18 research outputs found
Parametrized Stochastic Grammars for RNA Secondary Structure Prediction
We propose a two-level stochastic context-free grammar (SCFG) architecture
for parametrized stochastic modeling of a family of RNA sequences, including
their secondary structure. A stochastic model of this type can be used for
maximum a posteriori estimation of the secondary structure of any new sequence
in the family. The proposed SCFG architecture models RNA subsequences
comprising paired bases as stochastically weighted Dyck-language words, i.e.,
as weighted balanced-parenthesis expressions. The length of each run of
unpaired bases, forming a loop or a bulge, is taken to have a phase-type
distribution: that of the hitting time in a finite-state Markov chain. Without
loss of generality, each such Markov chain can be taken to have a bounded
complexity. The scheme yields an overall family SCFG with a manageable number
of parameters.Comment: 5 pages, submitted to the 2007 Information Theory and Applications
Workshop (ITA 2007
The set of realizations of a max-plus linear sequence is semi-polyhedral
We show that the set of realizations of a given dimension of a max-plus
linear sequence is a finite union of polyhedral sets, which can be computed
from any realization of the sequence. This yields an (expensive) algorithm to
solve the max-plus minimal realization problem. These results are derived from
general facts on rational expressions over idempotent commutative semirings: we
show more generally that the set of values of the coefficients of a commutative
rational expression in one letter that yield a given max-plus linear sequence
is a semi-algebraic set in the max-plus sense. In particular, it is a finite
union of polyhedral sets
Calibrating Generative Models: The Probabilistic Chomsky-SchĂĽtzenberger Hierarchy
A probabilistic Chomsky–Schützenberger 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
Nonnegativity Problems for Matrix Semigroups
The matrix semigroup membership problem asks, given square matrices
of the same dimension, whether lies in the semigroup
generated by . It is classical that this problem is undecidable
in general but decidable in case commute. In this paper we
consider the problem of whether, given , the semigroup
generated by contains a non-negative matrix. We show that in
case commute, this problem is decidable subject to Schanuel's
Conjecture. We show also that the problem is undecidable if the commutativity
assumption is dropped. A key lemma in our decidability result is a procedure to
determine, given a matrix , whether the sequence of matrices is ultimately nonnegative. This answers a problem posed by S. Akshay
(arXiv:2205.09190). The latter result is in stark contrast to the notorious
fact that it is not known how to determine effectively whether for any specific
matrix index the sequence is ultimately nonnegative
(which is a formulation of the Ultimate Positivity Problem for linear
recurrence sequences)