280 research outputs found
Factorizations of languages and commutativity conditions
Representations of languages as a product (catenation) of languages are investigated, where the factor languages are "prime", that is, cannot be decomposed further in a nontrivial manner. In general, such prime decompositions do not necessarily exist. If they exist, they are not necessarily unique - the number of factors can vary even exponentially. The paper investigates prime decompositions, as well as the commuting of the factors, especially for the case of finite languages. In particular, a technique about commuting is developed in Section 4, where the factorization of languages L1 and L2 is discussed under the assumption L1L2 = L2L1
Factoring the Adjoint and Maximal Cohen--Macaulay Modules over the Generic Determinant
A question of Bergman asks whether the adjoint of the generic square matrix
over a field can be factored nontrivially as a product of square matrices. We
show that such factorizations indeed exist over any coefficient ring when the
matrix has even size. Establishing a correspondence between such factorizations
and extensions of maximal Cohen--Macaulay modules over the generic determinant,
we exhibit all factorizations where one of the factors has determinant equal to
the generic determinant. The classification shows not only that the
Cohen--Macaulay representation theory of the generic determinant is wild in the
tame-wild dichotomy, but that it is quite wild: even in rank two, the
isomorphism classes cannot be parametrized by a finite-dimensional variety over
the coefficients. We further relate the factorization problem to the
multiplicative structure of the \Ext--algebra of the two nontrivial rank-one
maximal Cohen--Macaulay modules and determine it completely.Comment: 44 pages, final version of the work announced in math.RA/0408425, to
appear in the American Journal of Mathematic
Well-Pointed Coalgebras
For endofunctors of varieties preserving intersections, a new description of
the final coalgebra and the initial algebra is presented: the former consists
of all well-pointed coalgebras. These are the pointed coalgebras having no
proper subobject and no proper quotient. The initial algebra consists of all
well-pointed coalgebras that are well-founded in the sense of Osius and Taylor.
And initial algebras are precisely the final well-founded coalgebras. Finally,
the initial iterative algebra consists of all finite well-pointed coalgebras.
Numerous examples are discussed e.g. automata, graphs, and labeled transition
systems
Exploiting Causal Independence in Bayesian Network Inference
A new method is proposed for exploiting causal independencies in exact
Bayesian network inference. A Bayesian network can be viewed as representing a
factorization of a joint probability into the multiplication of a set of
conditional probabilities. We present a notion of causal independence that
enables one to further factorize the conditional probabilities into a
combination of even smaller factors and consequently obtain a finer-grain
factorization of the joint probability. The new formulation of causal
independence lets us specify the conditional probability of a variable given
its parents in terms of an associative and commutative operator, such as
``or'', ``sum'' or ``max'', on the contribution of each parent. We start with a
simple algorithm VE for Bayesian network inference that, given evidence and a
query variable, uses the factorization to find the posterior distribution of
the query. We show how this algorithm can be extended to exploit causal
independence. Empirical studies, based on the CPCS networks for medical
diagnosis, show that this method is more efficient than previous methods and
allows for inference in larger networks than previous algorithms.Comment: See http://www.jair.org/ for any accompanying file
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