The Symbolic Dynamics Of Multidimensional Tiling Systems

We prove a multidimensional version of the theorem that every shift of finite type has a power that can be realized as the same power of a tiling system. We also show that the set of entropies of tiling systems equals the set of entropies of shifts of finite type

Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpi\'nski carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system

A graph isomorphism condition and equivalence of reaction systems

Algorithms and the Foundations of Software technolog

Is Every Irreducible Shift of Finite Type Flow Equivalent to a Renewal System?

Is every irreducible shift of finite type flow equivalent to a renewal system? For the first time, this variation of a classic problem formulated by Adler is investigated, and several partial results are obtained in an attempt to find the range of the Bowen--Franks invariant over the set of renewal systems of finite type. In particular, it is shown that the Bowen--Franks group is cyclic for every member of a class of renewal systems known to attain all entropies realised by shifts of finite type, and several classes of renewal systems with non--trivial values of the invariant are constructed.Comment: 22 pages, 5 figures. For the conference proceedings of Operator Algebra and Dynamics, NordForsk Network Closing Conference, 15-20 May 2012, Gj\'aargar{\eth}ur, Faroe Island

Exact Synchronization for Finite-State Sources

We analyze how an observer synchronizes to the internal state of a finite-state information source, using the epsilon-machine causal representation. Here, we treat the case of exact synchronization, when it is possible for the observer to synchronize completely after a finite number of observations. The more difficult case of strictly asymptotic synchronization is treated in a sequel. In both cases, we find that an observer, on average, will synchronize to the source state exponentially fast and that, as a result, the average accuracy in an observer's predictions of the source output approaches its optimal level exponentially fast as well. Additionally, we show here how to analytically calculate the synchronization rate for exact epsilon-machines and provide an efficient polynomial-time algorithm to test epsilon-machines for exactness.Comment: 9 pages, 6 figures; now includes analytical calculation of the synchronization rate; updates and corrections adde

A New Euler's Formula for DNA Polyhedra

DNA polyhedra are cage-like architectures based on interlocked and interlinked DNA strands. We propose a formula which unites the basic features of these entangled structures. It is based on the transformation of the DNA polyhedral links into Seifert surfaces, which removes all knots. The numbers of components , of crossings , and of Seifert circles are related by a simple and elegant formula: . This formula connects the topological aspects of the DNA cage to the Euler characteristic of the underlying polyhedron. It implies that Seifert circles can be used as effective topological indices to describe polyhedral links. Our study demonstrates that, the new Euler's formula provides a theoretical framework for the stereo-chemistry of DNA polyhedra, which can characterize enzymatic transformations of DNA and be used to characterize and design novel cages with higher genus

Forbidding - Enforcing Conditions in DNA Self-assembly of Graphs

Starting from a set of strands (or other types of building blocks) a variant of forbidding-enforcing systems for graphs which models DNA self-assembly is proposed. All possible outcomes of the self-assembly process comply with necessary constraints arising from the physical and chemical properties of DNA. A set of forbidding and enforcing rules that describe these constraints are presented

Forbidding-Enforcing Graphs for Self-Assembly

A variant of forbidding-enforcing systems on graphs is proposed in order to model the DNA self assembly process starting from given substructures