We prove that if a set $X \subseteq \Z^2$ weakly self-assembles at temperature 1 in a deterministic tile assembly system satisfying a natural condition known as \emph{pumpability}, then $X$ is a finite union of semi-doubly periodic sets. This shows that only the most simple of infinite shapes and patterns can be constructed using pumpable temperature 1 tile assembly systems, and gives evidence for the thesis that temperature 2 or higher is required to carry out general-purpose computation in a tile assembly system. Finally, we show that general-purpose computation \emph{is} possible at temperature 1 if negative glue strengths are allowed in the tile assembly model.

Limitations of Self-Assembly at Temperature 1

Fractal Dimensions of Networks

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-assemblyof 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 cannotstrictly 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 Winfree counters, coupled with measured delay and corner-turning operations. We verify our strict self-assembly using the local determinism method of Soloveichik and Winfree (2005).

Strict Self-assembly of Discrete Sierpinski Triangles

The purpose of this paper is to prove that the spectrum of the non-self-adjoint one-particle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433–6443) has interior points. We do this by first recalling that the spectrum of this random operator is the union of the set of `∞ eigenvalues of all infinite matrices with the same structure. We then construct an infinite matrix of this structure for which every point of the open unit disk is an `∞ eigenvalue, this following from the fact that the components of the eigenvector are polynomials in the spectral parameter whose non-zero coefficients are ±1’s, forming the pattern of an infinite discrete Sierpinski triangle. Mathematics subject classification (2000): Primary 47B80; Secondary 47A10, 47B36.

/pdf/eigenvalue-problem-meets-sierpinski-triangle-computing-the-1cvob4jt5a.pdf

Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non–self–adjoint random operator

The problem of testing membership in the subset of the natural numbers produced at the output gate of a (∩,∪, - ,+,×} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case (∩,∪,+,×) is shown NEXPTIME-complete, the cases (∪,∩, - ,×), (∪,∩, ×}, {∪,∩,+} are shown PSPACE-complete, the case (∪,+) is shown NP-complete, the case (∩,+) is shown C=L-complete, and several other cases are resolved. Interesting auxiliary problems are used, such as testing nonemptyness for union-intersection-concatenation circuits, and expressing each integer, drawn from a set given as input, as powers of relatively prime integers of one's choosing. Our results extend in nontrivial ways past work by Stockmeyer and Meyer (1973), Wagner (1984) and Yang (2000).

The complexity of membership problems for circuits over sets of natural numbers

The zeta-dimension of a set A of positive integers is the infimum s such that the sum of the reciprocals of the s-th powers of the elements of A is finite Zeta-dimension serves as a fractal dimension on the positive integers that extends naturally usefully to discrete lattices such as the set of all integer lattice points in d-dimensional space This paper reviews the origins of zeta-dimension (which date to the eighteenth and nineteenth centuries) and develops its basic theory, with particular attention to its relationship with algorithmic information theory New results presented include extended connections between zeta-dimension and classical fractal dimensions, a gale characterization of zeta-dimension, and a theorem on the zeta-dimensions of pointwise sums and products of sets of positive integers

Zeta-Dimension

We study population protocols: networks of anonymous agents whose pairwise interactions are chosen uniformly at random. The size counting problem is that of calculating the exact number n of agents in the population, assuming no leader (each agent starts in the same state). We give the first protocol that solves this problem in sublinear time.
The protocol converges in O(log n log log n) time and uses O(n^60) states (O(1) + 60 log n bits of memory per agent) with probability 1-O((log log n)/n). The time to converge is also O(log n log log n) in expectation. Crucially, unlike most published protocols with omega(1) states, our protocol is uniform: it uses the same transition algorithm for any population size, so does not need an estimate of the population size to be embedded into the algorithm.

https://drops.dagstuhl.de/opus/volltexte/2018/9835/pdf/LIPIcs-DISC-2018-46.pdf/

Brief Announcement: Exact Size Counting in Uniform Population Protocols in Nearly Logarithmic Time

This paper introduces two complexity-theoretic formulations of Bennett's logical depth: finite-state depthand polynomial-time depth. It is shown that for both formulations, trivial and random infinite sequences are shallow, and a slow growth lawholds, implying that deep sequences cannot be created easily from shallow sequences. Furthermore, the E analogue of the halting language is shown to be polynomial-time deep, by proving a more general result: every language to which a nonnegligible subset of E can be reduced in uniform exponential time is polynomial-time deep.

Feasible Depth

This paper introduces two complexity-theoretic formulations of Bennett's logical depth: finite-state depth and polynomial-time depth It is shown that for both formulations, trivial and random infinite sequences are shallow, and a slow growth law holds, implying that deep sequences cannot be created easily from shallow sequences Furthermore, the E analogue of the halting language is shown to be polynomial-time deep, by proving a more general result: every language to which a nonnegligible subset of E can be reduced in uniform exponential time is polynomial-time deep

Pushdown dimension

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Zeta-Dimension

Brief Announcement: Exact Size Counting in Uniform Population Protocols in Nearly Logarithmic Time

Feasible Depth

Feasible Depth

Pushdown dimension