1,129 research outputs found
Finite state verifiers with constant randomness
We give a new characterization of as the class of languages
whose members have certificates that can be verified with small error in
polynomial time by finite state machines that use a constant number of random
bits, as opposed to its conventional description in terms of deterministic
logarithmic-space verifiers. It turns out that allowing two-way interaction
with the prover does not change the class of verifiable languages, and that no
polynomially bounded amount of randomness is useful for constant-memory
computers when used as language recognizers, or public-coin verifiers. A
corollary of our main result is that the class of outcome problems
corresponding to O(log n)-space bounded games of incomplete information where
the universal player is allowed a constant number of moves equals NL.Comment: 17 pages. An improved versio
Minimum Description Length Induction, Bayesianism, and Kolmogorov Complexity
The relationship between the Bayesian approach and the minimum description
length approach is established. We sharpen and clarify the general modeling
principles MDL and MML, abstracted as the ideal MDL principle and defined from
Bayes's rule by means of Kolmogorov complexity. The basic condition under which
the ideal principle should be applied is encapsulated as the Fundamental
Inequality, which in broad terms states that the principle is valid when the
data are random, relative to every contemplated hypothesis and also these
hypotheses are random relative to the (universal) prior. Basically, the ideal
principle states that the prior probability associated with the hypothesis
should be given by the algorithmic universal probability, and the sum of the
log universal probability of the model plus the log of the probability of the
data given the model should be minimized. If we restrict the model class to the
finite sets then application of the ideal principle turns into Kolmogorov's
minimal sufficient statistic. In general we show that data compression is
almost always the best strategy, both in hypothesis identification and
prediction.Comment: 35 pages, Latex. Submitted IEEE Trans. Inform. Theor
Delzant's T-invariant, Kolmogorov complexity and one-relator groups
We prove that ``almost generically'' for a one-relator group Delzant's
-invariant (which measures the smallest size of a finite presentation for a
group) is comparable in magnitude with the length of the defining relator. The
proof relies on our previous results regarding isomorphism rigidity of generic
one-relator groups and on the methods of the theory of Kolmogorov-Chaitin
complexity. We also give a precise asymptotic estimate (when is fixed and
goes to infinity) for the number of isomorphism classes of
-generator one-relator groups with a cyclically reduced defining relator of
length : Here
means that .Comment: A revised version, to appear in Comment. Math. Hel
How the Dimension of Space Affects the Products of Pre-Biotic Evolution: The Spatial Population Dynamics of Structural Complexity and The Emergence of Membranes
We show that autocatalytic networks of epsilon-machines and their population
dynamics differ substantially between spatial (geographically distributed) and
nonspatial (panmixia) populations. Generally, regions of spacetime-invariant
autocatalytic networks---or domains---emerge in geographically distributed
populations. These are separated by functional membranes of complementary
epsilon-machines that actively translate between the domains and are
responsible for their growth and stability. We analyze both spatial and
nonspatial populations, determining the algebraic properties of the
autocatalytic networks that allow for space to affect the dynamics and so
generate autocatalytic domains and membranes. In addition, we analyze
populations of intermediate spatial architecture, delineating the thresholds at
which spatial memory (information storage) begins to determine the character of
the emergent auto-catalytic organization.Comment: 9 pages, 7 figures, 2 tables;
http://cse.ucdavis.edu/~cmg/compmech/pubs/ss.ht
Inkdots as advice for finite automata
We examine inkdots placed on the input string as a way of providing advice to
finite automata, and establish the relations between this model and the
previously studied models of advised finite automata. The existence of an
infinite hierarchy of classes of languages that can be recognized with the help
of increasing numbers of inkdots as advice is shown. The effects of different
forms of advice on the succinctness of the advised machines are examined. We
also study randomly placed inkdots as advice to probabilistic finite automata,
and demonstrate the superiority of this model over its deterministic version.
Even very slowly growing amounts of space can become a resource of meaningful
use if the underlying advised model is extended with access to secondary
memory, while it is famously known that such small amounts of space are not
useful for unadvised one-way Turing machines.Comment: 14 page
From quantum cellular automata to quantum lattice gases
A natural architecture for nanoscale quantum computation is that of a quantum
cellular automaton. Motivated by this observation, in this paper we begin an
investigation of exactly unitary cellular automata. After proving that there
can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in
one dimension, we weaken the homogeneity condition and show that there are
nontrivial, exactly unitary, partitioning cellular automata. We find a one
parameter family of evolution rules which are best interpreted as those for a
one particle quantum automaton. This model is naturally reformulated as a two
component cellular automaton which we demonstrate to limit to the Dirac
equation. We describe two generalizations of this automaton, the second of
which, to multiple interacting particles, is the correct definition of a
quantum lattice gas.Comment: 22 pages, plain TeX, 9 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages); minor typographical
corrections and journal reference adde
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
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