James I. Lathrop

Strict self-assembly of discrete Sierpinski triangles

Winfree (1998) showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004). Precisely speaking, the above self-assemblies tile completely filled-in, two-dimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict self-assembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else. We first prove that the standard discrete Sierpinski triangle cannot strictly self-assemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, non-strict self-assemblies, our strict self-assembly algorithm makes extensive, recursive use of optimal counters, coupled with measured delay and corner-turning operations. We verify our strict self-assembly using the local determinism method of Soloveichik and Winfree (2007).

Finite-state dimension

Classical Hausdorff dimension (sometimes called fractal dimension) was recently effectivized using gales (betting strategies that generalize martingales), thereby endowing various complexity classes with dimension structure and also defining the constructive dimensions of individual binary (infinite) sequences. In this paper we use gales computed by multi-account finite-state gamblers to develop the finite-state dimensions of sets of binary sequences and individual binary sequences. The theorem of Eggleston (Quart. J. Math. Oxford Ser. 20 (1949) 31-36) relating Hausdorff dimension to entropy is shown to hold for finite-state dimension, both in the space of all sequences and in the space of all rational sequences (binary expansions of rational numbers). Every rational sequence has finite-state dimension 0, but every rational number in [0,1] is the finite-state dimension of a sequence in the low-level complexity class AC0. Our main theorem shows that the finite-state dimension of a sequence is precisely the infimum of all compression ratios achievable on the sequence by information-lossless finite-state compressors.

Computability and Complexity in Self-assembly

This paper explores the impact of geometry on computability and complexity in Winfree’s model of nanoscale self-assembly. We work in the two-dimensional tile assembly model, i.e., in the discrete Euclidean plane ℤ×ℤ. Our first main theorem says that there is a roughly quadratic function f such that a set A⊆ℤ+ is computably enumerable if and only if the set XA={(f(n),0)∣n∈A}—a simple representation of A as a set of points on the x-axis—self-assembles in Winfree’s sense. In contrast, our second main theorem says that there are decidable sets D⊆ℤ×ℤ that do not self-assemble in Winfree’s sense.Our first main theorem is established by an explicit translation of an arbitrary Turing machine M to a modular tile assembly system

Computational depth and reducibility

\mathcal{T}_{M}

Recursive computational depth

, together with a proof that

A Fully Characterized Test Suite for Genetic Programming

\mathcal{T}_{M}

Requirements analysis for a product family of DNA nanodevices

carries out concurrent simulations of M on all positive integer inputs.

Strict Self-assembly of Discrete Sierpinski Triangles

This paper reviews and investigates Bennetts notions of strong and weak computational depth (also called logical depth) for infinite binary sequences. Roughly, an infinite binary sequence x is defined to be weakly useful if every element of nonnegligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennetts observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost every infinite binary sequence is weakly deep, but not strongly deep.

Self-assembly of the Discrete Sierpinski Carpet and Related Fractals

In the 1980s, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an objects structure. In particular, Bennett identified the classes of weakly deep and strongly deep sequences, and showed that the halting problem is strongly deep. Juedes, Lathrop, and Lutz subsequently extended this result by defining the class of weakly useful sequences, and proving that every weakly useful sequence is strongly deep.

Automated requirements analysis for a molecular watchdog timer

We present a family of related test problems for genetic programming. These test problems form a very simple test environment that nevertheless possesses some degree of algorithmic subtlety. We term this genetic programming environment plus-one-recall-store (PORS). This genetic programming environment has only a pair of terminals, 1 and recall, and a pair of operations, plus and store, together with a single memory location. We present an extensive mathematical characterization of the PORS environment and report experiments testing the benefits of incorporating expert knowledge into the initial population and into the operation of crossover. The experiments indicate that, in the test environment, expert knowledge is best incorporated only in the initial population. This is a welcome result as this is the computationally inexpensive choice of the two methods of incorporating expert knowledge tested.