Results 1  10
of
121
Algorithmic SelfAssembly of DNA
, 1998
"... How can molecules compute? In his early studies of reversible computation, Bennett imagined an enzymatic Turing Machine which modified a heteropolymer (such as DNA) to perform computation with asymptotically low energy expenditures. Adleman's recent experimental demonstration of a DNA computation, ..."
Abstract

Cited by 104 (6 self)
 Add to MetaCart
How can molecules compute? In his early studies of reversible computation, Bennett imagined an enzymatic Turing Machine which modified a heteropolymer (such as DNA) to perform computation with asymptotically low energy expenditures. Adleman's recent experimental demonstration of a DNA computation, using an entirely different approach, has led to a wealth of ideas for how to build DNAbased computers in the laboratory, whose energy efficiency, information density, and parallelism may have potential to surpass conventional electronic computers for some purposes. In this thesis, I examine one mechanism used in all designs for DNAbased computer  the selfassembly of DNA by hybridization and formation of the double helix  and show that this mechanism alone in theory can perform universal computation. To do so, I borrow an important result in the mathematical theory of tiling: Wang showed how jigsawshaped tiles can be designed to simulate the operation of any Turing Machine. I propose...
Tiling with Polyominoes and Combinatorial Group Theory
 JOURNAL OF COMBINATORIAL THEORY, SERIES A 53, 1833208
, 1990
"... When can a given finite region consisting of cells in a regular lattice (triangular, square, or hexagonal) in [w ’ be perfectly tiled by tiles drawn from a finite set of tile shapes? This paper gives necessary conditions for the existence of such tilings using boundary inuariants, which are combinat ..."
Abstract

Cited by 83 (0 self)
 Add to MetaCart
When can a given finite region consisting of cells in a regular lattice (triangular, square, or hexagonal) in [w ’ be perfectly tiled by tiles drawn from a finite set of tile shapes? This paper gives necessary conditions for the existence of such tilings using boundary inuariants, which are combinatorial grouptheoretic invariants associated to the boundaries of the tile shapes and the regions to be tiled. Boundary invariants are used to solve problems concerning the tiling of triangularshaped regions of hexagons in the hexagonal lattice with certain tiles consisting of three hexagons. Boundary invariants give stronger conditions for nonexistence of tilings than those obtainable by weighting or coloring arguments. This is shown by considering whether or not a region has a signed tiling, which is a placement of tiles assigned weights 1 orI, such that all cells in the region are covered with total weight 1 and all cells outside with total weight 0. Any coloring (or weighting) argument that proves nonexistence of a tiling of a region also proves nonexistence of any signed tiling of the region as well. A partial converse holds: if a simply connected region has no signed tiling by simply connected tiles, then there is a generalized coloring argument proving that no signed tiling exists. There exist regions possessing a signed tiling which can be shown to have no perfect tiling using boundary invariants.
Simulations of Computing by SelfAssembly
, 1998
"... Winfree (1996) proposed a Turinguniversal model of DNA selfassembly. In this abstract model, DNA doublecrossover molecules selfassemble to form an algorithmicallypatterned twodimensional lattice. Here, we develop a more realistic model based on the thermodynamics and kinetics of oligonucleo ..."
Abstract

Cited by 69 (15 self)
 Add to MetaCart
Winfree (1996) proposed a Turinguniversal model of DNA selfassembly. In this abstract model, DNA doublecrossover molecules selfassemble to form an algorithmicallypatterned twodimensional lattice. Here, we develop a more realistic model based on the thermodynamics and kinetics of oligonucleotide hydridization. Using a computer simulation, we investigate what physical factors influence the error rates, i.e., when the more realistic model deviates from the ideal of the abstract model. We find, in agreement with rules of thumb for crystal growth, that the lowest error rates occur at the melting temperature when crystal growth is slowest, and that error rates can be made arbitrarily low by decreasing concentration and increasing binding strengths. 1 Introduction Early work in DNA computing (Adleman 1994; Lipton 1995; Boneh et al. 1996; Ouyang et al. 1997) showed how computations can be accomplished by first creating a combinatorial library of DNA and then, through successiv...
COMPLEXITY OF SELFASSEMBLED SHAPES
, 2007
"... The connection between selfassembly and computation suggests that a shape can be considered the output of a selfassembly “program,” a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest selfassembly program that builds a shape and the shape’s descrip ..."
Abstract

