Damien Woods

P-completeness of cellular automaton rule 110

We show that the problem of predicting t steps of the 1D cellular automaton Rule 110 is P-complete. The result is found by showing that Rule 110 simulates deterministic Turing machines in polynomial time. As a corollary we find that the small universal Turing machines of Mathew Cook run in polynomial time, this is an exponential improvement on their previously known simulation time overhead.

The complexity of small universal Turing machines: A survey

We survey some work concerned with small universal Turing machines, cellular automata, tag systems, and other simple models of computation. For example, it has been an open question for some time as to whether the smallest known universal Turing machines of Minsky, Rogozhin, Baiocchi and Kudlek are efficient (polynomial time) simulators of Turing machines. These are some of the most intuitively simple computational devices and previously the best known simulations were exponentially slow. We discuss recent work that shows that these machines are indeed efficient simulators. As a related result, we also find that Rule 110, a well-known elementary cellular automaton, is also efficiently universal. We also review a large number of old and new universal program size results, including new small universal Turing machines and new weakly, and semi-weakly, universal Turing machines. We then discuss some ideas for future work arising out of these, and other, results.

Four Small Universal Turing Machines

We present universal Turing machines with state-symbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bi-tag system and are the smallest known single-tape universal Turing machines with 5, 4, 3 and 2-symbols, respectively. Our 5-symbolmachine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin. Also, all of the universalmachines we present here simulate Turing machines in polynomial time.

Active self-assembly of algorithmic shapes and patterns in polylogarithmic time

We describe a computational model for studying the complexity of self-assembled structures with active molecular components. Our model captures notions of growth and movement ubiquitous in biological systems. The model is inspired by biologys fantastic ability to assemble biomolecules that form systems with complicated structure and dynamics, from molecular motors that walk on rigid tracks and proteins that dynamically alter the structure of the cell during mitosis, to embryonic development where large scale complicated organisms efficiently grow from a single cell. Using this active self-assembly model, we show how to efficiently self-assemble shapes and patterns from simple monomers. For example we show how to grow a line of monomers in time and number of monomer states that is merely logarithmic in its length. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time polylogarithmic in the size of the shape plus roughly the time required to run a Turing machine deciding whether or not a given pixel is in the shape. We do this while keeping the number of monomer types logarithmic in shape size, plus monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the fundamental efficiency advantage of active self-assembly over passive self-assembly and motivates experimental effort to construct self-assembly systems with active molecular components.

An optical model of computation

We prove computability and complexity results for an original model of computation called the continuous space machine. Our model is inspired by the theory of Fourier optics. We prove our model can simulate analog recurrent neural networks, thus establishing a lower bound on its computational power. We also define a Θ(log2n) unordered search algorithm with our model.

The computational power of membrane systems under tight uniformity conditions

We apply techniques from complexity theory to a model of biological cellular membranes known as membrane systems or P-systems. Like Boolean circuits, membrane systems are defined as uniform families of computational devices. To date, polynomial time uniformity has been the accepted uniformity notion for membrane systems. Here, we introduce the idea of using AC0-uniformity and investigate the computational power of membrane systems under these tighter conditions. It turns out that the computational power of some systems is lowered from P to NL when using AC0-semi-uniformity, so we argue that this is a more reasonable uniformity notion for these systems as well as others. Interestingly, other P-semi-uniform systems that are known to be lower-bounded by P are shown to retain their P lower-bound under the new tighter semi-uniformity condition. Similarly, a number of membrane systems that are known to solve PSPACE-complete problems retain their computational power under tighter uniformity conditions.

INTRINSIC UNIVERSALITY IN SELF-ASSEMBLY

We show that the Tile Assembly Model exhibits a strong notion of universality where the goal is to give a single tile assembly system that simulates the behavior of any other tile assembly system. We give a tile assembly system that is capable of simulating a very wide class of tile systems, including itself. Specifically, we give a tile set that simulates the assembly of any tile assembly system in a class of systems that we call \emph{locally consistent}: each tile binds with exactly the strength needed to stay attached, and that there are no glue mismatches between tiles in any produced assembly. Our construction is reminiscent of the studies of \emph{intrinsic universality} of cellular automata by Ollinger and others, in the sense that our simulation of a tile system

Small fast universal turing machines

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A cargo-sorting DNA robot

by a tile system

Small Semi-Weakly Universal Turing Machines

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