Jack H. Lutz

The quantitative structure of exponential time

Recent results on the internal, measure-theoretic structure of the exponential time complexity classes E=DTIME(2/sup linear/) and E/sub 2/=DTIME(2/sup polynomial/) are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, /sub <or=/m/sup P/-reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are discussed.<<ETX>>

Category and measure in complexity classes

This paper presents resource-bounded category and resource-bounded measure—two new tools for computational complexity theory—and some applications of these tools to the structure theory of exponent...

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.

Cook versus Karp-Levin: separating completeness notions if NP is not small

Under the hypothesis that NP does not have p-measure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is shown that there is a language that is ≤ T P -complete (“Cook complete”), but not ≤ m P -complete (“Karp-Levin complete”), for NP. This conclusion, widely believed to be true, is not known to follow from P ≠ NP or other traditional complexity-theoretic hypotheses.

The Complexity and Distribution of Hard Problems

Measure-theoretic aspects of the

Measure, Stochasticity, and the Density of Hard Languages

\leq^{\rm P}_{\rm m}

Gales and the Constructive Dimension of Individual Sequences

-reducibility structure of the exponential time complexity classes E=DTIME(

Weak completeness in E and E 2

2^{\rm linear}

Computability and Complexity in Self-assembly

) and