17,017 research outputs found
Exploring the concept of interaction computing through the discrete algebraic analysis of the BelousovâZhabotinsky reaction
Interaction computing (IC) aims to map the properties of integrable low-dimensional non-linear dynamical systems to the discrete domain of finite-state automata in an attempt to reproduce in software the self-organizing and dynamically stable properties of sub-cellular biochemical systems. As the work reported in this paper is still at the early stages of theory development it focuses on the analysis of a particularly simple chemical oscillator, the Belousov-Zhabotinsky (BZ) reaction. After retracing the rationale for IC developed over the past several years from the physical, biological, mathematical, and computer science points of view, the paper presents an elementary discussion of the Krohn-Rhodes decomposition of finite-state automata, including the holonomy decomposition of a simple automaton, and of its interpretation as an abstract positional number system. The method is then applied to the analysis of the algebraic properties of discrete finite-state automata derived from a simplified Petri net model of the BZ reaction. In the simplest possible and symmetrical case the corresponding automaton is, not surprisingly, found to contain exclusively cyclic groups. In a second, asymmetrical case, the decomposition is much more complex and includes five different simple non-abelian groups whose potential relevance arises from their ability to encode functionally complete algebras. The possible computational relevance of these findings is discussed and possible conclusions are drawn
Chasing diagrams in cryptography
Cryptography is a theory of secret functions. Category theory is a general
theory of functions. Cryptography has reached a stage where its structures
often take several pages to define, and its formulas sometimes run from page to
page. Category theory has some complicated definitions as well, but one of its
specialties is taming the flood of structure. Cryptography seems to be in need
of high level methods, whereas category theory always needs concrete
applications. So why is there no categorical cryptography? One reason may be
that the foundations of modern cryptography are built from probabilistic
polynomial-time Turing machines, and category theory does not have a good
handle on such things. On the other hand, such foundational problems might be
the very reason why cryptographic constructions often resemble low level
machine programming. I present some preliminary explorations towards
categorical cryptography. It turns out that some of the main security concepts
are easily characterized through the categorical technique of *diagram
chasing*, which was first used Lambek's seminal `Lecture Notes on Rings and
Modules'.Comment: 17 pages, 4 figures; to appear in: 'Categories in Logic, Language and
Physics. Festschrift on the occasion of Jim Lambek's 90th birthday', Claudia
Casadio, Bob Coecke, Michael Moortgat, and Philip Scott (editors); this
version: fixed typos found by kind reader
Calibrating Generative Models: The Probabilistic Chomsky-SchĂźtzenberger Hierarchy
A probabilistic ChomskyâSchĂźtzenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning
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
Equations of States in Statistical Learning for a Nonparametrizable and Regular Case
Many learning machines that have hierarchical structure or hidden variables
are now being used in information science, artificial intelligence, and
bioinformatics. However, several learning machines used in such fields are not
regular but singular statistical models, hence their generalization performance
is still left unknown. To overcome these problems, in the previous papers, we
proved new equations in statistical learning, by which we can estimate the
Bayes generalization loss from the Bayes training loss and the functional
variance, on the condition that the true distribution is a singularity
contained in a learning machine. In this paper, we prove that the same
equations hold even if a true distribution is not contained in a parametric
model. Also we prove that, the proposed equations in a regular case are
asymptotically equivalent to the Takeuchi information criterion. Therefore, the
proposed equations are always applicable without any condition on the unknown
true distribution
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
Virtually all questions that one can ask about the behavioral and structural
complexity of a stochastic process reduce to a linear algebraic framing of a
time evolution governed by an appropriate hidden-Markov process generator. Each
type of question---correlation, predictability, predictive cost, observer
synchronization, and the like---induces a distinct generator class. Answers are
then functions of the class-appropriate transition dynamic. Unfortunately,
these dynamics are generically nonnormal, nondiagonalizable, singular, and so
on. Tractably analyzing these dynamics relies on adapting the recently
introduced meromorphic functional calculus, which specifies the spectral
decomposition of functions of nondiagonalizable linear operators, even when the
function poles and zeros coincide with the operator's spectrum. Along the way,
we establish special properties of the projection operators that demonstrate
how they capture the organization of subprocesses within a complex system.
Circumventing the spurious infinities of alternative calculi, this leads in the
sequel, Part II, to the first closed-form expressions for complexity measures,
couched either in terms of the Drazin inverse (negative-one power of a singular
operator) or the eigenvalues and projection operators of the appropriate
transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht
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