58,369 research outputs found
Combinatorics of branchings in higher dimensional automata
We explore the combinatorial properties of the branching areas of execution
paths in higher dimensional automata. Mathematically, this means that we
investigate the combinatorics of the negative corner (or branching) homology of
a globular -category and the combinatorics of a new homology theory
called the reduced branching homology. The latter is the homology of the
quotient of the branching complex by the sub-complex generated by its thin
elements. Conjecturally it coincides with the non reduced theory for higher
dimensional automata, that is -categories freely generated by
precubical sets. As application, we calculate the branching homology of some
-categories and we give some invariance results for the reduced
branching homology. We only treat the branching side. The merging side, that is
the case of merging areas of execution paths is similar and can be easily
deduced from the branching side.Comment: Final version, see
http://www.tac.mta.ca/tac/volumes/8/n12/abstract.htm
Complex and Adaptive Dynamical Systems: A Primer
An thorough introduction is given at an introductory level to the field of
quantitative complex system science, with special emphasis on emergence in
dynamical systems based on network topologies. Subjects treated include graph
theory and small-world networks, a generic introduction to the concepts of
dynamical system theory, random Boolean networks, cellular automata and
self-organized criticality, the statistical modeling of Darwinian evolution,
synchronization phenomena and an introduction to the theory of cognitive
systems.
It inludes chapter on Graph Theory and Small-World Networks, Chaos,
Bifurcations and Diffusion, Complexity and Information Theory, Random Boolean
Networks, Cellular Automata and Self-Organized Criticality, Darwinian
evolution, Hypercycles and Game Theory, Synchronization Phenomena and Elements
of Cognitive System Theory.Comment: unformatted version of the textbook; published in Springer,
Complexity Series (2008, second edition 2010
Partially ordered models
We provide a formal definition and study the basic properties of partially
ordered chains (POC). These systems were proposed to model textures in image
processing and to represent independence relations between random variables in
statistics (in the later case they are known as Bayesian networks). Our chains
are a generalization of probabilistic cellular automata (PCA) and their theory
has features intermediate between that of discrete-time processes and the
theory of statistical mechanical lattice fields. Its proper definition is based
on the notion of partially ordered specification (POS), in close analogy to the
theory of Gibbs measure. This paper contains two types of results. First, we
present the basic elements of the general theory of POCs: basic geometrical
issues, definition in terms of conditional probability kernels, extremal
decomposition, extremality and triviality, reconstruction starting from
single-site kernels, relations between POM and Gibbs fields. Second, we prove
three uniqueness criteria that correspond to the criteria known as bounded
uniformity, Dobrushin and disagreement percolation in the theory of Gibbs
measures.Comment: 54 pages, 11 figures, 6 simulations. Submited to Journal of Stat.
Phy
ALMA: Automata Learner using Modulo 2 Multiplicity Automata
We present ALMA (Automata Learner using modulo 2 Multiplicity Automata), a
Java-based tool that can learn any automaton accepting regular languages of
finite or infinite words with an implementable membership query function. Users
can either pass as input their own membership query function, or use the
predefined membership query functions for modulo 2 multiplicity automata and
non-deterministic B\"uchi automata. While learning, ALMA can output the state
of the observation table after every equivalence query, and upon termination,
it can output the dimension, transition matrices, and final vector of the
learned modulo 2 multiplicity automaton. Users can test whether a word is
accepted by performing a membership query on the learned automaton.
ALMA follows the polynomial-time learning algorithm of Beimel et. al.
(Learning functions represented as multiplicity automata. J. ACM 47(3), 2000),
which uses membership and equivalence queries and represents hypotheses using
modulo 2 multiplicity automata. ALMA also implements a polynomial-time learning
algorithm for strongly unambiguous B\"uchi automata by Angluin et. al.
(Strongly unambiguous B\"uchi automata are polynomially predictable with
membership queries. CSL 2020), and a minimization algorithm for modulo 2
multiplicity automata by Sakarovitch (Elements of Automata Theory. 2009)
Rules and derivations in an elementary logic course
When teaching an elementary logic course to students who have a general
scientific background but have never been exposed to logic, we have to face the
problem that the notions of deduction rule and of derivation are completely new
to them, and are related to nothing they already know, unlike, for instance,
the notion of model, that can be seen as a generalization of the notion of
algebraic structure. In this note, we defend the idea that one strategy to
introduce these notions is to start with the notion of inductive definition
[1]. Then, the notion of derivation comes naturally. We also defend the idea
that derivations are pervasive in logic and that defining precisely this notion
at an early stage is a good investment to later define other notions in proof
theory, computability theory, automata theory, ... Finally, we defend the idea
that to define the notion of derivation precisely, we need to distinguish two
notions of derivation: labeled with elements and labeled with rule names. This
approach has been taken in [2]
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