387 research outputs found
On Nonnegative Integer Matrices and Short Killing Words
Let be a natural number and a set of -matrices
over the nonnegative integers such that the joint spectral radius of
is at most one. We show that if the zero matrix is a product
of matrices in , then there are with . This result has applications in
automata theory and the theory of codes. Specifically, if
is a finite incomplete code, then there exists a word of
length polynomial in such that is not a factor of any
word in . This proves a weak version of Restivo's conjecture.Comment: This version is a journal submission based on a STACS'19 paper. It
extends the conference version as follows. (1) The main result has been
generalized to apply to monoids generated by finite sets whose joint spectral
radius is at most 1. (2) The use of Carpi's theorem is avoided to make the
paper more self-contained. (3) A more precise result is offered on Restivo's
conjecture for finite code
On Probabilistic Parallel Programs with Process Creation and Synchronisation
We initiate the study of probabilistic parallel programs with dynamic process
creation and synchronisation. To this end, we introduce probabilistic
split-join systems (pSJSs), a model for parallel programs, generalising both
probabilistic pushdown systems (a model for sequential probabilistic procedural
programs which is equivalent to recursive Markov chains) and stochastic
branching processes (a classical mathematical model with applications in
various areas such as biology, physics, and language processing). Our pSJS
model allows for a possibly recursive spawning of parallel processes; the
spawned processes can synchronise and return values. We study the basic
performance measures of pSJSs, especially the distribution and expectation of
space, work and time. Our results extend and improve previously known results
on the subsumed models. We also show how to do performance analysis in
practice, and present two case studies illustrating the modelling power of
pSJSs.Comment: This is a technical report accompanying a TACAS'11 pape
Bisimilarity of Pushdown Systems is Nonelementary
Given two pushdown systems, the bisimilarity problem asks whether they are
bisimilar. While this problem is known to be decidable our main result states
that it is nonelementary, improving EXPTIME-hardness, which was the previously
best known lower bound for this problem. Our lower bound result holds for
normed pushdown systems as well
Computing Least Fixed Points of Probabilistic Systems of Polynomials
We study systems of equations of the form X1 = f1(X1, ..., Xn), ..., Xn =
fn(X1, ..., Xn), where each fi is a polynomial with nonnegative coefficients
that add up to 1. The least nonnegative solution, say mu, of such equation
systems is central to problems from various areas, like physics, biology,
computational linguistics and probabilistic program verification. We give a
simple and strongly polynomial algorithm to decide whether mu=(1, ..., 1)
holds. Furthermore, we present an algorithm that computes reliable sequences of
lower and upper bounds on mu, converging linearly to mu. Our algorithm has
these features despite using inexact arithmetic for efficiency. We report on
experiments that show the performance of our algorithms.Comment: Published in the Proceedings of the 27th International Symposium on
Theoretical Aspects of Computer Science (STACS). Technical Report is also
available via arxiv.or
Minimisation of Multiplicity Tree Automata
We consider the problem of minimising the number of states in a multiplicity
tree automaton over the field of rational numbers. We give a minimisation
algorithm that runs in polynomial time assuming unit-cost arithmetic. We also
show that a polynomial bound in the standard Turing model would require a
breakthrough in the complexity of polynomial identity testing by proving that
the latter problem is logspace equivalent to the decision version of
minimisation. The developed techniques also improve the state of the art in
multiplicity word automata: we give an NC algorithm for minimising multiplicity
word automata. Finally, we consider the minimal consistency problem: does there
exist an automaton with states that is consistent with a given finite
sample of weight-labelled words or trees? We show that this decision problem is
complete for the existential theory of the rationals, both for words and for
trees of a fixed alphabet rank.Comment: Paper to be published in Logical Methods in Computer Science. Minor
editing changes from previous versio
Computing the Least Fixed Point of Positive Polynomial Systems
We consider equation systems of the form X_1 = f_1(X_1, ..., X_n), ..., X_n =
f_n(X_1, ..., X_n) where f_1, ..., f_n are polynomials with positive real
coefficients. In vector form we denote such an equation system by X = f(X) and
call f a system of positive polynomials, short SPP. Equation systems of this
kind appear naturally in the analysis of stochastic models like stochastic
context-free grammars (with numerous applications to natural language
processing and computational biology), probabilistic programs with procedures,
web-surfing models with back buttons, and branching processes. The least
nonnegative solution mu f of an SPP equation X = f(X) is of central interest
for these models. Etessami and Yannakakis have suggested a particular version
of Newton's method to approximate mu f.
We extend a result of Etessami and Yannakakis and show that Newton's method
starting at 0 always converges to mu f. We obtain lower bounds on the
convergence speed of the method. For so-called strongly connected SPPs we prove
the existence of a threshold k_f such that for every i >= 0 the (k_f+i)-th
iteration of Newton's method has at least i valid bits of mu f. The proof
yields an explicit bound for k_f depending only on syntactic parameters of f.
We further show that for arbitrary SPP equations Newton's method still
converges linearly: there are k_f>=0 and alpha_f>0 such that for every i>=0 the
(k_f+alpha_f i)-th iteration of Newton's method has at least i valid bits of mu
f. The proof yields an explicit bound for alpha_f; the bound is exponential in
the number of equations, but we also show that it is essentially optimal.
Constructing a bound for k_f is still an open problem. Finally, we also provide
a geometric interpretation of Newton's method for SPPs.Comment: This is a technical report that goes along with an article to appear
in SIAM Journal on Computing
Convergence Thresholds of Newton's Method for Monotone Polynomial Equations
Monotone systems of polynomial equations (MSPEs) are systems of fixed-point
equations where
each is a polynomial with positive real coefficients. The question of
computing the least non-negative solution of a given MSPE arises naturally in the analysis of stochastic models such as stochastic
context-free grammars, probabilistic pushdown automata, and back-button
processes. Etessami and Yannakakis have recently adapted Newton's iterative
method to MSPEs. In a previous paper we have proved the existence of a
threshold for strongly connected MSPEs, such that after iterations of Newton's method each new iteration computes at least 1 new
bit of the solution. However, the proof was purely existential. In this paper
we give an upper bound for as a function of the minimal component
of the least fixed-point of . Using this result we
show that is at most single exponential resp. linear for strongly
connected MSPEs derived from probabilistic pushdown automata resp. from
back-button processes. Further, we prove the existence of a threshold for
arbitrary MSPEs after which each new iteration computes at least new
bits of the solution, where and are the width and height of the DAG of
strongly connected components.Comment: version 2 deposited February 29, after the end of the STACS
conference. Two minor mistakes correcte
Stabilization of Branching Queueing Networks
Queueing networks are gaining attraction for the performance analysis of parallel computer systems. A Jackson network is a set of interconnected servers, where the completion of a job at server i may result in the creation of a new job for server j. We propose to extend Jackson networks by "branching" and by "control" features. Both extensions are new and substantially expand the modelling power of Jackson networks. On the other hand, the extensions raise computational questions, particularly concerning the stability of the networks, i.e, the ergodicity of the underlying Markov chain. We show for our extended model that it is decidable in polynomial time if there exists a controller that achieves stability. Moreover, if such a controller exists, one can efficiently compute a static randomized controller which stabilizes the network in a very strong sense; in particular, all moments of the queue sizes are finite
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