14 research outputs found
Probabilistic regular graphs
Deterministic graph grammars generate regular graphs, that form a structural
extension of configuration graphs of pushdown systems. In this paper, we study
a probabilistic extension of regular graphs obtained by labelling the terminal
arcs of the graph grammars by probabilities. Stochastic properties of these
graphs are expressed using PCTL, a probabilistic extension of computation tree
logic. We present here an algorithm to perform approximate verification of PCTL
formulae. Moreover, we prove that the exact model-checking problem for PCTL on
probabilistic regular graphs is undecidable, unless restricting to qualitative
properties. Our results generalise those of EKM06, on probabilistic pushdown
automata, using similar methods combined with graph grammars techniques.Comment: In Proceedings INFINITY 2010, arXiv:1010.611
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
On the power of randomized multicounter machines
AbstractOne-way two-counter machines represent a universal model of computation. Here we consider the polynomial-time classes of multicounter machines with a constant number of reversals and separate the computational power of nondeterminism, randomization and determinism. For instance, we show that polynomial-time one-way multicounter machines, with error probability tending to zero with growing input length, can recognize languages that cannot be accepted by polynomial-time nondeterministic two-way multicounter machines with a bounded number of reversals. A similar result holds for the comparison of determinism and one-sided-error randomization, and of determinism and Las Vegas randomization
Recursive Concurrent Stochastic Games
We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent
analysis of recursive simple stochastic games to a concurrent setting where the
two players choose moves simultaneously and independently at each state. For
multi-exit games, our earlier work already showed undecidability for basic
questions like termination, thus we focus on the important case of single-exit
RCSGs (1-RCSGs).
We first characterize the value of a 1-RCSG termination game as the least
fixed point solution of a system of nonlinear minimax functional equations, and
use it to show PSPACE decidability for the quantitative termination problem. We
then give a strategy improvement technique, which we use to show that player 1
(maximizer) has \epsilon-optimal randomized Stackless & Memoryless (r-SM)
strategies for all \epsilon > 0, while player 2 (minimizer) has optimal r-SM
strategies. Thus, such games are r-SM-determined. These results mirror and
generalize in a strong sense the randomized memoryless determinacy results for
finite stochastic games, and extend the classic Hoffman-Karp strategy
improvement approach from the finite to an infinite state setting. The proofs
in our infinite-state setting are very different however, relying on subtle
analytic properties of certain power series that arise from studying 1-RCSGs.
We show that our upper bounds, even for qualitative (probability 1)
termination, can not be improved, even to NP, without a major breakthrough, by
giving two reductions: first a P-time reduction from the long-standing
square-root sum problem to the quantitative termination decision problem for
finite concurrent stochastic games, and then a P-time reduction from the latter
problem to the qualitative termination problem for 1-RCSGs.Comment: 21 pages, 2 figure
Tableaux for Policy Synthesis for MDPs with PCTL* Constraints
Markov decision processes (MDPs) are the standard formalism for modelling
sequential decision making in stochastic environments. Policy synthesis
addresses the problem of how to control or limit the decisions an agent makes
so that a given specification is met. In this paper we consider PCTL*, the
probabilistic counterpart of CTL*, as the specification language. Because in
general the policy synthesis problem for PCTL* is undecidable, we restrict to
policies whose execution history memory is finitely bounded a priori.
Surprisingly, no algorithm for policy synthesis for this natural and
expressive framework has been developed so far. We close this gap and describe
a tableau-based algorithm that, given an MDP and a PCTL* specification, derives
in a non-deterministic way a system of (possibly nonlinear) equalities and
inequalities. The solutions of this system, if any, describe the desired
(stochastic) policies.
Our main result in this paper is the correctness of our method, i.e.,
soundness, completeness and termination.Comment: This is a long version of a conference paper published at TABLEAUX
2017. It contains proofs of the main results and fixes a bug. See the
footnote on page 1 for detail
Model Checking Probabilistic Pushdown Automata
We consider the model checking problem for probabilistic pushdown automata
(pPDA) and properties expressible in various probabilistic logics. We start
with properties that can be formulated as instances of a generalized random
walk problem. We prove that both qualitative and quantitative model checking
for this class of properties and pPDA is decidable. Then we show that model
checking for the qualitative fragment of the logic PCTL and pPDA is also
decidable. Moreover, we develop an error-tolerant model checking algorithm for
PCTL and the subclass of stateless pPDA. Finally, we consider the class of
omega-regular properties and show that both qualitative and quantitative model
checking for pPDA is decidable
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