Nondeterministic Algorithm for Breaking Diffie-Hellman Key Exchange using Self-Assembly of DNA Tiles

The computation based on DNA tile self-assembly has been demonstrated to be scalable, which is consider as a promising technique for computation. In this work, I first show how the tile self-assembly process can be used for computing the modular multiplication by mainly constructing three small systems including addition system, subtraction system and comparing system which can also be parallely implemented the discrete logarithm problem in the finite field GF(p). Then the nondeterministic algorithm is successfully performed to break Diffie-Hellman key exchange with the computation time complexity of Θ(p), and the probability of finding the successful solutions among many parallel executions is proved to be arbitrarily close to 1

Small Tile Sets That Compute While Solving Mazes

We ask the question of how small a self-assembling set of tiles can be yet have interesting computational behaviour. We study this question in a model where supporting walls are provided as an input structure for tiles to grow along: we call it the Maze-Walking Tile Assembly Model. The model has a number of implementation prospects, one being DNA strands that attach to a DNA origami substrate. Intuitively, the model suggests a separation of signal routing and computation: The input structure (maze) supplies a routing diagram, and the programmer's tile set provides the computational ability. We ask how simple the computational part can be. We give two tiny tile sets that are computationally universal in the Maze-Walking Tile Assembly Model. The first has four tiles and simulates Boolean circuits by directly implementing NAND, NXOR and NOT gates. Our second tile set has 6 tiles and is called the Collatz tile set as it produces patterns found in binary/ternary representations of iterations of the Collatz function. Using computer search we find that the Collatz tile set is expressive enough to encode Boolean circuits using blocks of these patterns. These two tile sets give two different methods to find simple universal tile sets, and provide motivation for using pre-assembled maze structures as circuit wiring diagrams in molecular self-assembly based computing.ISSN:1868-896

Parallel Computation Using Active Self-assembly

We study the computational complexity of the recently proposed nubots model of molecular-scale self-assembly. The model generalizes asynchronous cellular automaton to have non-local movement where large assemblies of molecules can be moved around, analogous to millions of molecular motors in animal muscle effecting the rapid movement of large arms and legs. We show that nubots is capable of simulating Boolean circuits of polylogarithmic depth and polynomial size, in only polylogarithmic expected time. In computational complexity terms, any problem from the complexity class NC is solved in polylogarithmic expected time on nubots that use a polynomial amount of workspace. Along the way, we give fast parallel algorithms for a number of problems including line growth, sorting, Boolean matrix multiplication and space-bounded Turing machine simulation, all using a constant number of nubot states (monomer types). Circuit depth is a well-studied notion of parallel time, and our result implies that nubots is a highly parallel model of computation in a formal sense. Thus, adding a movement primitive to an asynchronous non-deterministic cellular automation, as in nubots, drastically increases its parallel processing abilities

Communication-Efficient Jaccard Similarity for High-Performance Distributed Genome Comparisons

The Jaccard similarity index is an important measure of the overlap of two sets, widely used in machine learning, computational genomics, information retrieval, and many other areas. We design and implement SimilarityAtScale, the first communication-efficient distributed algorithm for computing the Jaccard similarity among pairs of large datasets. Our algorithm provides an efficient encoding of this problem into a multiplication of sparse matrices. Both the encoding and sparse matrix product are performed in a way that minimizes data movement in terms of communication and synchronization costs. We apply our algorithm to obtain similarity among all pairs of a set of large samples of genomes. This task is a key part of modern metagenomics analysis and an evergrowing need due to the increasing availability of high-throughput DNA sequencing data. The resulting scheme is the first to enable accurate Jaccard distance derivations for massive datasets, using largescale distributed-memory systems. We package our routines in a tool, called GenomeAtScale, that combines the proposed algorithm with tools for processing input sequences. Our evaluation on real data illustrates that one can use GenomeAtScale to effectively employ tens of thousands of processors to reach new frontiers in large-scale genomic and metagenomic analysis. While GenomeAtScale can be used to foster DNA research, the more general underlying SimilarityAtScale algorithm may be used for high-performance distributed similarity computations in other data analytics application domains

