze the possible existence of disordered crystals. We give the first such models which are disordered in the sense of having no discrete spectrum. * Research supported in part by a grant from the Israel Science and Technology Ministry ** Research supported in part by NSF Grant No. DMS-9001475 1.

Formalized study of self-assembly has led to the definition of the tile assembly model, a highly distributed parallel model of computation that may be implemented using molecules or a large computer network such as the Internet. Previously, I defined deterministic and nondeterministic computation in the tile assembly model and showed how to add, multiply and factor. Here, I extend the notion of computation to include deciding subsets of the natural numbers, and present a system that decides SubsetSum, a well-known NP-complete problem. The computation is nondeterministic and each parallel assembly executes in time linear in the input. The system requires only a constant number of different tile types: 49. I describe mechanisms for finding the successful solutions among the many parallel assemblies and explore bounds on the probability of such a nondeterministic system succeeding and prove that probability can be made arbitrarily close to one.

In this paper we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new counterexample in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial recursive functionals from X into Y and vice versa. The Medvedev degree of X is the equivalence class of X under Medvedev equivalence. There is an extensive recursion-theoretic literature on the lattice of Medvedev degrees of nonempty Π 0 1 subsets of {0, 1} N.Thislattice is known as Ps. WeprovethatPsconsists precisely of the Medvedev degrees of 2-dimensional subshifts of finite type. We use this result to obtain an infinite collection of 2-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible. Definition 1. Let A be a finite set of symbols. The full 2-dimensional shift on A is the dynamical system consisting of the natural action of Z2 on the compact set AZ2. A 2-dimensional subshift is a nonempty closed set X ⊆ AZ2 which is invariant under the action of Z2. A 2-dimensional subshift X is said to be of finite type if it is defined by a finite set of forbidden configurations. An interesting paper on 2-dimensional subshifts of finite type is Mozes [22]. A standard reference for the 1-dimensional case is the book of Lind/Marcus [20], which also includes an appendix [20, §13.10] on the 2-dimensional case.

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Introduction In this appendix, we prove the undecidability of the following problems: ffl the constrained domino problem (proved undecidable by Wang): give a tile set and a tile as input, ask whether it is possible to form a tiling of the plane containing the given tile; ffl the unconstrained domino problem (Berger's Theorem): the input is a tile set and the question is whether one can tile the plane with it; ffl the periodic domino problem (Berger Gurevich Koryakov): the input is also a tile set, but the question is whether it can be used to form a periodic tiling of the plane. In order to prove these results, we present some recursive transformations of Turing machines into tile sets. These constructions are not independent, thus the reader may not understand the last one if he could not understand the first ones. The last construction also provides a direct proof of the recursive inseparability result of Berger, Gurevich a

We derive lower bounds on the capacity of certain two-dimensional constraints by considering bounds on the entropy of measures induced by bit-stuffing encoders. A more detailed analysis of a previously proposed bit-stuffing encoder for (d � 1)-RLL constraints on the square lattice yields improved lower bounds on the capacity for all d 2. This encoding approach is extended to (d � 1)-RLL constraints on the hexagonal lattice, and a similar analysis yields lower bounds on capacity ford 2. For the hexagonal (1 � 1)-RLL constraint, the exact coding ratio of the bit-stuffing encoder is calculated and is shown to be within 0:5 % of the (known) capacity. Finally, alower bound is presented on the coding ratio of a bit-stuffing encoder for the constraint on the square lattice where each bit is equal to at least one of its four closest neighbors, thereby providing a lower bound on the capacity of this constraint.