Cited by 61 (4 self)
 Add to MetaCart
The connection between selfassembly and computation suggests that a shape can be considered the output of a selfassembly “program,” a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest selfassembly program that builds a shape and the shape’s descriptional (Kolmogorov) complexity should be related. We show that when using a notion of a shape that is independent of scale, this is indeed so: in the tile assembly model, the minimal number of distinct tile types necessary to selfassemble a shape, at some scale, can be bounded both above and below in terms of the shape’s Kolmogorov complexity. As part of the proof, we develop a universal constructor for this model of selfassembly that can execute an arbitrary Turing machine program specifying how to grow a shape. Our result implies, somewhat counterintuitively, that selfassembly of a scaledup version of a shape often requires fewer tile types. Furthermore, the independence of scale in selfassembly theory appears to play the same crucial role as the independence of running time in the theory of computability. This leads to an elegant formulation of languages of shapes generated by selfassembly. Considering functions from bit strings to shapes, we show that the runningtime complexity, with respect to Turing machines, is polynomially equivalent to the scale complexity of the same function implemented via selfassembly by a finite set of tile types. Our results also hold for shapes defined by Wang tiling—where there is no sense of a selfassembly process—except that here time complexity must be measured with respect to nondeterministic Turing machines.
The Convenience of Tilings
 In Complexity, Logic, and Recursion Theory
, 1997
"... Tiling problems provide for a very simple and transparent mechanism for encoding machine computations. This gives rise to rather simple master reductions showing various versions of the tiling problem complete for various complexity classes. We investigate the potential for using these tiling proble ..."
Abstract

Cited by 36 (0 self)
 Add to MetaCart
Tiling problems provide for a very simple and transparent mechanism for encoding machine computations. This gives rise to rather simple master reductions showing various versions of the tiling problem complete for various complexity classes. We investigate the potential for using these tiling problems in subsequent reductions showing hardness of the combinatorial problems that really matter. We ilustrate our approach by means of three examples: a short reduction chain to the Knapsack problem followed by a Hilbert 10 reduction using similar ingredients. Finally we reprove the Deterministic Exponential Time lowerbound for satisfiablility in Propositional Dynamic Logic. The resulting reductions are relatively simple; they do however infringe on the principle of orthogonality of reductions since they abuse extra structure in the instances of the problems reduced from which results from the fact that these instances were generated by a master reduction previously. 1 Introduction This paper...
Feature Logics
 HANDBOOK OF LOGIC AND LANGUAGE, EDITED BY VAN BENTHEM & TER MEULEN
, 1994
"... Feature logics form a class of specialized logics which have proven especially useful in classifying and constraining the linguistic objects known as feature structures. Linguistically, these structures have their origin in the work of the Prague school of linguistics, followed by the work of Chom ..."
Abstract

Cited by 33 (0 self)
 Add to MetaCart
Feature logics form a class of specialized logics which have proven especially useful in classifying and constraining the linguistic objects known as feature structures. Linguistically, these structures have their origin in the work of the Prague school of linguistics, followed by the work of Chomsky and Halle in The Sound Pattern of English [16]. Feature structures have been reinvented several times by computer scientists: in the theory of data structures, where they are known as record structures, in artificial intelligence, where they are known as frame or slotvalue structures, in the theory of data bases, where they are called "complex objects", and in computati
Experimental Progress in Computation by SelfAssembly of DNA Tilings
, 1999
"... Approaches to DNAbased computing by selfassembly require the use of DNA nanostructures, called tiles, that have efficient chemistries, expressive computational power, and convenient input and output (I/O) mechanisms. We have designed two new classes of DNA tiles, TAO and TAE, both of which contain ..."
Abstract

Cited by 27 (14 self)
 Add to MetaCart
Approaches to DNAbased computing by selfassembly require the use of DNA nanostructures, called tiles, that have efficient chemistries, expressive computational power, and convenient input and output (I/O) mechanisms. We have designed two new classes of DNA tiles, TAO and TAE, both of which contain three doublehelices linked by strand exchange. Structural analysis of a TAO molecule has shown that the molecule assembles efficiently from its four component strands. Here we demonstrate a novel method for I/O whereby multiple tiles assemble around a singlestranded (input) scaffold strand. Computation by tiling theoretically results in the formation of structures that contain singlestranded (output) reported strands, which can then be isolated for subsequent steps of computation if necessary. We illustrate the advantages of TAO and TAE designs by detailing two examples of massively parallel arithmetic: construction of complete XOR and addition tables by linear assemblies of DNA t...
Limit(quasi)periodic Point Sets as Quasicrystals With PAdic Internal Spaces
, 1998
"... . Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead spaces with padic topologies or even with mixed Euclidean/p ..."
Abstract

Cited by 24 (12 self)
 Add to MetaCart
. Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead spaces with padic topologies or even with mixed Euclidean/padic topologies. We show that a number of well known tilings precisely fit this form, including the chair tiling and the Robinson square tilings. Thus the scope of the cut and project formalism is considerably larger than is usually supposed. Applying the powerful consequences of model sets we derive the diffractive nature of these tilings. z Heisenberg Fellow padic quasicrystals 2 1. Introduction The cut and project method of constructing nonperiodic point sets, as developed by Peter Kramer and others in the early eigthies [10, 11, 9, 3, 12], is one of the basic tools in the mathematical study of quasicrystals and aperiodic order. The intuition behind their use is that quasiperiodic po...
On the Products of Linear Modal Logics
 JOURNAL OF LOGIC AND COMPUTATION
, 2001
"... We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz: ..."
Abstract

Cited by 24 (9 self)
 Add to MetaCart
We study twodimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like K4:3, S4:3, GL:3, Grz:3, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman [7]. We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square K4:3 K4:3 of the minimal liner logic using nonstructural Gabbaytype inference rules.