Computational Complexity in Tile Self-Assembly

One of the most fundamental and well-studied problems in Tile Self-Assembly is the Unique Assembly Verification (UAV) problem. This algorithmic problem asks whether a given tile system uniquely assembles a specific assembly. The complexity of this problem in the 2-Handed Assembly Model (2HAM) at a constant temperature is a long-standing open problem since the model was introduced. Previously, only membership in the class coNP was known and that the problem is in P if the temperature is one (τ = 1). The problem is known to be hard for many generalizations of the model, such as allowing one step into the third dimension or allowing the temperature of the system to be a variable, but the most fundamental version has remained open. In this Thesis I will cover verification problems in different models of self-assembly leading to the proof that the UAV problem in the 2HAM is hard even with a small constant temperature (τ = 2), and finally answer the complexity of this problem (open since 2013). Further, this result proves that UAV in the staged self-assembly model is coNP-complete with a single bin and stage (open since 2007), and that UAV in the q-tile model is also coNP-complete (open since 2004). We reduce from Monotone Planar 3-SAT with Neighboring Variable Pairs, a special case of 3SAT recently proven to be NP-hard

COMPUTER SIMULATION AND COMPUTABILITY OF BIOLOGICAL SYSTEMS

The ability to simulate a biological organism by employing a computer is related to the ability of the computer to calculate the behavior of such a dynamical system, or the "computability" of the system.* However, the two questions of computability and simulation are not equivalent. Since the question of computability can be given a precise answer in terms of recursive functions, automata theory and dynamical systems, it will be appropriate to consider it first. The more elusive question of adequate simulation of biological systems by a computer will be then addressed and a possible connection between the two answers given will be considered. A conjecture is formulated that suggests the possibility of employing an algebraic-topological, "quantum" computer (Baianu, 1971b) for analogous and symbolic simulations of biological systems that may include chaotic processes that are not, in genral, either recursively or digitally computable. Depending on the biological network being modelled, such as the Human Genome/Cell Interactome or a trillion-cell Cognitive Neural Network system, the appropriate logical structure for such simulations might be either the Quantum MV-Logic (QMV) discussed in recent publications (Chiara, 2004, and references cited therein)or Lukasiewicz Logic Algebras that were shown to be isomorphic to MV-logic algebras (Georgescu et al, 2001)

Distributed Memory, GPU Accelerated Fock Construction for Hybrid, Gaussian Basis Density Functional Theory

With the growing reliance of modern supercomputers on accelerator-based architectures such a GPUs, the development and optimization of electronic structure methods to exploit these massively parallel resources has become a recent priority. While significant strides have been made in the development of GPU accelerated, distributed memory algorithms for many-body (e.g. coupled-cluster) and spectral single-body (e.g. planewave, real-space and finite-element density functional theory [DFT]), the vast majority of GPU-accelerated Gaussian atomic orbital methods have focused on shared memory systems with only a handful of examples pursuing massive parallelism on distributed memory GPU architectures. In the present work, we present a set of distributed memory algorithms for the evaluation of the Coulomb and exact-exchange matrices for hybrid Kohn-Sham DFT with Gaussian basis sets via direct density-fitted (DF-J-Engine) and seminumerical (sn-K) methods, respectively. The absolute performance and strong scalability of the developed methods are demonstrated on systems ranging from a few hundred to over one thousand atoms using up to 128 NVIDIA A100 GPUs on the Perlmutter supercomputer.Comment: 45 pages, 9 figure

Small tile sets that compute while solving mazes

We ask the question of how small a self-assembling set of tiles can be yet have interesting computational behaviour. We study this question in a model where supporting walls are provided as an input structure for tiles to grow along: we call it the Maze-Walking Tile Assembly Model. The model has a number of implementation prospects, one being DNA strands that attach to a DNA origami substrate. Intuitively, the model suggests a separation of signal routing and computation: the input structure (maze) supplies a routing diagram, and the programmer’s tile set provides the computational ability. We ask how simple the computational part can be. We give two tiny tile sets that are computationally universal in the Maze-Walking Tile Assembly Model. The first has four tiles and simulates Boolean circuits by directly implementing NAND, NXOR and NOT gates. Our second tile set has 6 tiles and is called the Collatz tile set as it produces patterns found in binary/ternary representations of iterations of the Collatz function. Using computer search we find that the Collatz tile set is expressive enough to encode Boolean circuits using blocks of these patterns. These two tile sets give two different methods to find simple universal tile sets, and provide motivation for using pre-assembled maze structures as circuit wiring diagrams in molecular self-assembly based computing

