394 research outputs found
Randomized Two-Process Wait-Free Test-and-Set
We present the first explicit, and currently simplest, randomized algorithm
for 2-process wait-free test-and-set. It is implemented with two 4-valued
single writer single reader atomic variables. A test-and-set takes at most 11
expected elementary steps, while a reset takes exactly 1 elementary step. Based
on a finite-state analysis, the proofs of correctness and expected length are
compressed into one table.Comment: 9 pages, 4 figures, LaTeX source; Submitte
Algorithmic Statistics
While Kolmogorov complexity is the accepted absolute measure of information
content of an individual finite object, a similarly absolute notion is needed
for the relation between an individual data sample and an individual model
summarizing the information in the data, for example, a finite set (or
probability distribution) where the data sample typically came from. The
statistical theory based on such relations between individual objects can be
called algorithmic statistics, in contrast to classical statistical theory that
deals with relations between probabilistic ensembles. We develop the
algorithmic theory of statistic, sufficient statistic, and minimal sufficient
statistic. This theory is based on two-part codes consisting of the code for
the statistic (the model summarizing the regularity, the meaningful
information, in the data) and the model-to-data code. In contrast to the
situation in probabilistic statistical theory, the algorithmic relation of
(minimal) sufficiency is an absolute relation between the individual model and
the individual data sample. We distinguish implicit and explicit descriptions
of the models. We give characterizations of algorithmic (Kolmogorov) minimal
sufficient statistic for all data samples for both description modes--in the
explicit mode under some constraints. We also strengthen and elaborate earlier
results on the ``Kolmogorov structure function'' and ``absolutely
non-stochastic objects''--those rare objects for which the simplest models that
summarize their relevant information (minimal sufficient statistics) are at
least as complex as the objects themselves. We demonstrate a close relation
between the probabilistic notions and the algorithmic ones.Comment: LaTeX, 22 pages, 1 figure, with correction to the published journal
versio
Binary Lambda Calculus and Combinatory Logic
We introduce binary representations of both lambda calculus
and combinatory logic terms, and demonstrate their simplicity
by providing very compact parser-interpreters for these binary
languages.
We demonstrate their application to Algorithmic Information Theory
with several concrete upper bounds on program-size complexity,
including an elegant self-delimiting code for binary strings
On the Complexity of the Single Individual SNP Haplotyping Problem
We present several new results pertaining to haplotyping. These results
concern the combinatorial problem of reconstructing haplotypes from incomplete
and/or imperfectly sequenced haplotype fragments. We consider the complexity of
the problems Minimum Error Correction (MEC) and Longest Haplotype
Reconstruction (LHR) for different restrictions on the input data.
Specifically, we look at the gapless case, where every row of the input
corresponds to a gapless haplotype-fragment, and the 1-gap case, where at most
one gap per fragment is allowed. We prove that MEC is APX-hard in the 1-gap
case and still NP-hard in the gapless case. In addition, we question earlier
claims that MEC is NP-hard even when the input matrix is restricted to being
completely binary. Concerning LHR, we show that this problem is NP-hard and
APX-hard in the 1-gap case (and thus also in the general case), but is
polynomial time solvable in the gapless case.Comment: 26 pages. Related to the WABI2005 paper, "On the Complexity of
Several Haplotyping Problems", but with more/different results. This papers
has just been submitted to the IEEE/ACM Transactions on Computational Biology
and Bioinformatics and we are awaiting a decision on acceptance. It differs
from the mid-August version of this paper because here we prove that 1-gap
LHR is APX-hard. (In the earlier version of the paper we could prove only
that it was NP-hard.
Kolmogorov Random Graphs and the Incompressibility Method
We investigate topological, combinatorial, statistical, and enumeration
properties of finite graphs with high Kolmogorov complexity (almost all graphs)
using the novel incompressibility method. Example results are: (i) the mean and
variance of the number of (possibly overlapping) ordered labeled subgraphs of a
labeled graph as a function of its randomness deficiency (how far it falls
short of the maximum possible Kolmogorov complexity) and (ii) a new elementary
proof for the number of unlabeled graphs.Comment: LaTeX 9 page
On the vector space of the automatic reals
AbstractA sequence (an)n ⩾ 0 is said to be k-automatic if an is a finite-state function of the base-k digits of n. We say a real number is (k, b)-automatic if its fractional part has a base-b expansion that forms a k-automatic sequence, and we denote the set of all such numbers as L(k,b). Lehr (Theoret. Comput. Sci. 108 (1993) 385–391) proved that L(k, b) forms a vector space over Q. In this paper we give a shortened version of the proof of Lehr's result and, answering a question of Bach, show that the dimension of the vector space L(k, b) is infinite.We also give an example of a transcendental number such that all of its positive powers are automatic. The proof requires examining the coefficient of Xn in the formal power series (X + X2 + X4 + X8 + …)r. Along the way we are led to examine several sequences of independent combinatorial interest.Finally, solving an open problem, we show that the automatic reals are not closed under (1) product; (2) squaring; and (3) reciprocal
Cuckoo Cycle: a memory bound graph-theoretic proof-of-work
We introduce the first graph-theoretic proof-of-work system,
based on finding small cycles or other structures in large random graphs.
Such problems are trivially verifiable and arbitrarily scalable,
presumably requiring memory linear in graph size to solve efficiently.
Our cycle finding algorithm uses one bit per edge, and up to one bit per node.
Runtime is linear in graph size and dominated by random access latency,
ideal properties for a memory bound proof-of-work.
We exhibit two alternative algorithms that allow for a memory-time trade-off
(TMTO)---decreased memory usage, by a factor , coupled with increased runtime, by a factor .
The constant implied in gives a notion of memory-hardness, which is shown to be dependent
on cycle length, guiding the latter\u27s choice. Our algorithms are shown to parallelize reasonably well
- …