Dataflow Programming Paradigms for Computational Chemistry Methods

The transition to multicore and heterogeneous architectures has shaped the High Performance Computing (HPC) landscape over the past decades. With the increase in scale, complexity, and heterogeneity of modern HPC platforms, one of the grim challenges for traditional programming models is to sustain the expected performance at scale. By contrast, dataflow programming models have been growing in popularity as a means to deliver a good balance between performance and portability in the post-petascale era. This work introduces dataflow programming models for computational chemistry methods, and compares different dataflow executions in terms of programmability, resource utilization, and scalability. This effort is driven by computational chemistry applications, considering that they comprise one of the driving forces of HPC. In particular, many-body methods, such as Coupled Cluster methods (CC), which are the gold standard to compute energies in quantum chemistry, are of particular interest for the applied chemistry community. On that account, the latest development for CC methods is used as the primary vehicle for this research, but our effort is not limited to CC and can be applied across other application domains. Two programming paradigms for expressing CC methods into a dataflow form, in order to make them capable of utilizing task scheduling systems, are presented. Explicit dataflow, is the programming model where the dataflow is explicitly specified by the developer, is contrasted with implicit dataflow, where a task scheduling runtime derives the dataflow. An abstract model is derived to explore the limits of the different dataflow programming paradigms

Theory and Practice of Computing with Excitable Dynamics

Reservoir computing (RC) is a promising paradigm for time series processing. In this paradigm, the desired output is computed by combining measurements of an excitable system that responds to time-dependent exogenous stimuli. The excitable system is called a reservoir and measurements of its state are combined using a readout layer to produce a target output. The power of RC is attributed to an emergent short-term memory in dynamical systems and has been analyzed mathematically for both linear and nonlinear dynamical systems. The theory of RC treats only the macroscopic properties of the reservoir, without reference to the underlying medium it is made of. As a result, RC is particularly attractive for building computational devices using emerging technologies whose structure is not exactly controllable, such as self-assembled nanoscale circuits. RC has lacked a formal framework for performance analysis and prediction that goes beyond memory properties. To provide such a framework, here a mathematical theory of memory and information processing in ordered and disordered linear dynamical systems is developed. This theory analyzes the optimal readout layer for a given task. The focus of the theory is a standard model of RC, the echo state network (ESN). An ESN consists of a fixed recurrent neural network that is driven by an external signal. The dynamics of the network is then combined linearly with readout weights to produce the desired output. The readout weights are calculated using linear regression. Using an analysis of regression equations, the readout weights can be calculated using only the statistical properties of the reservoir dynamics, the input signal, and the desired output. The readout layer weights can be calculated from a priori knowledge of the desired function to be computed and the weight matrix of the reservoir. This formulation explicitly depends on the input weights, the reservoir weights, and the statistics of the target function. This formulation is used to bound the expected error of the system for a given target function. The effects of input-output correlation and complex network structure in the reservoir on the computational performance of the system have been mathematically characterized. Far from the chaotic regime, ordered linear networks exhibit a homogeneous decay of memory in different dimensions, which keeps the input history coherent. As disorder is introduced in the structure of the network, memory decay becomes inhomogeneous along different dimensions causing decoherence in the input history, and degradation in task-solving performance. Close to the chaotic regime, the ordered systems show loss of temporal information in the input history, and therefore inability to solve tasks. However, by introducing disorder and therefore heterogeneous decay of memory the temporal information of input history is preserved and the task-solving performance is recovered. Thus for systems at the edge of chaos, disordered structure may enhance temporal information processing. Although the current framework only applies to linear systems, in principle it can be used to describe the properties of physical reservoir computing, e.g., photonic RC using short coherence-length light